Eftychia Vasili

Antonio Dominguez-Meijide

Manuel Flores-León

Mohammed Al-Azzani

Angeliki Kanellidi

Ronald Melki

Leonidas Stefanis

• Tiago

Fleming Outeiro

Endogenous Levels of Alpha-Synuclein Modulate Seeding and Aggregation in Cultured Cells

Keywords: PD, LBs, aSyn, Aggregation, Phosphorylation Abbreviations aSyn Alpha-synuclein DAPI 4′6′-Diamidino-2-phenylindol Dox Doxycycline

Parkinson's disease is a progressive neurodegenerative disorder characterized by the accumulation of misfolded alphasynuclein in intraneuronal inclusions known as Lewy bodies and Lewy neurites. Multiple studies strongly implicate the levels of alpha-synuclein as a major risk factor for the onset and progression of Parkinson's disease. Alpha-synuclein pathology spreads progressively throughout interconnected brain regions but the precise molecular mechanisms underlying the seeding of alpha-synuclein aggregation are still unclear. Here, using stable cell lines expressing alpha-synuclein, we examined the correlation between endogenous alpha-synuclein levels and the seeding propensity by exogenous alpha-synuclein preformed fibrils. We applied biochemical approaches and imaging methods in stable cell lines expressing alpha-synuclein and in primary neurons to determine the impact of alpha-synuclein levels on seeding and aggregation. Our results indicate that the levels of alpha-synuclein define the pattern and severity of aggregation and the extent of p-alpha-synuclein deposition, likely explaining the selective vulnerability of different cell types in synucleinopathies. The elucidation of the cellular processes involved in the pathological aggregation of alpha-synuclein will enable the identification of novel targets and the development of therapeutic strategies for Parkinson's disease and other synucleinopathies.

Introduction

Misfolding and aggregation of aSyn into large intraneuronal inclusions called Lewy bodies (LBs) and Lewy neurites (LNs) is a common feature and a defining hallmark of Parkinson's disease (PD) and other related neurodegenerative diseases known as synucleinopathies [START_REF] Rodrigues E Silva | Who was the man who discovered the "Lewy bodies"?[END_REF]. Neuropathologically, aSyn pathology seems to progress throughout interconnected brain regions in a pattern consistent with the "prion hypothesis" for the spreading of protein pathology [START_REF] Braak | Staging of brain pathology related to sporadic Parkinson's disease[END_REF]. Cell-to-cell transmission of aSyn pathology involves the conversion of the endogenous protein into a pathological conformation, thus propagating the pathology. However, the exact mechanisms involved remain unclear. Mounting evidence suggests that the intrinsically disordered monomeric aSyn assembles into small soluble oligomers, with increased β-sheet content that are prone to further aggregation and that may, along the way, result in the formation of toxic species [START_REF] Choi | Crucial role of protein oligomerization in the pathogenesis of Alzheimer's and Parkinson's diseases[END_REF][START_REF] Uversky | Neuropathology, biochemistry, and biophysics of alpha-synuclein aggregation[END_REF].

Several in vivo studies described the injection of aSyncontaining brain tissue homogenates from patients or from transgenic mice, or of in vitro assembled preformed fibrils (PFFs) of recombinant aSyn into mouse brains, blood, or into other organs. The injected animals develop aggregates containing endogenous aSyn phosphorylated on Ser129 in brain regions directly or indirectly connected with the injection site, time-dependent neuronal loss, neuroinflammation, and behavioral alterations [START_REF] Chung | Modeling alpha-synuclein propagation with preformed fibril injections[END_REF].

In addition, small seeds of PFFs are internalized in primary neuronal cultures and promote aggregation of endogenous aSyn phosphorylated on Ser129, alongside with progressive impairment in neuronal network function that eventually leads to neuronal death [START_REF] Volpicelli-Daley | Addition of exogenous alpha-synuclein preformed fibrils to primary neuronal cultures to seed recruitment of endogenous alpha-synuclein to Lewy body and Lewy neurite-like aggregates[END_REF][START_REF] Volpicelli-Daley | Exogenous alpha-synuclein fibrils induce Lewy body pathology leading to synaptic dysfunction and neuron death[END_REF]. These PFF-based models have been extensively used to investigate the mechanisms by which aSyn seeds are transferred from donor to recipient cells, thereby spreading throughout the brain. However, the precise molecular mechanisms involved in the uptake, seeding, and aggregation of aSyn are still elusive.

In our study, we tested the hypothesis that the susceptibility to the accumulation of pathological aSyn might be influenced by the endogenous levels of expression of the protein. This arises from the fact that, in the human brain, different cell populations appear to be particularly sensitive in PD. To test our hypothesis, we utilized stable cell lines expressing wild type (WT) aSyn, incubated with a low concentration of PFFs in order to recapitulate aSyn seeding. The development of our model aimed at the comparative study among cell lines with different endogenous levels of aSyn in order to investigate the impact on aSyn uptake and formation of endogenously aggregated aSyn species.

Materials and Methods

Recombinant Protein Preparation

For the production of recombinant aSyn protein from E.coli, we followed established procedures [START_REF] Dominguez-Meijide | Effects of pharmacological modulators of alpha-synuclein and tau aggregation and internalization[END_REF][START_REF] Szego | Cytosolic trapping of a mitochondrial heat shock protein is an early pathological event in synucleinopathies[END_REF]. For the purification of recombinant aSyn, the protein was first filtered through a 0.22-µm syringe filter and then applied to Hi Trap Q HP anion exchange column (GE Healthcare Life Sciences) equilibrated with 25 mm Tris, and pH 7.6. aSyn was eluted by running a linear gradient of elution buffer (3 M NaCl in 25 mm Tris, pH 7.6) up to 30% elution buffer. aSynenriched fractions (analyzed by SDS-PAGE and Coomassie staining/destaining) were collected, combined, and concentrated with 3-kDa molecular weight cut-off Amicon Ultra Centrifuge filters according to Manufacturer's instructions. Proteins were further purified by Superdex 75 column (GE Healthcare Life Sciences) equilibrated with 25 mM Tris, pH 7.6, and 100 mm NaCl. Pure fractions corresponding to monomeric state (as determined by Native Gel, Coomassie staining/destaining) were combined, concentrated at a final concentration of 5 mg/ml, and stored at -80 °C until further use. PFFs were prepared from the monomers according to standard protocols [START_REF] Volpicelli-Daley | Addition of exogenous alpha-synuclein preformed fibrils to primary neuronal cultures to seed recruitment of endogenous alpha-synuclein to Lewy body and Lewy neurite-like aggregates[END_REF].

Generation of the Inducible Stable Cell Line for aSyn Oligomerization Based on Bimolecular Fluorescence Complementation (BiFC)

Naïve HEK293 cells were first transfected with the pTet-Off (Clontech) plasmid that expresses the regulatory protein tTA (tetracycline-controlled transactivator, tTA), which contain a neomycin-resistance gene for the selection. Selection with G418 (500 μg/ml; Millipore) was used for the selection of resistant colonies. Inducibility of resistant colonies was determined after transient transfection with the firefly luciferase vector pTRE-Tight-Luc, in presence or absence of Doxycycline (Dox, 2 μg/ml; Sigma-Aldrich). A clonal HEK293-pTet-Off cell line was selected and used for the generation of the different pTRE stable cell lines (pTRE-Tight-VN-aSyn and pTRE-Tight-aSyn-VC). Transient cotransfection for the pTRE-Tight-VN-aSyn and pTRE-Tight-aSyn-VC vectors was used to verify the formation of the BiFC oligomers. The VN and VC aSyn constructs were generated from initial plasmids containing the VN and VC fused aSyn proteins, digested with the SnaBI/XbaI enzymes and cloned into the pTRE-Tight vector (Clontech) digested before with SmaI/XbaI. The positive clones were verified by DNA sequencing. Subsequently, the VN and VC fused aSyn vectors were co-transfected with a linear selection marker for Hygromycin (Clontech). Selection was performed with 50 μg/ml Hygromycin (Roche). Resistant clones were picked 1 3 and aSyn was examined by immunoblotting in the presence or absence of Dox (2 μg/mL added for more than 5 days) using a specific antibody against aSyn (anti-aSyn Syn-1 mouse, BD Transduction Laboratories).

Generation of HEK293-aSynEGFP Stable Cell Line

HEK293 were transfected with a plasmid encoding human WT aSyn fused to EGFP, at the C-terminus, driven by the cytomegalovirus (CMV) promoter. The plasmid contained a selection marker for the antibiotic geneticin (G418) which was used for the selection of the stable transformants. Protein expression was confirmed by Western blot analysis and fluorescence microscopy. A clonal HEK293-aSynEGFP cell line was selected and used for subsequent experiments.

Construction of Lentivirus Vectors

Full-length human aSyn was PCR-amplified and sub cloned into a 3rd-generation lentiviral plasmid with hUbC-driven EGFP, where the original GFP cassette has been replaced by the aSyn gene including the Kozak sequence. pFUGW vector containing only the GFP cassette was used for control experiments (Addgene) [START_REF] Lois | Germline transmission and tissue-specific expression of transgenes delivered by lentiviral vectors[END_REF].

Cell Culture and Cell Treatment

HEK293 cells were maintained in DMEM media supplemented with 10% Fetal bovine serum gold (FBS) (PAA) and 1% penicillin-streptomycin (PAN). The cells were grown at 37 °C in an atmosphere of 5% CO 2 . For the seeding experiments, aSyn monomers and PFFs were diluted in PBS, fragmented by sonication in case of PFFs [START_REF] Volpicelli-Daley | Addition of exogenous alpha-synuclein preformed fibrils to primary neuronal cultures to seed recruitment of endogenous alpha-synuclein to Lewy body and Lewy neurite-like aggregates[END_REF], and then added to the cells at final concentration of 100 nM. Control cells were exposed to vehicle only (PBS). Cells were further incubated for 48 h.

Primary Cultures of Cortical Neurons

Primary cortical neurons were prepared from embryonic day E16-E17 mouse brains. Briefly, after dissection, the tissue was washed three times with Hank's Balanced Salt Solution (HBSS; Gibco) and digested by Trypsin (Gibco) for 15 min at 37 °C, followed by addition of 100 μl DNase I (10 mg/ ml; Roche) and 100 μl FBS (Life Technologies), mixed by inverting and centrifuge at 500 rpm for 5 min. After centrifuge trypsin solution was removed and replaced with 1 ml of FBS. Cortices were then dissociated by mechanical trituration with glass pasteur pipettes. Dissociated cells were transferred in fresh culture medium (Neurobasal medium, Gibco), containing 2% B27 supplement (Gibco), 0.5 mM L-glutamine (200 Mm Gibco), and 1% penicillin/streptomycin (Gibco) and plated in 24-or 6-well plates previously coated with Poly-L-ornithin (0,1 mg/ml, Sigma-Aldrich) at a density of 250,000 cells/mL. Cultures at DIV5 were treated with recombinant aSyn (monomers and PFFs; final concentration of 100 nM) and long-term incubated until DIV25. No further media changes or addition of recombinant protein was made until the end of the culture. After mild trypsinization with 0.5% Trypsin-EDTA for 5 min to remove the excess of unbound material, cells were washed with PBS and fixed with 4% paraformaldehyde (PFA) for 20 min at RT. Fixed cultures were immunostained and subsequently analyzed. In addition, cells were lysed following the Triton soluble/insoluble biochemical assay described below. To verify the specificity of effects in our study, two batches of PFFs were used from two different labs. Protein concentration was estimated by the Bradford assay. Primary neuronal cultures were infected at DIV5 with the lentivirus pFUGW coding for WT aSyn or GFP.

