Christophe Bignon
email: christophe.bignon@afmb.univ-mrs.fr

Francesca Troilo

Stefano Gianni

Sonia Longhi
email: sonia.longhi@afmb.univ-mrs.fr

Partner-Mediated Polymorphism of an Intrinsically Disordered Protein

Keywords: Measles virus, NTAIL, XD, major inducible heat shock protein (hsp70), induced folding

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The measles virus (MeV) nucleoprotein consists of a structured region (NCORE, aa 1-400) [START_REF] Gutsche | Near-atomic cryo-EM structure of the helical measles virus nucleocapsid[END_REF] and a C-terminal domain (NTAIL, aa 401-525) that is intrinsically disordered [START_REF] Longhi | The Cterminal domain of the measles virus nucleoprotein is intrinsically disordered and folds upon binding to the C-terminal moiety of the phosphoprotein[END_REF][START_REF] Bourhis | The C-terminal domain of measles virus nucleoprotein belongs to the class of intrinsically disordered proteins that fold upon binding to their physiological partner[END_REF]. Intrinsically disordered proteins (IDPs) or regions (IDRs) are ubiquitous proteins/regions that lack highly populated secondary and tertiary structure in the absence of a partner or ligand [START_REF] Dunker | Protein disorder and the evolution of molecular recognition: theory, predictions and observations[END_REF][START_REF] Wright | Intrinsically unstructured proteins: re-assessing the protein structurefunction paradigm[END_REF][START_REF] Uversky | What does it mean to be natively unfolded?[END_REF][START_REF] Uversky | Natively unfolded proteins: a point where biology waits for physics[END_REF][START_REF] Habchi | Introducing Protein Intrinsic Disorder[END_REF]. As for many IDPs [START_REF] Wright | Linking folding and binding[END_REF], including the homologous NTAIL domains from the cognate Henipaviruses [START_REF] Blocquel | Compaction and binding properties of the intrinsically disordered C-terminal domain of Henipavirus nucleoprotein as unveiled by deletion studies[END_REF][START_REF] Habchi | Characterization of the interactions between the nucleoprotein and the phosphoprotein of Henipaviruses[END_REF][START_REF] Communie | Atomic resolution description of the interaction between the nucleoprotein and phosphoprotein of Hendra virus[END_REF][START_REF] Baronti | Dynamics of the intrinsically disordered C-terminal domain of the Nipah virus nucleoprotein and interaction with the X domain of the phosphoprotein as unveiled by NMR spectroscopy[END_REF], MeV NTAIL undergoes folding upon binding to the X domain (XD) of the viral phosphoprotein [START_REF] Bourhis | The intrinsically disordered C-terminal domain of the measles virus nucleoprotein interacts with the C-terminal domain of the phosphoprotein via two distinct sites and remains predominantly unfolded[END_REF][START_REF] Blocquel | Interaction between the C-terminal domains of measles virus nucleoprotein and phosphoprotein: A tight complex implying one binding site[END_REF][START_REF] Dosnon | Demonstration of a folding after binding mechanism in the recognition between the measles virus NTAIL and X domains[END_REF]. Interaction with the partner triggers α-helical folding within a Molecular Recognition Element (MoRE, aa 486-502) located within one (Box2, aa 489-506) out of three conserved NTAIL regions (Figure 1) [START_REF] Bourhis | The C-terminal domain of measles virus nucleoprotein belongs to the class of intrinsically disordered proteins that fold upon binding to their physiological partner[END_REF][START_REF] Bourhis | The intrinsically disordered C-terminal domain of the measles virus nucleoprotein interacts with the C-terminal domain of the phosphoprotein via two distinct sites and remains predominantly unfolded[END_REF][START_REF] Johansson | Crystal structure of the measles virus phosphoprotein domain responsible for the induced folding of the C-terminal domain of the nucleoprotein[END_REF][START_REF] Kingston | Structural basis for the attachment of a paramyxoviral polymerase to its template[END_REF][START_REF] Belle | Mapping alphahelical induced folding within the intrinsically disordered C-terminal domain of the measles virus nucleoprotein by site-directed spin-labeling EPR spectroscopy[END_REF][START_REF] Bernard | Interaction between the Cterminal domains of N and P proteins of measles virus investigated by NMR[END_REF][START_REF] Bischak | Probing structural transitions in the intrinsically disordered C-terminal domain of the measles virus nucleoprotein by vibrational spectroscopy of cyanylated cysteines[END_REF][START_REF] Gely | Solution structure of the C-terminal X domain of the measles virus phosphoprotein and interaction with the intrinsically disordered C-terminal domain of the nucleoprotein[END_REF][START_REF] Kavalenka | Conformational analysis of the partially disordered measles virus NTAIL-XD complex by SDSL EPR spectroscopy[END_REF][START_REF] Jensen | Intrinsic disorder in measles virus nucleocapsids[END_REF]. Binding of NTAIL to XD results in a "fuzzy" complex [START_REF] Fuxreiter | Fuzzy interactome: the limitations of models in molecular biology[END_REF][START_REF] Tompa | Fuzzy complexes: polymorphism and structural disorder in proteinprotein interactions[END_REF][START_REF] Fuxreiter | Fuzziness: linking regulation to protein dynamics[END_REF][START_REF] Miskei | FuzDB: database of fuzzy complexes, a tool to develop stochastic structure-function relationships for protein complexes and higher-order assemblies[END_REF], i.e. the regions preceding and following the MoRE remain both conspicuously disordered [START_REF] Bourhis | The intrinsically disordered C-terminal domain of the measles virus nucleoprotein interacts with the C-terminal domain of the phosphoprotein via two distinct sites and remains predominantly unfolded[END_REF][START_REF] Belle | Mapping alphahelical induced folding within the intrinsically disordered C-terminal domain of the measles virus nucleoprotein by site-directed spin-labeling EPR spectroscopy[END_REF][START_REF] Kavalenka | Conformational analysis of the partially disordered measles virus NTAIL-XD complex by SDSL EPR spectroscopy[END_REF][START_REF] D'urzo | Molecular Basis for Structural Heterogeneity of an Intrinsically Disordered Protein Bound to a Partner by Combined ESI-IM-MS and Modeling[END_REF]. Although the MoRE is partly pre-structured in the free form [START_REF] Belle | Mapping alphahelical induced folding within the intrinsically disordered C-terminal domain of the measles virus nucleoprotein by site-directed spin-labeling EPR spectroscopy[END_REF][START_REF] Bischak | Probing structural transitions in the intrinsically disordered C-terminal domain of the measles virus nucleoprotein by vibrational spectroscopy of cyanylated cysteines[END_REF][START_REF] Gely | Solution structure of the C-terminal X domain of the measles virus phosphoprotein and interaction with the intrinsically disordered C-terminal domain of the nucleoprotein[END_REF][START_REF] Jensen | Intrinsic disorder in measles virus nucleocapsids[END_REF][START_REF] Morin | Assessing induced folding of an intrinsically disordered protein by site-directed spin-labeling EPR spectroscopy[END_REF][START_REF] Wang | Multiscaled exploration of coupled folding and binding of an intrinsically disordered molecular recognition element in measles virus nucleoprotein[END_REF], computational and experimental studies revealed that MeV NTAIL folds according to a folding after binding mechanism [START_REF] Dosnon | Demonstration of a folding after binding mechanism in the recognition between the measles virus NTAIL and X domains[END_REF][START_REF] Wang | Multiscaled exploration of coupled folding and binding of an intrinsically disordered molecular recognition element in measles virus nucleoprotein[END_REF]. It should be reminded indeed that although conformational selection requires that the IDP populates (at least partly) the bound state in the free form, which therefore presages the final bound conformation [START_REF] Fuxreiter | Preformed structural elements feature in partner recognition by intrinsically unstructured proteins[END_REF][START_REF] Lee | Understanding pre-structured motifs (PreSMos) in intrinsically unfolded proteins[END_REF], the pre-existence of folded structures in an IDP does not necessarily commit to a conformational selection mechanism [START_REF] Wang | Multiscaled exploration of coupled folding and binding of an intrinsically disordered molecular recognition element in measles virus nucleoprotein[END_REF][START_REF] Hammes | Conformational selection or induced fit: a flux description of reaction mechanism[END_REF][START_REF] Wang | Exploration of multi-state conformational dynamics and underlying global functional landscape of maltose binding protein[END_REF]. To further complicate the scenario, many IDPs display a high extent of plasticity with respect to the partner, i.e. they can fold into different structures upon binding to different partners (see [START_REF] Habchi | Introducing Protein Intrinsic Disorder[END_REF] and references therein cited). This polymorphism thus raises the question as to which is the role of preconfigured MoREs in the recognition process, and