SDS-PAGE and Immunoblotting

After treatment, cells were washed with PBS and lysed on ice in radio-immunoprecipitation assay buffer (RIPA) (50 mM Tris pH 8.0, 150 mM NaCl, 0.1% Sodium-Dodecyl-Sulfate (SDS), 1% Nonidet P40, 0.5% Sodium-Deoxycholate, protease inhibitors, Roche). Lysates were centrifuged at 10,000 rpm and 4 °C for 10 min and post-nuclear supernatants were kept. Protein concentration was determined using the Bradford assay (Bio-Rad). All samples were measured in triplicate. For RIPA samples, equal protein amounts of denatured samples (5 min at 95 °C in 5 × protein sample buffer; 125 mM of 1 M Tris HCl pH 6.8, 4% SDS 0,5% Bromophenol blue, 4 mM EDTA 20% Glycerol 10% β-Mercaptoethanol) were subjected to SDS-PAGE on 12% separating gels with 7% stacking gels, using Tris-Glycine SDS 0.5% running buffer (250 mM Tris, 200 mM Glycine, 1% SDS, pH 8.3). The SDS insoluble fractions were denatured at 42 °C for 15 min. The transfer was carried out to 0.45 μm nitrocellulose membranes for 20 min per membrane at constant 25 mA in a semi-dry transfer chamber Trans-Blot® Turbo™ Transfer Solution from Bio-Rad (Bio-Rad). Membranes were blocked in 5% (w/v) skim milk (Fluka, Sigma-Aldrich) dissolved in 1 × TBS-Tween (50 mM Tris (hydroxymethyl)-aminomethane (TRIS) supplemented with 0.05% (v/v) Tween-20) for 1 h at RT. Incubation with the primary antibodies (anti-aSyn Syn-1 mouse, BD Transduction Laboratories 1:1000; anti-pS129-α-syn, Rabbit Abcam; mouse anti-β-actin, 1:10.000, Sigma) was performed overnight at 4 °C in 5% Albumin Bovine Fraction V (BSA)/ TBS-Tween. Secondary antibodies (anti-mouse and anti-rabbit IgG, 1∶10,000 in TBS-Tween) were applied after three times washing in TBS-Tween, for 1 h at room temperature (RT). Membranes were visualized using Fusion Fx (Vilber Lourmat) with Immobilon Western Chemiluminescent HRP 1 3 Substrate (Merck Millipore). Protein levels were quantified using ImageJ and normalized to the β-actin levels.

Triton Soluble/Insoluble Biochemical Analysis

HEK293 cells were treated with aSyn monomers and PFFs for 48 h, harvested with trypsin, and washed with ice cold PBS and diluted 1:10 trypsin. The cell pellet was collected and solubilized in 1% triton buffer (50 mM tris pH 7.6, 150 mM NaCl, 2 mM EDTA, 1% triton, protease and phosphatase inhibitors), followed by 30-min incubation on ice. Cell lysates were then centrifuged at 13,000 g for 30 min at 4 °C, and the supernatant was collected as the Triton soluble fraction. The pellet was washed with ice cold PBS. Insoluble fractions were prepared by resuspending the pellets in 2% SDS soluble buffer (50 mM tris pH 7.6, 150 mM NaCl, 2 mM EDTA, 2% SDS, protease and phosphatase inhibitors), sonicated and centrifuged at 13,000 g for 10 min after incubation for 30 min at RT. The collected supernatant represents the SDS soluble fraction. The protein concentration was determined by the Bradford protein assay according to the manufacturer's instructions, and the samples were then subjected to Western blot analysis. In order to estimate the effect of the levels of aSyn on seeding, we ran all samples in the same blot, or in pairs (HEK293-EGFP with HEK293-aSynEGFP, and HEK293 naive with HEK293-aSynBiFC).

Filter Trap and Dot-Blot Assay

For the filter trap assay, equal protein amounts of recombinant aSyn monomers and PFFs were adjusted to equal volumes (diluted in PBS) and subsequently subjected to vacuum filtration through a 96-well dot blot apparatus (Bio-Rad) with an acetate cellulose membrane; immunoblotting (IB) was then performed with antibody against total aSyn (Syn-1). As a loading control experiment, the same samples were subjected to vacuum filtration through a 96-well dot blot apparatus with a nitrocellulose membrane followed by immunoblotting (IB) with Syn-1 antibody. Results shown are representative of 3 independent experiments.

Mitochondrial Fractionation

Mitochondria were isolated as previously described [START_REF] Fernandez-Vizarra | Isolation of mitochondria for biogenetical studies: an update[END_REF], with some modifications. All buffers were prepared fresh before the isolation and placed in ice at least 15 min before harvesting the cells. Briefly, cells were harvested by trypsinization, pelleted and washed with cold-PBS buffer twice at 600 g for 5 min at RT. After the last wash, PBS was discarded and one volume of the cell pellet of IB 0.1 × buffer (NaCl 2.5 mM, MgCl 2 0.5 mM, TRIS 3.5 mM pH = 7.8) was added. The cells were resuspended gently and homogenized with 15 strokes using a Heidolph D-91126 Type: RZR1 homogenizer at 1500 rpm. The sample was transferred to Eppendorf 1.5-mL tubes and 1/10 of the initial cell pellet volume of IB 10 × Buffer (NaCl 0.25 M, MgCl 2 50 mM, TRIS 0.35 M pH = 7.8) was added. A microcentrifuge was used to pellet the debris, unbroken cells, and nuclei at 1200 rpm for 3 min at 4 °C. The supernatant that contained mitochondria and cytoplasmic fraction was collected. An additional centrifuge step was performed at 1200 g for 3 min at 4 °C to obtain a cleaner fraction and avoid contamination of "heavy material." The supernatant was collected again. Finally, the collected supernatant was centrifuged at 1500 rpm for 2 min at 4 °C. The obtained pellet was the mitochondrial fraction and the supernatant was the cytoplasmic fraction, respectively. After collection of the cytoplasmic fraction, the mitochondrial fraction was resuspended in RIPA buffer. Protein concentration was determined by the Bradford protein assay and the samples were then subjected to Western blot analysis.

Immunocytochemistry

For ICC, cells were washed with PBS and diluted 1:10 Trypsin, fixed with 4% paraformaldehyde (PFA) for 20 min at RT, and followed by a permeabilization step with 0.5% Triton X-100 (Sigma-Aldrich) for 20 min at RT. After blocking in 1.5% normal goat serum (PAA)/DPBS for 1 h, cells were incubated with primary antibody. Primary antibodies used were as follows: anti-aSyn Syn-1 mouse, BD Transduction Laboratories (1:1000); anti-pS129-α-syn Rabbit Abcam (1:1000), incubated overnight at 4 °C and secondary antibody (Alexa Fluor 488 donkey anti-mouse IgG and/or Alexa Fluor 555 goat anti rabbit IgG, (Life Technologies-Invitrogen, Carlsbad, CA, USA) for 2 h at RT. Nuclei were stained with 4′6′-diamidino-2-phenylindol (DAPI, Sigma-Aldrich) (1∶5000 in DPBS) for 10 min. After a final wash, coverslips were mounted by using Mowiol (Sigma-Aldrich) and subjected to fluorescence microscopy. Images were analyzed using LAS AF v.2.2.1 (Leica Microsystems) software.

RT-QuIC Assay

Real-time quaking-induced conversion (RT-QuIC) was performed as a modification of previously proposed methods for different amyloid proteins [START_REF] Orrú | Factors that improve RT-QuIC detection of prion seeding activity[END_REF][START_REF] Schmitz | The real-time quaking-induced conversion assay for detection of human prion disease and study of other protein misfolding diseases[END_REF] and optimized in our lab. In brief, a reaction mixture with the following composition was made: 150 mM NaCl, 1 mM EDTA, 10 μM ThioT, 70 μM SDS, 10 μg/ml of monomeric aSyn and fibrillar aSyn [START_REF] Volpicelli-Daley | Addition of exogenous alpha-synuclein preformed fibrils to primary neuronal cultures to seed recruitment of endogenous alpha-synuclein to Lewy body and Lewy neurite-like aggregates[END_REF] at a final concentration of 0.1 μg/ml, 1 μg/ml, 10 μg/ml and 100 μg/ml in PBS buffer. A total of 100 μl of this reaction mixture was pipetted in black 96-well plates (Corning Incorporated, WA, USA) in triplicates. Additionally, cell lysates were added to the mixture to a final protein concentration of 0.1 µM. Finally, the plates were covered 1 3 with sealing tape and incubated in a plate reader (41 °C, Infinite M200 fluorescence plate reader, Tecan, Hamburg, Germany) for 250 amplification cycles (1-min orbital shaking at 432 rpm; 2-min incubation; measurement of fluorescence intensity at 480 nm). Endpoint fluorescent intensities were normalized by baseline values.

Statistical Analyses

Statistical analyses were performed using the Student's t-test or 2-way ANOVA for comparisons of independent variables. The data are presented as mean ± standard deviations and represent results from at least 3 independent experiments. Differences were considered statistically significant when *p < 0.05, **p < 0.01, and ***p < 0.001.

Results

Exogenous aSyn Preformed Fibrils (PFFs) Induce aSyn Aggregation and Accumulation of Endogenous Phosphorylated pS129aSyn Inclusions

The overall objective of our study was to investigate the effect of extracellular aSyn PFFs on the seeding and aggregation of endogenous aSyn expressed at variable levels in cells. For this purpose, we employed simple but highly tractable cellular bioreactors (stable human embryonic kidney (HEK293) cells) expressing aSyn fused with EGFP (HEK293-aSynEGFP), to assess the effects in living cells. Individual clones from both stable cell lines were selected for our experiments. Cells were treated with exogenous aSyn monomers or PFFs, characterized by immunoblot and electron microscopy before and after sonication (Supplementary Fig. 1A andB) [START_REF] Volpicelli-Daley | Addition of exogenous alpha-synuclein preformed fibrils to primary neuronal cultures to seed recruitment of endogenous alpha-synuclein to Lewy body and Lewy neurite-like aggregates[END_REF], at a final concentration of 100 nM, for 48 h. Control cells were exposed to vehicle only (PBS). We also quality-checked aSyn monomers and fibrils using a filter trap assay (Fig. 1A). To ensure we loaded comparable amounts of protein, a dot-blot was run in parallel with the same samples. Immunofluorescence analyses of treated cells showed that PFFs induced the formation of aSyn inclusions that were EGFP-positive, confirming the seeding of the endogenously expressed aSynEGFP by the PFFs (Fig. 1B), as this was not observed in PBS-treated cells. Quantification of the aSynEGFP inclusions revealed a progressive and time-dependent process that resulted in almost 25% of the cells displaying aSyn inclusions (Fig. 1C). No aggregates were detected after treatment with 100 nM of aSyn monomers.

To further validate our results and assess reproducibility between different sources of aSyn, we repeated the experiments with aSyn monomers and PFFs prepared independently, in a different laboratory (Mon2, PFFs2, Supplementary Fig. 1C). We found no differences between the results obtained with the two different sources PFFs (Supplementary Fig. 1C). All experiments included in the subsequent experiments were performed using PFFs1.

Next, we used the real-time quaking-induced conversion (RT-QuIC) assay, an ultrasensitive biochemical assay based on the binding of thioflavin T (ThT) by amyloid fibrils, to estimate the seeding activity of aSyn [START_REF] Groveman | Rapid and ultra-sensitive quantitation of disease-associated alpha-synuclein seeds in brain and cerebrospinal fluid by alphaSyn RT-QuIC[END_REF][START_REF] Han | RT-QuIC-based detection of alpha-synuclein seeding activity in brains of dementia with Lewy Body patients and of a transgenic mouse model of synucleinopathy[END_REF][START_REF] Fairfoul | Alpha-synuclein RT-QuIC in the CSF of patients with alpha-synucleinopathies[END_REF]. We used the HEK293-aSynEGFP cells and a stable cell line reporting on aSyn oligomerization based on the bimolecular fluorescence complementation (BiFC) assay HEK293-aSynBiFC [START_REF] Lazaro | Systematic comparison of the effects of alpha-synuclein mutations on its oligomerization and aggregation[END_REF]18] (Supplementary Fig. 2). RT-QuIC from the different cell lines non-treated with aSyn monomers and PFFs (n = 4) showed lower ThT fluorescence intensity than that of cells treated with monomers and PFFs (Supplementary Fig. 2A). We verified that the addition of PFFs induced seeding of monomeric aSyn leading to an increase in the overall normalized ThT fluorescence levels consisted with higher aggregation levels [START_REF] Volpicelli-Daley | Addition of exogenous alpha-synuclein preformed fibrils to primary neuronal cultures to seed recruitment of endogenous alpha-synuclein to Lewy body and Lewy neurite-like aggregates[END_REF] (Supplementary Fig. 2B). Importantly, the initial ThT fluorescence intensity values were not significantly different among groups of the different cell lines, suggesting the absence of additional unbound PFFs in the reaction buffer (Supplementary Fig. 2C). Addition of PFFs also led to shorter lag times, i.e., earlier and faster aggregation, consistent with the seeding effect (Supplementary Fig. 3D). As expected, higher concentrations of PFFs resulted in shorter lag phases and higher overall normalized ThT fluorescence levels. Next, cells were incubated with aSyn monomers or PFFs and reaction mixtures of cell lysates to final protein concentrations of 100 nM and were tested using the RT-QuIC assay. Interestingly, we observed stronger aSyn aggregation with cell lysates followed by treatment with aSyn PFFs, as demonstrated by the increase in ThT signal with the different concentrations. Importantly, the kinetics of conversion was identical between the cell lines seeded with cell lysates prepared after treatment with PFFs, unlike those seeded with lysates prepared from cells treated with aSyn monomers, with almost no amplification (Fig. 1D and Supplementary Fig. 2D). In addition, immunostaining experiments against Serine 129 confirmed the accumulation of endogenous phosphorylated pS129aSyn inclusions in both HEK293-aSynBiFC and HEK293-aSynEGFP cell lines (Fig. 1E). In contrast, treatment with aSyn monomers did not induce the accumulation of pS129aSyn (Supplementary Fig. 3A andB). Overall, we concluded that addition of aSyn PFFs induced aSyn seeding and accumulation of pS129aSyn aggregates. These data provide evidence that the extent to which endogenous aSyn aggregates depends on the expression levels of aSyn.

3

Different Expression Levels of aSyn Define the Seeding of Exogenously Added aSyn PFF Seeds

Focusing on the comparative analysis of the seeding effect between the different cell lines, we examined whether the different expression levels of aSyn modulated the uptake and seeding of exogenously-added aSyn PFFs. Therefore, cell lysates from the different cell lines were fractionated using Triton-X-and SDS-containing buffers. The seeding capacity of HEK293-aSynBiFC and HEK293-aSynEGFP stable cell lines was compared with that of naïve HEK293 and HEK293-EGFP cell lines. We found that high molecular weight (HMW) aSyn species were detected in PFF-treated cells in the SDS-soluble fraction in both stable cell lines expressing aSyn, whereas mostly monomeric species were detected in the Triton X-100 soluble fraction (Fig. 2A). The comparison between the HEK293-EGFP stable cell line and the HEK293-aSynEGFP line revealed, as expected, higher levels of aSyn in the later cells, demonstrating the contribution of the endogenous levels towards seeding. In addition, we detected an increase in phosphorylated aSyn on S129 in the SDS soluble fraction (Fig. 2B), in agreement with the immunostaining results (Fig. 1E). Consistently, we observed identical effects when comparing the HEK293-aSynBiFC cell line with naïve HEK293 cells (Fig. 2C and Supplementary Fig. 4A andB). Overall, we demonstrate that there is a clear correlation between the endogenous of aSyn and the seeding efficiency followed by the accumulation of extracellular pathological assemblies.

Treatment with aSyn PFFs Results to Accumulation of Truncated aSyn in Mitochondria

Mitochondrial dysfunction has long been implicated in PD, and several studies suggest that aSyn may associate with mitochondria and with mitochondrial proteins [START_REF] Szego | Cytosolic trapping of a mitochondrial heat shock protein is an early pathological event in synucleinopathies[END_REF][START_REF] Lin | The overcrowded crossroads: mitochondria, alpha-synuclein, and the endo-lysosomal system interaction in Parkinson's disease[END_REF][START_REF] Ortega | Alpha-synuclein over-expression induces increased iron accumulation and redistribution in iron-exposed neurons[END_REF]. Furthermore, pathogenic phospho-aSyn aggregates bind to mitochondria leading to cellular respiration defects [START_REF] Wang | Pathogenic alpha-synuclein aggregates preferentially bind to mitochondria and affect cellular respiration[END_REF]. Therefore, we examined whether aSyn could be detected in Immunoblot analysis against pS129 and Syn-1 resulted in a shift of the SDSsoluble aSyn in higher molecular weight species. This SDS-soluble aSyn fraction was significantly enriched in the cell lines overexpressing aSyn (HEK293-aSynBiFC and HEK293-aSynEGFP) compared to the control cell lines (HEK293 and HEK293-EGFP, respectively). High molecular weight species with aSyn PFFs were also positive for the phospho Ser 129 aSyn antibody. β-Actin was used as a loading control. C Densitometry of the SDS soluble fractions confirmed the significant difference between the cell lines (n = 3, mean ± SD) mitochondria, and whether the localization of aSyn in mitochondria was altered upon treatment with aSyn monomers or PFFs. Using subcellular fractionation, and using VDAC1 as a mitochondrial marker and β-actin as a cytosolic marker, we confirmed the presence of aSyn in the mitochondrial fraction. As expected, we detected a strong increase of aSyn-EGFP in the mitochondrial fraction upon overexpression. Furthermore, we observed a robust increase of truncated aSyn, mostly after treatment with aSyn PFFs (Fig. 3A-D). Treatment with PFFs promoted the localization of aSyn in the mitochondrial fraction in both HEK293-aSynEGFP and HEK293-EGFP cells. Surprisingly, no obvious difference was observed on the accumulation of aSyn in the mitochondria in cells expressing aSynGFP comparing to cells expressing only GFP (Fig. 3B). In addition, we examined if treatment with aSyn PFFs affected the levels of mitochondrial proteins such as OPA1, DRP1, HSP27, Mitofusin, and TRAP1. Immunoblot analyses showed that increased localization of aSyn in mitochondria did not alter the levels of mitochondrial proteins (Supplementary Fig. 5A andB).

Increased Levels of aSyn Enable Accelerated Seeding by aSyn PFFs in Primary Neuronal Cultures

Next, we examined the seeding effect of aSyn PFFs in primary cortical neurons. Briefly, we treated primary neurons at DIV5 with aSyn monomers or PFFs and incubated the cultures until DIV25 (Fig. 4A). Using immunocytochemistry, we detected the accumulation of pS129aSyn upon treatment with PFFs (Fig. 4B). Consistently, we observed the presence of HMWs of aSyn and increased accumulation of pS129aSyn in the SDS-soluble fraction, particularly after treatment with aSyn PFFs (Fig. 4C andD).

Finally, we assessed the seeding of exogenous aSyn in primary cortical neurons on the seeding by exogenous aSyn seeds in primary cortical neurons transduced with lentiviral vectors encoding aSyn or GFP as a control (Fig. 5A andB). Interestingly, we found increased accumulation of phosphorylated pS129aSyn aggregates in aSyn WT-infected neurons at DIV25. This was absent in cells treated with aSyn monomers or with PBS (Supplementary Fig. 6 and7). On the other hand, we observed lower pS129aSyn accumulation in GFP-infected cells, similar to what was observed with non-infected cells, indicating that phosphorylation of aSyn at Ser129 depends on endogenous aSyn levels. These data are in line with those observed using stable cell lines and demonstrate that the levels of aSyn influence the extent of seeding and the formation of pathological aSyn assemblies.

Discussion

Duplications or triplications of the SNCA gene are associated with early onset familial forms of PD, as well as motor and cognitive dysfunction [START_REF] Singleton | Alpha-synuclein locus triplication causes Parkinson's disease[END_REF][START_REF] Fuchs | Phenotypic variation in a large Swedish pedigree due to SNCA duplication and triplication[END_REF]. In fact, there is a clear dosage effect showing that an increase in SNCA copies leads to earlier disease onset and faster progression [START_REF] Fuchs | Phenotypic variation in a large Swedish pedigree due to SNCA duplication and triplication[END_REF][START_REF] Ferese | Four copies of SNCA responsible for autosomal dominant Parkinson's disease in two Italian siblings[END_REF][START_REF] Konno | Autosomal dominant Parkinson's disease caused by SNCA duplications[END_REF].