suggests that the conformation sampled in solution will not necessarily be favored upon binding to a partner [START_REF] Wright | Linking folding and binding[END_REF][START_REF] Oldfield | Flexible nets: disorder and induced fit in the associations of p53 and 14-3-3 with their partners[END_REF]. In line with previous reports that pointed out that structural disorder is a determinant of protein interactivity [START_REF] Dunker | Flexible nets[END_REF][START_REF] Uversky | Showing your ID: intrinsic disorder as an ID for recognition, regulation and cell signaling[END_REF][START_REF] Haynes | Intrinsic disorder is a common feature of hub proteins from four eukaryotic interactomes[END_REF], beyond XD NTAIL binds to various viral and cellular proteins [START_REF] De | Involvement of actin microfilaments in the transcription/replication of human parainfluenza virus type 3: possible role of actin in other viruses[END_REF][START_REF] Moyer | Host cell proteins required for measles virus reproduction[END_REF][START_REF] Laine | Measles virus nucleoprotein binds to a novel cell surface receptor distinct from FcgRII via its C-terminal domain: role in MV-induced immunosuppression[END_REF][START_REF] Laine | Measles virus nucleoprotein induces cell proliferation arrest and apoptosis through NTAIL/NR and NCORE/FcgRIIB1 interactions, respectively[END_REF][START_REF] Iwasaki | The matrix protein of measles virus regulates viral RNA synthesis and assembly by interacting with the nucleocapsid protein[END_REF][START_REF] Sato | Morbillivirus nucleoprotein possesses a novel nuclear localization signal and a CRM1-independent nuclear export signal[END_REF][START_REF] Tenoever | Recognition of the Measles Virus Nucleocapsid as a Mechanism of IRF-3 Activation[END_REF][START_REF] Colombo | The interaction between the measles virus nucleoprotein and the Interferon Regulator Factor 3 relies on a specific cellular environment[END_REF][START_REF] Watanabe | Peroxiredoxin 1 is required for efficient transcription and replication of measles virus[END_REF] including the major inducible heat shock protein 70 (hsp70) [START_REF] Zhang | Identification and characterization of a regulatory domain on the carboxyl terminus of the measles virus nucleocapsid protein[END_REF][START_REF] Zhang | Hsp72 recognizes a P binding motif in the measles virus N protein C-terminus[END_REF][START_REF] Couturier | High Affinity Binding between Hsp70 and the C-Terminal Domain of the Measles Virus Nucleoprotein Requires an Hsp40 Co-Chaperone[END_REF]. In a previous study, we have shown that progressively shortening the NTAIL fuzzy appendage from residue 401 to residue 481 results in a non-monotonic increase of its binding strength towards both XD and hsp70 [START_REF] Gruet | Fuzzy regions in an intrinsically disordered protein impair protein-protein interactions[END_REF]. This finding might reflect a similar binding mechanism in spite of the notable differences between XD and hsp70 in terms of size, fold and origin. Albeit the major hsp70-binding site has been already mapped to Box2 [START_REF] Zhang | Hsp72 recognizes a P binding motif in the measles virus N protein C-terminus[END_REF][START_REF] Couturier | High Affinity Binding between Hsp70 and the C-Terminal Domain of the Measles Virus Nucleoprotein Requires an Hsp40 Co-Chaperone[END_REF], no structural information is available for the NTAIL/hsp70 complex. It therefore remains to be established whether the MoRE undergoes α-helical folding also upon binding to hsp70 and whether the interaction relies on the same NTAIL residues. Deciphering the molecular mechanisms governing the NTAIL/hsp70 interaction is of broad relevance in light of its well-documented functional impact on viral transcription and replication [START_REF] Zhang | Identification and characterization of a regulatory domain on the carboxyl terminus of the measles virus nucleocapsid protein[END_REF][START_REF] Zhang | Hsp72 recognizes a P binding motif in the measles virus N protein C-terminus[END_REF] and on the innate immune response [START_REF] Kim | hsp70 and a novel axis of type I interferon-dependent antiviral immunity in the measles virus-infected brain[END_REF]. In the present paper, we have addressed these questions using mutational studies combined with a protein complementation assay based on green fluorescent protein (GFP) reconstitution [START_REF] Wilson | Detecting protein-protein interactions with GFPfragment reassembly[END_REF][START_REF] Gruet | Dissecting partner recognition by an intrinsically disordered protein using descriptive random mutagenesis[END_REF].