Here, we used simple cellular models to investigate the relationship between the cellular uptake of exogenously-applied aSyn PFFs and the pathological conversion of endogenously expressed of aSyn into aggregated species. We observed an increased percentage of cells with inclusions and more insoluble aSyn in cells expressing aSynEGFP. Also, we detected an increase in phosphorylated aSyn in the SDS-soluble fraction, confirming seeded aggregation. Similar results were obtained when we used aSynBiFC cells. However, the levels of pS129 aSyn aggregates were lower in the later cells. Given the nature of the models, this limited detection of pS129 aSyn inclusions in the aSynBiFC cells suggests that aggregation precedes the pathological accumulation of pS129 aSyn deposits, as previously reported using an inducible model of in SH-SY5Y cells. In this study, the formation of insoluble pS129 aSyn inclusions was observed when a greater amount of fibrils was applied to the cells, and for a longer incubation period [START_REF] Pantazopoulou | Distinct alpha-synuclein species induced by seeding are selectively cleared by the Lysosome or the Proteasome in neuronally differentiated SH-SY5Y cells[END_REF]. Consistently, we found that using a lower amount of PFF seeds rarely resulted in the detection of pS129 aSyn-positive aggregates (data not shown). Using the RT-QuIC assay, we detected seeding in reactions where the cell lines were treated 5 aSyn overexpression in primary cortical neurons results to a robust increase of pS129 under treatment with aSyn PFFs. A Infection of primary cortical neurons at DIV5 with lentiviral vector for aSyn WT (pFUGW-aSyn). As a control, cells were infected with viruses encoding GFP (pFUGW-GFP). The same day, aSyn monomers and PFFs were added and incubated till DIV25, were neurons fixed and counterstained with aSyn (green) and pS129 (red) antibod-ies. For the pFUGW-GFP neurons, we checked the GFP fluorescence (green) and stained for pS129 (red). Indicative immunofluorescence showing the formation of phospho-Ser129 only at PFF-treated neurons for the indicated time. DAPI was used for nuclear staining (Scale bar 50 μm). B Indicative immunofluorescence of neurons counterstained with Map2 (green) and pS129 (red) antibodies. DAPI was used for nuclear staining (Scale bar 50 μm) 1 3 with aSyn PFFs but not in reactions where the cell lines were treated with monomeric aSyn. In accord with recent literature, the mitochondria of HEK293-aSynEGFP cells were enriched in aSynEGFP when compared to those from control HEK293-EGFP cells [START_REF] Shavali | Mitochondrial localization of alpha-synuclein protein in alpha-synuclein overexpressing cells[END_REF][START_REF] Park | Alpha-synuclein-induced mitochondrial dysfunction is mediated via a sirtuin 3-dependent pathway[END_REF]. In addition, fractionation of mitochondrial and cytosolic extracts revealed the presence of aSyn in mitochondria, mostly as truncated forms. These data demonstrate that differences in the endogenous levels of aSyn may render the cells more susceptible to the seeding effects of aggregated (PFFs) aSyn. This is consistent with a recent study showing that the accumulation of aSyn significantly differed between primary neuronal cultures from various brain regions characterized by different expression levels of aSyn [START_REF] Courte | The expression level of alpha-synuclein in different neuronal populations is the primary determinant of its prion-like seeding[END_REF].

Primary hippocampal neuronal cultures exhibit different aSyn expression profiles among cells since these cultures are composed of several neuronal and non-neuronal cells [START_REF] Luna | Differential alpha-synuclein expression contributes to selective vulnerability of hippocampal neuron subpopulations to fibril-induced toxicity[END_REF]. In this study, the authors demonstrate that the subpopulation of cells with higher intracellular levels of aSyn is more prone to seeding, exhibiting the highest pathology and selective vulnerability [START_REF] Luna | Differential alpha-synuclein expression contributes to selective vulnerability of hippocampal neuron subpopulations to fibril-induced toxicity[END_REF]. This is further supported by the fact that abundant aSyn expression levels observed in neurons where brain regions were affected at early PD stages [START_REF] Taguchi | Differential expression of alpha-synuclein in hippocampal neurons[END_REF][START_REF] Van Der Perren | Longitudinal follow-up and characterization of a robust rat model for Parkinson's disease based on overexpression of alpha-synuclein with adeno-associated viral vectors[END_REF], reinforcing the hypothesis that an increase in aSyn expression may be a risk factor for neurodegeneration. Thus, the heterogeneity of endogenous aSyn levels is reflected in selective vulnerability among neuronal cells and provides an explanation for the selective vulnerability of the brain regions observed in synucleinopathies. Consistently, we found that overexpression of aSyn in primary neuronal cultures results in accelerated accumulation of pS129aSyn aggregates after exposure to PFFs.

Collectively, our study is consistent with the possibility that brain regions expressing higher levels of aSyn may contribute more strongly to the spreading of pathology. Alternatively, it could also be that, in such brain areas, the balance between aSyn S129 phosphorylation/dephosphorylation is shifted towards the earlier, thereby leading to the accumulation of higher levels of aSyn phosphorylated on S129.

In the future, a better understanding of the expression profile of aSyn will be important to better understand the physiological function of aSyn as well as the implicated mechanisms involved in pathological aggregation. Ultimately, it will also be important to understand how the levels of aSyn vary in different subcellular compartments and what pathological alterations (e.g., S129 phosphorylation) take place in such compartments (e.g., in the nuclei or mitochondria).

Conclusions

In conclusion, by using tractable stable cell lines and primary neuronal cultures with different expression levels of aSyn, we demonstrate that the endogenous aSyn levels are a key element that defines the seeding by extracellular aSyn PFFs. Our study puts forward models that will now be useful for addressing additional molecular mechanisms regulating the spreading and pathological accumulation of aSyn and for the identification of novel targets for therapeutic intervention.

Fig. 1

 1 Fig. 1 Exogenous aSyn preformed fibrils (PFFs) induce aSyn aggregation. A Filter trap and dot blot assay from recombinant (100 nM) aSyn both diluted in PBS. B Indicative immunofluorescence of the HEK293-aSynEGFP stable cell line stained with anti-aSyn antibody, after treatment with aSyn monomers and PFFs (100 nM/48 h) shows to formation of aSyn aggregates after treatment with PFFs (Scale bar 25 μm). C Quantification of the cells with aSyn inclusions shows a progressive formation of inclusions over time in cells treated with

Fig. 2

 2 Fig.2aSyn fibrils seed endogenous aSyn to different extent depending on the endogenous levels. A, B Immunoblot for the Triton-X soluble and SDS-soluble samples extracted from the different cell lines (HEK293, HEK293-aSynBiFC, HEK293-EGFP, and HEK293-aSynEGFP) treated with 100 nM of aSyn monomers and PFFs diluted in PBS. Cells treated with PBS were used as the control. Immunoblot analysis against pS129 and Syn-1 resulted in a shift of the SDSsoluble aSyn in higher molecular weight species. This SDS-soluble

Fig. 3

 3 Fig. 3 Treatment with aSyn fibrils induces accumulation of truncated aSyn in mitochondria. A Immunoblot of the mitochondrial extracts from HEK293-EGFP and HEK293-aSynEGFP stable cell lines shows increased localization of aSyn truncated (indicated with arrow) after treatment with aSyn PFFs. VDAC was used as a loading control. B Densitometric analysis of fractionated mitochondrial lysates (n = 3, mean ± SD). C Immunoblot of the cytosolic extracts from HEK293-EGFP and HEK293-aSynEGFP stable cell lines. was used as a loading control. D Densitometric analysis of fractionated cytoplasmic lysates (n = 3, mean ± SD)

Fig. 4

 4 Fig. 4 aSyn fibrils seed endogenous aSyn in primary neuronal cultures. A Seeding model in primary cortical neurons. At DIV5 aSyn, monomers and PFFS were added and incubated until DIV25, when neurons fixed and counterstained with Map2 (green) and pS129 (red) antibodies. B Representative immunofluorescence showing the formation of phospho-Ser129 only at PFF-treated neurons for the indicated time. DAPI was used for nuclear staining (Scale bar 50 μm).

  Fig.5aSyn overexpression in primary cortical neurons results to a robust increase of pS129 under treatment with aSyn PFFs. A Infection of primary cortical neurons at DIV5 with lentiviral vector for aSyn WT (pFUGW-aSyn). As a control, cells were infected with viruses encoding GFP (pFUGW-GFP). The same day, aSyn monomers and PFFs were added and incubated till DIV25, were neurons fixed and counterstained with aSyn (green) and pS129 (red) antibod-

Molecular Neurobiology (2022) 59:1273-1284

Availability of Data and Materials

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Supplementary Information

The online version contains supplementary material available at https:// doi. org/ 10. 1007/ s12035-021-02713-2.

Author Contribution TFO and EV designed the study. EV performed cell cultures, treatments, prepared samples, immunoblotting, immunofluorescence, assisted in the interpretation of data, and wrote the manuscript. ADM performed the RT-QuiC experiments. MFL performed the mitochondrial fractionation experiments. MAA synthesized and characterized the fibrils. AK performed experiments and provided assistance with biochemical assays. EV and TFO wrote the manuscript. All authors revised the manuscript.

Funding TFO, RM, and LS are supported by the EU/EFPIA/Innovative Medicines Initiative [START_REF] Braak | Staging of brain pathology related to sporadic Parkinson's disease[END_REF] Joint Undertaking (IMPRIND grant no. 116060). ADM is supported by a postdoctoral fellowship from the Galician Government (Programa de axuda á etapa posdoutoral, XUGA, GAIN, ED481B 2017/053). TFO is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy -EXC 2067/1-390729940, and SFB1286 (Project B8).

Declarations

Ethics Approval and Consent to Participate Not applicable.

Consent for Publication Not applicable.

Competing Interest The authors declare no competing interests.

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Ronald Melki
email: ronald.melki@cnrs.fr.

Amyloid polymorphs and pathological diversities

The presence of protein aggregates within the central nervous system is intimately associated to debilitating neurodegenerative diseases. While the aggregation of proteins, that share no primary structure identity, is understandable in different diseases, that of a given protein yielding distinct pathologies is counterintuitive. This short review relates molecular and mechanistic processes to the observed pathological diversity.

Introduction

The aggregation of many proteins and polypeptides into assemblies of fibrillar nature is the hallmark of numerous diseases. Tau, A-beta, alpha-synuclein, PrP, huntingtin, serpins, aggregation is for instance intimately associated to tauopathies, such as Alzheimer's disease, synucleinopathies, such as Parkinson's disease, spongiform encephalopathies, such as Creutzfeldt-Jacob disease, Huntington's disease and different forms of dementias. These proteins share no primary structure identity and this may account for the different diseases they cause. However, taken individually, all of these proteins are tightly associated to different diseases. For instance, Tau aggregation is involved in Alzheimer's disease but also Pick's disease, progressive supranuclear palsy, globular glial tauopathy, aging related tau astrogliopathy, chronic traumatic encephalopathy, argyrophilic grain disease, primary agerelated tauopathy. Alpha-synuclein aggregation is involved in Parkinson's disease, Lewy body dementia and the two forms of multiple system atrophy. There must be explanations for that. This short review provides plausible molecular and mechanistic events that may account for the observed pathological diversity.

Amyloid formation and structural diversity

Most of the proteins listed above are in total or part "natively unfolded", a term used to reflect their dynamics. These proteins in total or part adopt multiple conformation, not necessarily extended, that does not allow determining an average conformation unless upon binding a ligand ranging from a small molecule to a ligand [START_REF] Ferreon | Interplay of alphasynuclein binding and conformational switching probed by single-molecule fluorescence[END_REF][START_REF] Frimpong | Characterization of intrinsically disordered proteins with electrospray ionization mass spectrometry: conformational heterogeneity of a-synuclein[END_REF][START_REF] Weinreb | NACP, a protein implicated in Alzheimer's disease and learning, is natively unfolded[END_REF][START_REF] Eliezer | Conformational properties of alpha-synuclein in its free and lipid-associated states[END_REF][START_REF] Uversky | A protein-chameleon: conformational plasticity of α-synuclein, a disordered protein involved in neurodegenerative disorders[END_REF][START_REF] Maiti | Raman spectroscopic characterization of secondary structure in natively unfolded proteins: alpha-synuclein[END_REF][START_REF] Sandal | Conformational equilibria in monomeric a-synuclein at the single-molecule level[END_REF][START_REF] Anderson | Interplay between desolvation and secondary structure in mediating cosolvent and temperature induced alphasynuclein aggregation[END_REF][START_REF] Burre | Properties of native brain alpha-synuclein[END_REF][START_REF] Lashuel | The many faces of alphasynuclein: from structure and toxicity to therapeutic target[END_REF][START_REF] Theillet | Structural disorder of monomeric alphasynuclein persists in mammalian cells[END_REF][START_REF] Ulmer | Structure and dynamics of micelle-bound human alphasynuclein[END_REF][START_REF] Jeganathan | The natively unfolded character of tau and its aggregation to Alzheimer-like paired helical filaments[END_REF][START_REF] Mukrasch | Structural polymorphism of 441-residue tau at single residue resolution[END_REF][START_REF] Narayanan | Automatic assignment of the intrinsically disordered protein Tau with 441residues[END_REF].

The ensemble of tertiary structures or conformational states these dynamic proteins or protein domains populate depends on their primary structure and is immense. Indeed, if we assume that each amino acid residue within a protein or protein domain made of 100 amino acid residues can adopt a limited number of conformations, for example 3 (1 trans and 2 gauche) with only 2 torsions each, the number of possible conformations such a protein or protein domain could adopt would surpass 3 198 (3 99x2 ) conformations. The different conformations within the ensemble of tertiary structures are in equilibrium (Fig. 1A), meaning that each conformation is populated for a given time [START_REF] Fersht | Structure and Mechanism in Protein Science: A Guide to Enzyme Catalysis and Protein Folding[END_REF]. If each conformation is populated for a very small fraction of a second, e.g. 10 -12 s, it would take a single molecule over the age of the universe to populate this limited set of possible conformations. The concentration and lifespan of each conformation are specific to the tertiary structure and are defined by intramolecular interactions between amino acid residues stabilized by hydrogen bonds, electrostatic and hydrophobic interactions. The latter depend on the amino acid composition of the protein, the distribution of the amino acid residues within the protein primary structure and the chemical and physical conditions surrounding the protein. This is why the ensemble of conformations such a protein or protein domain can explore is several orders of magnitude smaller than the number indicated above. As a consequence, the probability of a dynamic aggregation prone monomeric protein to populate conformers exposing amino acid stretches that allow them to establish well-defined inter-molecular interactions with molecules that are in a compatible conformation is far from negligible at any time. This leads to ordered aggregation.

As upon the crystallization of a protein, the nature of the interactions allowing the formation of the quaternary structures, e.g. dimers, trimers and higher molecular weight oligomeric species between molecules, of a given protein define the stability of the assemblies this protein forms [START_REF] Oosawa | Thermodynamics of the Polymerization of Protein[END_REF]. When a molecule adopts conformations incapable of establishing stable and highly complementary interaction with the seed elongating tips it cannot add on and be subsequently incorporated within the seed (Fig. 1B). However, a given protein in different conformations can establish different complementary interactions with molecules in compatible conformation. Thus, assemblies, including crystals, made of the same protein in different conformations can form (Fig. 1B).

Most of the proteins whose aggregation is associated to neurodegenerative diseases adopt beta-strands secondary structures rich conformations. The distribution of these beta strands is highly dependent on protein primary structure. The establishment of numerous hydrogen bonds between the molecules allows their highly ordered stacking into assemblies of fibrillar shapes named amyloids. The term amyloid signifies starchlike. This term was used in the 19th century to refer to deposits within the brain that stain pale blue with iodine and violet upon treatment with sulfuric acid as do starch deposits in plants. Amyloid fibrils are defined as fibrillar polypeptide aggregates with cross-beta conformation, a structure where the hydrogen bonds between two consecutive sheets are oriented parallel to the main fibril axis while the constituting beta-strands are oriented transversely to the main fibril axis [START_REF] Astbury | The X-ray interpretation of denaturation, the structure of the seed globulins[END_REF][START_REF] Makin | Structures for amyloid fibrils[END_REF]. This type of structure gives rise to a characteristic pattern of reflections in X-ray diffraction experiments consisting of a conserved 4.6-4.8 Angström meridional spacing and an equatorial spacing of about 10 Angström. The 4.6-4.8 Angström reflection comes from the distance between two hydrogen bonded strands and is invariant as it depends on the geometry of the polypeptide backbone. It is referred to as the "main chain spacing". The equatorial reflection at about 10 Angström comes from the packing distance between two juxtaposed beta-sheets [START_REF] Sunde | Common core structure of amyloid fibrils by synchrotron Xray diffraction[END_REF][START_REF] Fandrich | The behaviour of polyamino acids reveals an inverse side chain effect in amyloid structure formation[END_REF]. This distance can vary with the polypeptide amino acid composition as it depends on the orthogonal protrusion of the amino acid side chains from the plane of the sheet. It is worth noting that this reflection is not observed when the inter-sheet spacing is not regular.

Molecular consequence of amyloid fibrils structural diversity

Most dynamic proteins can populate different beta-strands rich conformations, the stacking of which, following the principles detailed above, yield amyloids with the characteristic 4.6-4.8 and ~10 Angström reflections and additional reflections originating from highly ordered domains stacks, when they exist. The distinct folds an amyloid core of a given fibril adopts defines the surfaces the fibrils expose to the solvent. It also determines whether the fibrillar scaffold is made of a single or multiple protofilaments. This has consequences on fibrils growth rates, nanomechanical properties, processing and degradation by the cellular machinery in charge of their clearance and ligands binding, in particular their cellular interactomes.

Indeed, the different tertiary structures or folds a given amyloid forming protein can adopt define the way the molecules stack into fibrils, the inter molecular interfaces between any two consecutive protein molecules and the number of intermolecular H-bonds within these interfaces [START_REF] Bousset | Structural and functional characterization of two alpha-synuclein strains[END_REF][START_REF] Guerrero-Ferreira | Cryo-EM structure of alpha-synuclein fibrils[END_REF], 2019;[START_REF] Schweighauser | Structures of αsynuclein filaments from multiple system atrophy[END_REF][START_REF] Zhao | Parkinson's disease-related phosphorylation at Tyr39 rearranges α-synuclein amyloid fibril structure revealed by cryo-EM[END_REF][START_REF] Li | Amyloid fibril structure of αsynuclein determined by cryo-electron microscopy[END_REF][START_REF] Li | Cryo-EM of fulllength α-synuclein reveals fibril polymorphs with a common structural kernel[END_REF][START_REF] Burger | Cryo-EM Structure of Alpha-Synuclein Fibrils Amplified by PMCA from PD and MSA Patient Brains bioRxiv[END_REF][START_REF] Fitzpatrick | Cryo-EM structures of tau filaments from Alzheimer's disease[END_REF][START_REF] Falcon | Structures of filaments from pick's disease reveal a novel tau protein fold[END_REF][START_REF] Falcon | Novel tau filament fold in chronic traumatic encephalopathy encloses hydrophobic molecules[END_REF]Zhang et al., 2020;[START_REF] Shi | Structure-based classification of tauopathies[END_REF]. This affects mostly fibrils growth through the incorporation of molecules in compatible conformations. Different folds for a given protein define the surfaces of the stacks they form. The latter allow or not defined interactions between different stacks of a protein in the same or different conformations yielding fibrils made of multiple protofilaments. Thus, the intrinsic architectures of fibrils made even from one given protein is defined by the amyloid forming folds its constituting protein populates. This in turn defines the amino acid stretches composing the solvent exposed polypeptide chains and their distribution in space at the external surfaces of the fibrils, the surfaces through which they interact with ligands ranging from small molecules to cell components [START_REF] Landureau | The differential solvent exposure of N-terminal residues provides "fingerprints" of alpha-synuclein fibrillar polymorphs[END_REF][START_REF] Caroux | Structural mapping techniques distinguish the surfaces of fibrillar 1N3R and 1N4R human tau[END_REF].

Amyloid structural diversity and differential tropism

The amino acid residues and peptide chains exposed to the solvent at the surface of amyloid protein fibrils involved in neurodegenerative diseases determine what cell type and cellular components they interact with. Indeed, the presence of receptors and their density at the surface of neuronal cells define whether an amyloid fibril can bind to a given cell and the tropism of the fibrils to the cells. These receptors are extracellular matrix, phospholipids and membrane proteins. Amyloid protein fibrils surfaces also define whether they may be post-translationally modified and as a consequence their interactomes that depend on post-translational modifications.

Indeed, Amyloids binding to the cell membrane has been shown to affect membrane fluidity and geometry/curvature [START_REF] Smith | Induction of negative curvature as a mechanism of cell toxicity by amyloidogenic peptides: the case of islet amyloid polypeptide[END_REF].

The lateral diffusion of amyloids in the plane of the plasma membrane together or independently of membranous components such as receptors, channels, adhesion molecules, contributes to the redistribution and/or coalescence of protein partners at the surface of the cell [START_REF] Rossy | The organisation of the cell membrane: do proteins rule lipids?[END_REF][START_REF] Shrivastava | Physico-pathologic mechanisms involved in neurodegeneration: Misfolded protein-plasma membrane interactions[END_REF].