Results and Discussion

Alanine scanning mutagenesis of the MoRE and effects on XD and hsp70 binding

In order to ascertain whether NTAIL binding to hsp70 is mediated by the same NTAIL residues as binding to XD, we targeted for alanine scanning mutagenesis the MoRE (Figure 1) and assessed the binding properties of the resulting variants towards both XD and hsp70 using a split-GFP reassembly assay. In this assay, wild-type (wt) NTAIL (or its variants) is fused to the N-terminal half of the green fluorescent protein (NGFP) and the binding partner under study is fused to the C-terminal half of GFP (CGFP). Upon co-expression in E. coli, if NTAIL interacts with the binding partner, the two GFP halves are brought to sufficiently close proximity to allow the reassembly of the two GFP halves thus leading to reconstitution of the fluorophore. The resulting fluorescence of the cell culture is measured, which is proportional to the interaction strength [START_REF] Wilson | Detecting protein-protein interactions with GFPfragment reassembly[END_REF]. Since our previous studies showed that the N-terminal fuzzy region of NTAIL spanning residues 401-485 is dispensable for binding, and actually even dampens it [START_REF] Gruet | Fuzzy regions in an intrinsically disordered protein impair protein-protein interactions[END_REF], we performed alanine scanning mutagenesis of the MoRE (aa 486-504) not in the context of full-length NTAIL (aa 401-525) but in the context of the N-terminal truncation variant 471 (aa 471-525) that displays a higher fluorescence (and hence higher interaction strength) with both XD and hsp70 than fulllength NTAIL [START_REF] Gruet | Fuzzy regions in an intrinsically disordered protein impair protein-protein interactions[END_REF]. Each residue of the MoRE was individually replaced with an alanine (or with a glycine when alanine was the wild-type residue) giving rise to 19 single-site variants, all derived from truncation variant 471 (Figure 1). Next we assessed their binding properties towards the two partners. The results obtained with XD and hsp70 are shown in Figures 2A and2B, respectively. Note that except for variant 486, which seems to be more sensitive than the others to bacterial proteolysis, the fluorescence variations reported in Figures 2A and 2B cannot be accounted for by differences in the expression levels of the NTAIL variants (Figure 2C). As shown in Figure 2A, except for variant 499 that exhibits a moderate fluorescence increase, all variants display a lower affinity for XD compared to wt471. The extent of fluorescence reduction varies greatly from one single variant to another. Variants can be roughly divided into three groups. The first one contains single variants bearing a substitution that has either no effect or only a marginal effect on XD binding (variants 487 to 489, 493, and 497). The single substitution borne by the second group significantly reduces but does not abolish the interaction (variants 486, 490, 492, 496, 498, 500, and 502 to 504). The single substitution of the third group of variants almost completely abolishes the interaction (variants 491, 494, 495, and 501), with fluorescence values close to that provided by the negative control NGFP alone. These findings can nicely be accounted for by structural data, in that the most critical MoRE residues are those whose side chains point directly towards the partner in the XD/MoRE complex [START_REF] Kingston | Structural basis for the attachment of a paramyxoviral polymerase to its template[END_REF] (pdb code 1T6O). They also nicely confirm our previous data obtained by random mutagenesis that identified positions 491, 494, 495, 498 and 501 as the most critical [START_REF] Gruet | Dissecting partner recognition by an intrinsically disordered protein using descriptive random mutagenesis[END_REF]. A notable discrepancy however concerns residue 497 whose replacement here with an alanine has only a negligible impact on XD binding, whereas its replacement with a glycine was found to lead to a significant reduction in binding as judged from split-GFP assays [START_REF] Gruet | Dissecting partner recognition by an intrinsically disordered protein using descriptive random mutagenesis[END_REF] and isothermal titration calorimetry (ITC) [START_REF] Bloyet | Modulation of reinitiation of measles virus transcription at intergenic regions by PXD to NTAIL binding strength[END_REF]. There are two possible explanations for these apparently contradictory results. The first one is that alanine promotes helix formation whereas glycine has the opposite effect, as confirmed in the next part of this study. The second one is related to the fact that in our previous study [START_REF] Gruet | Dissecting partner recognition by an intrinsically disordered protein using descriptive random mutagenesis[END_REF] the R497G substitution was used in the context of full-length NTAIL (residues 401 to 525) whereas it is used here in the context of NTAIL truncation variant 