The different surfaces amyloid fibrils expose to the solvent define their interactome. A-beta and alpha-synuclein interact with the α3 subunit of neuronal sodium/potassium pump [START_REF] Shrivastava | a-synuclein assemblies sequester neuronal a3-Na+/K+-ATPase and impair Na+ gradient[END_REF][START_REF] Ohnishi | Na, K-ATPase a3 is a death target of Alzheimer patient amyloid-b assembly[END_REF] while A-beta amyloids only to interact with α7-nicotinic acetylcholine receptors [START_REF] Wang | beta-Amyloid(1-42) binds to alpha7 nicotinic acetylcholine receptor with high affinity. Implications for Alzheimer's disease pathology[END_REF][START_REF] Dineley | Betaamyloid activates the mitogen-activated protein kinase cascade via hippocampal alpha7 nicotinic acetylcholine receptors: in vitro and in vivo mechanisms related to Alzheimer's disease[END_REF][START_REF] Snyder | Regulation of NMDA receptor trafficking by amyloid-beta[END_REF][START_REF] Dziewczapolski | Deletion of the alpha 7 nicotinic acetylcholine receptor gene improves cognitive deficits and synaptic pathology in a mouse model of Alzheimer's disease[END_REF], leukocyte immunoglobulin-like receptor B2 [START_REF] Kim | Human LilrB2 is a b-amyloid receptor and its murine homolog PirB regulates synaptic plasticity in an Alzheimer's model[END_REF], Ephrin B2 receptor [START_REF] Cissé | Reversing EphB2 depletion rescues cognitive functions in Alzheimer model[END_REF] and PrPc-mGluR5 complex [START_REF] Lauren | Cellular prion protein mediates impairment of synaptic plasticity by amyloid-beta oligomers[END_REF][START_REF] Renner | Deleterious effects of amyloid b oligomers acting as an extracellular scaffold for mGluR5[END_REF][START_REF] Um | Metabotropic glutamate receptor 5 is a coreceptor for Alzheimer ab oligomer bound to cellular prion protein[END_REF][START_REF] Haas | Therapeutic molecules and endogenous ligands regulate the interaction between brain cellular prion protein (PrPC) and metabotropic glutamate receptor 5 (mGluR5)[END_REF][START_REF] Haas | Oligomers of amyloid β prevent physiological activation of the cellular prion protein-metabotropic glutamate receptor 5 complex by glutamate in Alzheimer disease[END_REF][START_REF] Hu | mGlu5 receptors and cellular prion protein mediate amyloid-b-facilitated synaptic long-term depression in vivo[END_REF][START_REF] Casley | Up-regulation of astrocyte metabotropic glutamate receptor 5 by amyloid-b peptide[END_REF][START_REF] Lim | Amyloid beta deregulates astroglial mGluR5-mediated calcium signaling via calcineurin and Nf-kB[END_REF][START_REF] Shrivastava | b-amyloid and ATP-induced diffusional trapping of astrocyte and neuronal metabotropic glutamate type-5 receptors[END_REF]. Furthermore, while α-synuclein amyloids interact with neurexin-subunits [START_REF] Shrivastava | a-synuclein assemblies sequester neuronal a3-Na+/K+-ATPase and impair Na+ gradient[END_REF][START_REF] Mao | Pathological a-synuclein transmission initiated by binding lymphocyte-activation gene 3[END_REF], amyloid β precursor-like protein 1 [START_REF] Mao | Pathological a-synuclein transmission initiated by binding lymphocyte-activation gene 3[END_REF] and the Glucose Related Protein of 78 kDa [START_REF] Bellani | GRP78 clustering at the cell surface of neurons transduces the action of exogenous alpha-synuclein[END_REF], A-beta amyloids do not. This clearly shows that the surface properties of distinct amyloid proteins define their interaction with partner proteins at the surface of neuronal cells. As structurally distinct amyloid fibrils made of one given protein expose different amino acid residues and peptide chains to the solvent [START_REF] Landureau | The differential solvent exposure of N-terminal residues provides "fingerprints" of alpha-synuclein fibrillar polymorphs[END_REF][START_REF] Caroux | Structural mapping techniques distinguish the surfaces of fibrillar 1N3R and 1N4R human tau[END_REF] it appears that they will interact with different ensembles of protein partners at the surface of neuronal cells. The same holds true upon post-translational modifications of amyloid protein fibrils surfaces.

Relationship between amyloid fibrils diversity and the nature of pathology

After take up by neuronal cells, amyloids made of distinct proteins involved in different neurodegenerative diseases interact with cytosolic components ranging from organelles to proteins which nature is dictated by their surfaces. The same is true for structurally distinct amyloids made of one given protein. Thus, distinct amyloids redistribute and or trap into a non-functional state different partner proteins. They also compromise to different extents the integrity of intracellular membranous compartments [START_REF] Flavin | Endocytic vesicle rupture is a conserved mechanism of cellular invasion by amyloid proteins[END_REF][START_REF] Gribaudo | Propagation of α-synuclein strains within human reconstructed neuronal network[END_REF]. In addition, as the probability with which the different tertiary structures or folds compatible with the amyloid seed are populated defines the rate at which they elongate and multiply, amyloid fibrils structure determines the rate at which the functional form of an amyloid forming protein is exhausted in cells after aggregation and contributes to the speed at which a pathology progresses with time and its nature. The lateral surfaces of amyloids determine their capacity to bundle thus masking proteolytic cleavage sites which in turn defines their resistance to clearance by the cellular degradation machineries.

As detailed above, amino acid stretches exposed on the sides of different amyloids involved in neurodegenerative diseases define their interactomes. The full interactome of pathogenic protein aggregates whether in the cytoplasm or at the plasma membrane is far from being known. A number of extracellularly exposed membranous components have been identified. One kind of α-synuclein fibrillar amyloid has been shown to bind neurexin-subunits, α3-Na+/K+ -ATPase, amyloid β precursor-like protein 1, neurexins, apolipoprotein E, agrin and the Glucose Related Protein of 78 kDa [START_REF] Mao | Pathological a-synuclein transmission initiated by binding lymphocyte-activation gene 3[END_REF][START_REF] Shrivastava | a-synuclein assemblies sequester neuronal a3-Na+/K+-ATPase and impair Na+ gradient[END_REF][START_REF] Bellani | GRP78 clustering at the cell surface of neurons transduces the action of exogenous alpha-synuclein[END_REF]. Tau 1N3R and 1N4R amyloid fibrils interact both with muscarinic receptors [START_REF] Gomez-Ramos | Extracellular tau promotes intracellular calcium increase through M1 and M3 muscarinic receptors in neuronal cells[END_REF], Na+/K+ -ATPase, glutamate AMPA and NMDA receptors, glypicans, neurexins, plasma membrane calcium transporting ATPase, voltage dependent anion selective channel proteins with notable differences in the interactomes of 1N3R and 1N4R Tau amyloids (Shrivastava et al., 2019). One of the numerous Aβ amyloid polymorphs bind mGluR5s, α7-nicotinic acetylcholine receptors, leukocyte immunoglobulin-like receptor B2, α3-Na+/K+ -ATPase, Ephrin B2 receptor and PrPc [START_REF] Casley | Up-regulation of astrocyte metabotropic glutamate receptor 5 by amyloid-b peptide[END_REF][START_REF] Lim | Amyloid beta deregulates astroglial mGluR5-mediated calcium signaling via calcineurin and Nf-kB[END_REF][START_REF] Shrivastava | b-amyloid and ATP-induced diffusional trapping of astrocyte and neuronal metabotropic glutamate type-5 receptors[END_REF][START_REF] Wang | beta-Amyloid(1-42) binds to alpha7 nicotinic acetylcholine receptor with high affinity. Implications for Alzheimer's disease pathology[END_REF][START_REF] Dineley | Betaamyloid activates the mitogen-activated protein kinase cascade via hippocampal alpha7 nicotinic acetylcholine receptors: in vitro and in vivo mechanisms related to Alzheimer's disease[END_REF][START_REF] Snyder | Regulation of NMDA receptor trafficking by amyloid-beta[END_REF][START_REF] Dziewczapolski | Deletion of the alpha 7 nicotinic acetylcholine receptor gene improves cognitive deficits and synaptic pathology in a mouse model of Alzheimer's disease[END_REF][START_REF] Kim | Human LilrB2 is a b-amyloid receptor and its murine homolog PirB regulates synaptic plasticity in an Alzheimer's model[END_REF][START_REF] Ohnishi | Na, K-ATPase a3 is a death target of Alzheimer patient amyloid-b assembly[END_REF][START_REF] Cissé | Reversing EphB2 depletion rescues cognitive functions in Alzheimer model[END_REF][START_REF] Lauren | Cellular prion protein mediates impairment of synaptic plasticity by amyloid-beta oligomers[END_REF]. Further identification of fibrils interactomes will allow assessing modulators of aggregation by rapid screening methods [START_REF] Hideshima | Two-step screening method to identify α-synuclein aggregation inhibitors for Parkinson's disease[END_REF]. It will allow also a better understanding of their spread patterns as the presence and abundance of these proteins on the surface of neuronal cells define the tropism of distinct amyloid fibrils released from dying cells, where they form, for different neuronal cell populations within the central nervous system. In other words their target cell populations.

Conclusion

Altogether the surfaces of structurally distinct amyloid fibrils made of a given protein or proteins sharing no primary structure identity defines their interactomes and as a consequence tropism for defined neuronal populations within the central nervous system. This can lead to different distributions in distinct neurodegenerative diseases. The intrinsic architecture of amyloid fibrils and the structure their constituting protein adopt within the fibrillar scaffold define their ability to bundle, be post-translationally modified and grow by incorporation and depletion of the cellular pool of their functional cellular form, e.g. their resistance to the cellular clearance machinery and rate of multiplication, respectively. As diverse cells within the central nervous system differ among other things by the membrane proteins they express, differential binding of distinct polymorphs may lead to the pathophysiological characteristics of distinct pathologies. Crucial insights into the relationship between amyloid polymorphism and pathological diversities will come from the thorough identification of structurally distinct amyloids made of one given or different proteins involved in diverse neurodegenerative diseases.

Fig. 1 .

 1 Fig. 1. A, Proteins involved in neurodegenerative diseases such as Tau, A-beta, alpha-synuclein, PrP, huntingtin, serpins, etc … adopt multiple conformations in equilibrium with each other as represented here by bricks of different shapes and colors. The shapes represent the different conformations a polypeptide adopts. The colors are diverse given that when a conformation differ from another, the amino acid stretches exposed at the surface of the protein are distinct. B, A given aggregation prone protein in different conformations (bricks with different shapes) can establish different complementary interactions with molecules in compatible conformation. The nature of the interactions between molecules of a given protein (different ways the bricks are piled up) define the stability of the assemblies this protein forms.

Acknowledgement

Research carried out in R Melki laboratory is supported by grants from the Fondation pour la Recherche Medicale (contract ALZ201912009776), the EU Joint Programme on Neurodegenerative Disease Research and Agence National de la Recherche (contracts PROTEST-70, ANR-17-JPND-0005-01 and Trans-PathND, ANR-17-JPND-0002-02) and France Parkinson.


Mamadou Ndiaye

Yogendra Singh

Houcem Fourati

Mouhamadou Sy

Bassirou Lo

Kamel Boukheddaden
email: kamel.boukheddaden@uvsq.fr

Isomorphism between the Electro-Elastic Modelling of the Spin Transition and Ising-like Model with Competing Interactions: Elastic Generation of Self-Organized Spin States

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I. INTRODUCTION

Spin-crossover (SCO) compounds are extensively studied in the literature due to their rich physical properties covering optics [1][2][3][4] , spintronics [5][6][7][8][9][10][11] , magnetism [12][START_REF] Carlin | Magnetic Properties of Transition Metal Compounds[END_REF][START_REF] Kahn | Molecular Magnetism[END_REF][START_REF] Palacio | Magnetic Phenomena in Molecular Materials[END_REF] , elasticity [16][17][START_REF] Spiering | Elastic Interaction in Spin-Crossover Compounds[END_REF] , and statistical mechanics [START_REF] Boukheddaden | [END_REF]20 .

SCO materials are molecular solids containing a metallic central atom surrounded by ligands in an octahedral symmetry. Fe(II)-based SCO complexes exhibit two spin states, a paramagnetic high-spin state, stable at high temperature and a diamagnetic low-spin state, stable at low-temperature thanks to the ligand field energy of the ligands. It is important to mention for the non-specialist readers that the SCO solids switch between a paramagnetic and a diamagnetic state and do not show magnetic ordering. The main reason for the absence of magnetic ordering in the HS phase of SCO solids is the lack of covalent bonding between the metal ions, which are distant by 1 nm. Only a few experimental data obtained on the bimetallic coordination polymeric compound 21 have shown the existence of long-range magnetic ordering in SCO materials, which has been modeled using three states spin Hamiltonians 22,23,24 .

Under the effect of external perturbations, such as a variation of temperature [25][26][27][28][29][START_REF] Van Koningsbruggen | Iron(III) Spin Crossover Compounds[END_REF] , pressure [START_REF] Li | [END_REF][32][33][34][35][36] , light irradiation [36][37][38][39][40][41][START_REF] Koshihara | Optical Properties of Low-Dimensional Materials[END_REF][START_REF] Gütlich | [END_REF] , electric or magnetic field [44][START_REF] Bousseksou | The Spin Crossover Phenomenon under High Magnetic Field[END_REF][46][47][48][49][50] , or chemical composition 36,[51][52][53] etc, the SCO solids may switch reversibly between HS and LS configurations leading to first-order or continuous (gradual)

transformations. When the switching takes place through a first-order phase transition, this type of property materials. The switching between the HS and LS states is accompanied by a significant volume change ( ) as widely demonstrated by x-rays diffraction on single crystals [54][START_REF]Spin Crossover in Transition Metal Compounds I-III[END_REF][56][START_REF] Guionneau | [END_REF] . Imaging spin-transition by optical microscopy conducted first by Jeftic and Hauser in a [Fe(ptz)6](BF4)2 (where ptz=1-npropyltetrazole) large single crystals pointed out the important role of the thermo-chromic character of these materials 58 . Utilizing this property, Chong et al. revealed the existence of mechanical stresses during the spin transition, evidenced during the volume variation of these systems through the generation of inhomogeneous stresses causing irreversible damage to the crystals 58,59 . Ogawa et al. 60 reported these inhomogeneous features during the photo-excitation of [Fe(2-pic)3]Cl2.EtOH (pic=2-picolylamine) using a femtosecond (fs) laser pulse. Indeed, the study of the spatiotemporal aspects of the spin transition phenomenon shows that the nucleation and the growth of LS or HS phases proceed via a single domain process, with a well-defined and visible interface, which moves at constant velocity. It is demonstrated that this propagation process 61,62 is governed by the mechanical stress effects, through the contribution of the elastic interactions, 58,63 arising from the volume change at the transition. On the other hand, many experimental studies 37,[61][62][63][64][65][66][67][68][69][70][71] conducted on these complexes showed that a rich variety of spin transition behaviors can be generated. Among them, one may quote: (i) the gradual spin-transition, which could be described by a simple Boltzmann distribution of overall energy levels involved in weak cooperativity system 72,73,74 ; and (ii) sharp first-order transitions with hysteresis transition 75 . The last points concern (iii) incomplete transitions and (iv) two-step or multi-step transitions [76][77][78][79][80][81][82] . The latter, take place with the emergence of ordered intermediate phases, which are stabilized leading to the appearance of one or several plateaus in the thermal-dependence of the order parameters, as reported in several experimental studies, 83- 86 which demonstrated the existence of self-organization of the spin states in these regions. The spatial organizations of the spin states were found under the form of antiferro-like (checkerboard) structure 87,88 or may have more complex structures, like ordered alternate HS and LS sheets as in [Fe(2-pic)3]Cl2.EtOH (pic=2-picolylamine) 83 or modulated HS-LS patterns, where the modulation can be commensurate or incommensurate with the lattice parameter 84,85,86 as obtained re multiple transitions are observed during the thermal transition. It is important to notice that this multiplicity of transitions is often enhanced by the presence of inequivalent iron sites in the system, which then creates artificially additional plateaus.

From the theoretical point of view, the occurrence of multi-step transitions with self-organization of the spin states in the plateau region has been explained first, in elastic models, as due to a spontaneous elastic frustration 89 that appears in the lattice due to the existence of antagonist (or competing) elastic interactions.

Other phenomenological approaches based on the use of anisotropic next-nearest neighbour Ising (ANNNI) model have been also developed 90,91 and allowed to reproduce the double step transitions as well as the existence of spatial striped self-organization of the spin states along with the spin transition mechanism.

Recently, it has been shown that the behaviors of spin-crossover compounds, in solid state, can be obtained by using quite realistic microscopic models [92][93][94][95][96][97][98][99][100] In some, anharmonic Lennard-Jones or Morse potentials were implemented 105, 106, 107 , while in others mechanoelastic descriptions taking into account for internal pressure effects 108 are developed, as well as harmonic electroelastic models 92,93 including explicit spin-distortion coupling. In all these models, the spincrossover molecules are modeled as atoms with two electronic states 87,93,100 : a stable state at low temperature and a stable state at high temperature. In the case of the electro-elastic model, which will be used in the present work, these atoms are interconnected by springs whose elastic constants and equilibrium distances depend on their spin states 87,93 . This genuine electro-elastic model, considered here in the general case, takes into account the degree of freedom of the spin states and the positions, and leads to short-and long-range effects 93,109,110 whose competition produces self-organization of the spin states.

Indeed, several models outlined the interplay between the spin-crossover and the lattice properties which

II. ELECTRO-ELASTIC MODELLING

The electro-elastic model describing the spin transition phenomenon considers a 2D square lattice system of sites depicted in Fig. 1, in which each SCO molecule representing a lattice site is associated with a fictitious spin state, , where corresponds to HS state and to the LS state. The Ising description in this model ends here since all interactions between the spin states are taken into account through the local variation of the molecular volume accompanying the spin change.

Thus, the sites (nearest neighbors and next-nearest neighbors located along diagonals) are linked by harmonic springs 92,93,99 . Their associated elastic constants and the equilibrium distances depend on the spin states of the linked sites. The Hamiltonian describing such system in 2D is given by 92 , Here, corresponds to the average equilibrium distance between HS and LS phases, and , is the lattice misfit between the LS and the HS phases.

FIG
Due to the 2D character of the lattice, the nnn equilibrium bond length, connecting two spins and along diagonals, is assumed to be equal to, This model has been resolved in the past for several situations of SCO lattices (1D and 2D) including the case of nanoparticles 113 , nanostructures 114 . It also allows us to demonstrate that the multistability observed in a large number of SCO solids may result from the existence of elastic frustrations 93,96,98,115 between the degrees of freedom (distortions, rotations, expansions, and contractions) involved during the spin transition. Although very powerful, the presence of spin and position variables in the model makes generally its treatment quite heavy and requires long-time simulations. Indeed, the resolution of the model includes Monte Carlo simulations on the spin states and extensive molecular dynamics (or MC simulations)

for the lattice positions. Therefore, as an alternative approach, we propose to keep the elastic character of the model, which makes its originality and to solve it in the homogeneous case, by adopting a kind of meanfield elastic theory, which then absorbs the lattice variables, which become average quantities depending on the spin states. We checked that within this method, the simulation time is divided by fifty, with highquality results, as will be demonstrated below.

A. The homogeneous version of the electro-elastic Model and derivation of the Ising-like Hamiltonian

The electro-elastic model described above and used in our previous papers 92,93 is implemented under its general form. This model shows that the elastic constants, as well as the equilibrium distances between the nearest neighbors (nn) and next-nearest neighbors (nnn) sites, depend on their connected spin states leading to a mutual dependence between distances, elastic constants and spin states. This character highlights the complex interplay between a local field-like contribution, local short-range interactions and a complex long-range elastic contribution, whose sign depends on several parameters. On the other hand, the MC simulations used to study this model are conducted on the spin states and positions. In the way to simplify the study of such 2D square lattice system where the degree of freedom of spins and positions are both taken into account, we use a homogeneous electro-elastic model, which we describe in the following manner.

First, we adopt the situation of a lattice with a homogeneous lattice bond length between nn and between nnn. The Hamiltonian (1) becomes,

The idea here is to minimize first the total energy with respect to variable in order to reach the mechanical equilibrium. This is done by solving the equation: , which leads to the following expression of the relaxed bond length distance, where, and are the nn and nnn number lattice of bonds, respectively.

By replacing with its expression, given in ( 2), and considering the case of a large system, which gives , the expression of the homogeneous bond length in the minimum elastic energy, becomes after some rearrangements,

Where, is the average spin value which connects to the HS fraction, , through , and and are the average nn and nnn spin-spin correlations, respectively. It is important to notice that the dependence of the mechanically relaxed homogeneous bond length on average spin quantities will naturally introduce infinite long-range interaction inside the lattice. This clearly establishes the origin of the long-range nature of the elastic interactions in SCO models, and validates the use of mean-field approaches in describing SCO materials.

Inserting the expression instead of in ( 1) and rearranging all terms, we find after some simple algebra, that Hamiltonian (1) can be mapped under the form of an Ising-like Hamiltonian including shortand long-range interactions of elastic nature, whose expression is, where, , and are field-like, nn interactions and nnn (along diagonals) interaction terms, whose analytic expressions as a function of the elastic constants, average magnetization and correlations are,

The temperature-dependent energy, , whose expression is, of the SCO system, and so it will be omitted in the next developments.

Compared with the usual Ising-like model, Hamiltonian (7) contains true elastic contributions which impact the interaction parameters as well as the effective ligand field, . The present homogeneous Ising-like

Hamiltonian depends on lattice parameters, , as well as on the electronic order parameters, which express the macroscopic spin state configuration of the system. This way, the lattice variables have been integrated out, which considerably simplifies the resolution and the treatment of the Hamiltonian.

Another interesting aspect of the model relates to the sign of the interaction parameters. Indeed, depending on the model parameters, the nn and nnn interactions may be positive or negative, which competes with the ferroelastic long-range interactions emerging from the effective field [Eq. ( 8a)]. On one hand, the homogeneous model is given in Eq. ( 7), highlights, the link existing between spin states, and distances, indicates that the variation of distances affects the electronic spin states. While, on the other hand, the elastic constants being independent on the spin states any variation to them does not induce any change on the electronic spin states per site but affects the rigidity of the system. Otherwise, the change of distances and elastic constants results in a competition between the ferroelastic long-range field-like and antiferroelastic short-range interactions.

It is worth mentioning that the structure of Hamiltonian (7) recalls that of the well-known ANNNI model 116,117,118 since it includes short-range antiferro-like interactions competing with long-range ferrolike interactions. ANNNI model whose complex phase diagram shows many bifurcations is widely used to describe frustration and spin patterning, including spatially-modulated structures. Compared to the present Hamiltonian (7), the nnn antiferro interactions in ANNNI model are taken along and -axis of the crystal lattice. Therefore, they correspond to third 3 rd nearest sites in the case of Hamiltonian (7), in which the 2 nd neighbors are considered along the diagonals. Although, there is a clear difference in the interaction structure between the two models, we do think that their phase diagrams may show strong similarities, as indicated by the form of patterns obtained in the present simulations, due to the presence of similar long-range interactions and short-range antiferro coupling.

B. Monte Carlo technique and model parameter values

The study of the thermodynamic properties of the homogeneous electro-elastic 2D model ( 7) is performed on the 2D square lattice of size , with free boundary conditions. Each site interacts with its nearest and next-nearest neighbors. Initially, we prepare the system in the HS (resp. LS) phase by fixing all the spins to (resp. In the same way as beforehand, the increase of the nnn elastic constant from its previous value, 100 to and , results in the stabilization of the intermediate state until , leading to incomplete spin transitions, as presented in Fig. S1 of the supplemental material. Here also, complex structures mixing 1D ferroelastic structures along -and -directions are obtained in both cases. On the other hand, starting the simulations from the LS state also allows demonstrating the presence of a hidden stable LS state whose transition on heating to the intermediate state depends on parameter. These results are summarized in Fig. S2 of the supplemental material also demonstrate the appearance of complex structures around the thermal transitions from LS to the intermediate state and from the latter to the stable HS state. The phase diagram corresponding to the present case, , is reported in Fig. S3. It shows very similar trends with that of Fig. 8, with the existence of several regions of (i) two consecutive gradual transitions, (ii) a first-order transition followed with a gradual transition and at the end (iii) an incomplete first order transition, with a large plateau.

IV. CONCLUSION

To conclude, we have studied the thermal and spatiotemporal properties of a 2D spin-lattice system using a homogeneous electro-elastic model. We have shown that starting from the genuine electro-elastic model, in which we considered elastic constants independent of the spin states, and using the approximation

  Isomorphism between the electro-elastic modeling of the spin transition and Ising-like model with competing interactions: Elastic generation of self-organized spin states Mamadou Ndiaye, Yogendra Singh, Houcem Fourati, Mouhamadou Sy, Bassirou Lo, Kamel Boukheddaden To cite this version: Mamadou Ndiaye, Yogendra Singh, Houcem Fourati, Mouhamadou Sy, Bassirou Lo, et al.. Isomorphism between the electro-elastic modeling of the spin transition and Ising-like model with competing interactions: Elastic generation of self-organized spin states. Journal of Applied Physics, 2021, 129 (15), pp.153901. 10.1063/5.0045689. hal-03679113

  based on the elastic descriptions taking into account the volume change of the materials at the transition. These models include in fact the coupling between the spin state structure and the lattice degrees of freedom. Some of them are based on vibronic descriptions of the SCO phenomenon 101, 102, 103 while others used spin-phonon models 104 . In these two types of elastic models, the change of the lattice parameter at the transition has been omitted. Other kind of elastic models including volume change at the transition based on several types of potentials have been proposed.

  deform the lattice. This deformation of the lattice results from the variation of unit cell volume during the HS LS transition and is attributed to the long-range interactions, and to the non-homogeneous strains58,59,63,93,99, 109, 111 . On the other hand, it must be noticed that the resolution of the electro-elastic Hamiltonian, based on the change of position and spin degrees of freedom by Monte Carlo simulations is highly timeconsuming. This is due to the stochastic algorithm that imposes for each spin state change to relax mechanically the lattice by slight displacements of nodes irrespective whether this spin flip is accepted or rejected. Therefore, to reduce the execution time of this sequential resolution, we use, in this contribution, a homogeneous electro-elastic model to study the thermodynamic properties of spin-crossover compounds. This model is derived from the genuine electro-elastic Hamiltonian in the general case. For that, we first consider the elastic constants independent of the spin states, then the total elastic energy of the original 2D Hamiltonian is first minimized with respect to the lattice spacing and then re-expressed rigorously into an effective 2D Ising-like model, accounting for nearest and next-nearest neighbor interactions. The obtained model is solved by Monte Carlo simulations, based on Metropolis transition rates, from which we derive the thermal and spatiotemporal properties of this elastically homogeneous 2D lattice. Although the present Ising-like Hamiltonian, resulting from the general homogeneous elastic model, does not account for the lattice deformation, it is however more realistic than the usual Ising-like model or Drickamer 112 approaches, used to describe the thermal properties of SCO solids, in which the coupling parameters are purely phenomenological and miss the crucial information about their relation with the elastic interactions. The paper is organized as follows: in Sec. II, we first present the general version of the genuine 2D electro-elastic model for spin-crossover solids by considering the bond length distances and elastic constants depending on the spin states. Then, from this model, we deduce the homogeneous electro-elastic model equivalent to the Ising model combining long-range ferroelastic, and short-range nearest neighbors (nn) and next-nearest neighbors (nnn) antiferroelastic interactions. Finally, we describe the MC procedure and give the parameters used to solve the elastic Hamiltonian. Sec. III is devoted to the presentation and the discussion of the various thermal dependences of HS fractions and spatiotemporal organizations emerging from the simulations. In Sec. IV, we conclude and outline the possible extensions of the present work.

. 1 .

 1 FIG. 1. Left. An example of a non-relaxed spin-crossover lattice in the LS state. Right. Schematic representation of a central site of coordinates surrounded by four nearest and next-nearest neighbors.The coordinates of the neighbors , are respectively ( and )for nn and ( ), for the nnn.

FIG. 2 .FIG. 3 .FIG. 4 .

 234 FIG. 2. Thermal-dependence of the (a) HS fraction,

Figure 5 (Fig. 6 (FIG. 5 .FIG. 6 .FIG. 7 .

 56567 Fig. 6(b), where we repeated the thermal cycling by starting the Monte Carlo simulations from the LS state, for. There, a clear hidden LS state appeared, which becomes stable until , a

Figure 7 7 ,FIG. 8 .FIG. 9 .

 7789 Figure 7 depicts the dependence of the thermal width of the plateau, as a function of the nn elastic constant, Here, it is revealed that with (see inset of Fig. 7, indicating that during the thermal evolution, the plateau width follows a power law . The shape of the curve and the value of the exponent, close to , suggest the existence of a secondorder critical regime in the behavior of the thermal width of the plateau.

FollowingFig. 9 (

 9 Figs. 10(a) and 10(b) along with that of the total HS fraction, for the cooling and heating regimes respectively, in addition to the total HS fraction of the system, which is given for comparison. Thus on cooling from, , the two sublattices remain equivalent until , where a symmetry breaking occurs at this temperature for . Interestingly, we remark that until , the system self-organizes in the form of a ramified HS and LS stripes ordered along x-and y-directions, leading to a labyrinth formation. Below, , i.e. in the plateau regime, this ramified structure evolves towards a perfectly ordered pattern made of HS stripes ordered along the x-direction, reached around 90 K. Below this temperature, the LS state grows from the surface, as we see at on cooling in Fig. 9(c2), and again the ramified (labyrinth) HS-LS structure reappears before disappearing below A similar behaviour is also observed during heating where the ordered patterns are now obtained along the y-direction. Figures 10(a) and 10(b) which summarize these behaviours through the thermal dependences of the HS fraction of the two sublattices, and , shows two bifurcation points during cooling (resp. heating) between located at points E and F (resp. H and G) of coordinates ( ) and ( , ) [resp. ( , ) and ( , )], confirming the existence of a symmetry breaking between the two sublattices.

FIG. 10 .FIG. 12 .FIG. 13 .FIG. 14 .FIG. 15 .

 1012131415 FIG. 10. Thermal-dependence of HS fractions,

  of a homogeneous lattice parameter, significantly simplifies the model treatment. Indeed, the mechanical relaxation becomes analytic and the original model transforms naturally to an Ising-like model with longrange ferroelastic and antiferroelastic nn and nnn interactions. Somehow, this homogeneous treatment allows explaining the physical origin of the usual Ising-like model used to describe the thermal properties of SCO materials. In addition, this treatment allowed to identify the elastic origin of the short-range interactions as well as that of the long-range coupling. While the long-range contribution has the tendency to stabilize ordered ferroelastic or states, the short-range nn and nnn contribution lead to antiferroelastic structures along -and -directions, and along the diagonals, respectively. It is interesting to notice that the coexistence of competing long-range ferro-like and short-range nn and nnn antiferro-like interactions generates intrinsic elastic frustration in the lattice, resulting in the impossibility to realize simultaneously antiferroelastic order along the nn and nnn directions. This frustration creates an energetic competition between the elastic energies along -and -directions on one hand and along the diagonals on the other. The relevant parameters to control these competing energies are the nn and diagonal elastic constants, whose monitoring led to obtaining a rich variety of thermal behaviors of the high spin fraction, like two-and multi-step transitions (three and four-steps) with complex self-organizations of the spin states in the plateau regions. Among these, organization, checkerboard-like and stripes-like patterns have been obtained. The latter may be constituted of alternate HS or LS ferro-like stripes or alternate ferro (HS or LS) and antiferro-like chains. In this context, this work comes back to the first model one we developed for elastic frustration in SCO materials 89 , which also described the occurrence of multistep transitions using a different concept. Finally, this type of treatment opens the way to several extensions, among which the spin-dependence of the elastic constants, the use of anisotropic elastic interactions as well as the use of this model to develop elastic models under pressure for SCO materials. Compared to the usual electro-elastic models, this one presents the advantage of simplicity treatment, since the MC procedure is only applied on the spin states. Finally, it is worth to notice that the number of self-organized structures maybe highly increased if the range of the elastic interactions are increased. In the present study, only nearest neighbors and next-while one can increase this range to further far neighbors. This will increase the rate of frustration in the lattice, which will induce more complex structures and plateaus at the transition. The limiting case being where one considers directional interactions between all lattice sites.

ACKNOWLEDGMENTS

This work was supported by the French-Japan LIA (International Associate laboratory), the ANR project Mol-CoSM N° ANR-20-CE07-0028-02, the Universities of Versailles and Paris-Saclay-UPSAY, and the CNRS (Centre National de la Recherche Scientifique) and the French Embassy in Sénégal. We thank all of them for their financial support.

DATA AVAILABILITY

The data that supports the findings of this study are available within the article and its supplementary material.

SUPPLEMENTARY MATERIAL

See the supplemental material for : (i) additional data about self-organization of the spin states of Fig. 9, related to the conditions of obtaining incomplete spin transitions with the presence of complex structures mixing 1D ferroelastic structures along -and -directions, and for the evidence of hidden LS states when

NOTES

The authors declare no competing financial interest.