471 (residues 471 to 525). Since we have shown that the NTAIL N-terminal fuzzy appendage (residues 401 to 480) dampens the NTAIL/XD interaction [START_REF] Gruet | Fuzzy regions in an intrinsically disordered protein impair protein-protein interactions[END_REF], it is conceivable that the R497G substitution has a stronger effect when used in the context of full-length NTAIL than in the context of NTAIL truncation variant 471. In addition, the finding that the R497A substitution has only a moderate effect on binding may reflect the fact that preserving the Cβ of residue 497 is sufficient to maintain its stabilizing effect on complex formation, although the underlying mechanisms remain elusive. It is indeed difficult to rationalize the stabilizing effect of Ala497 in light of a previous molecular dynamics simulation study that revealed that Arg497 forms a water-mediated hydrogen bond with Tyr480 of XD [START_REF] Bloyet | Modulation of reinitiation of measles virus transcription at intergenic regions by PXD to NTAIL binding strength[END_REF]. The present results, which allow ranking NTAIL residues as 495>494>491>501>498 in terms of their role in binding to XD, are also in quite good agreement with recent mutational results obtained in the context of a Φ-value analysis of the NTAIL/XD binding reaction which identified residues 498, 495 and 494 (in that order) as most critical for binding [START_REF] Bonetti | Analyzing the folding and binding steps of an intrinsically disordered protein by protein engineering[END_REF]. The two studies however disagree as far as the role of residue 491 is concerned, in that the latter study unexpectedly showed that the S491A substitution leads to a KD that is close to that of wtNTAIL. The binding efficiencies of the single-site variants towards hsp70 are shown in Figure 2B. As previously reported [START_REF] Gruet | Fuzzy regions in an intrinsically disordered protein impair protein-protein interactions[END_REF], the interaction strength of variant 471 for hsp70 is about 7 times lower than that for XD (compare the y axis scale in Figures 2A and2B). Beyond this overall difference in interaction strength, the single-site variants behave quite differently towards hsp70. First, the substitutions of all the variants that exhibit a lower affinity compared to wt have only a marginal effect (variants 486, 491, 495, 496, 498, 499). In this respect, all these variants belong to group one described above for XD binding, with no variants falling in groups two or three. This means that, in contrast with the NTAIL/XD interaction, no single substitution on its own is able to abolish the NTAIL/hsp70 interaction. This observation suggests that no single residue is as critical for hsp70 binding as it is for XD binding, i.e. hsp70 is more tolerant of substitutions than XD, which implies that NTAIL interacts with XD and hsp70 using different mechanisms. The higher tolerance of hsp70 vis-à-vis NTAIL substitutions may be tightly connected with the lower affinity of the NTAIL/hsp70 interaction. One possible scenario could be that the NTAIL/XD interaction could rely on a few highly specific interactions, while hsp70 could make use of a larger number (possibly all) of MoRE residues in a less specific and more dynamic manner. Since IDPs have been shown to exhibit also static fuzziness [START_REF] Tompa | Fuzzy complexes: polymorphism and structural disorder in proteinprotein interactions[END_REF], i.e. the MoRE can remain highly dynamic at the surface of the partner (for examples see [START_REF] Communie | Atomic resolution description of the interaction between the nucleoprotein and phosphoprotein of Hendra virus[END_REF][START_REF] Baronti | Dynamics of the intrinsically disordered C-terminal domain of the Nipah virus nucleoprotein and interaction with the X domain of the phosphoprotein as unveiled by NMR spectroscopy[END_REF][START_REF] Mittag | Dynamic equilibrium engagement of a polyvalent ligand with a single-site receptor[END_REF]), it is also conceivable that the MoRE is weakly anchored at the surface of hsp70 through numerous transient and quasi-equivalent contacts. Support of this hypothesis comes from recent studies that showed that hsp70 binds its substrates promiscuously leading to fuzzy chaperone-substrate ensembles endowed with a high conformational heterogeneity [START_REF] Rosenzweig | Promiscuous binding by Hsp70 results in conformational heterogeneity and fuzzy chaperone-substrate ensembles[END_REF][START_REF] Sekhar | Conformational heterogeneity in the Hsp70 chaperone-substrate ensemble identified from analysis of NMR-detected titration data[END_REF]. The second notable difference with respect to XD binding is that the majority of variants (i.e., variants 487, 489, 490, 492 to 494, 497, and 500 to 504) bind hsp70 better than wt471, indicating that the NTAIL sequence can be improved in terms of hsp70 binding. By contrast, in a previous random mutagenesis study, where substitutions were randomly introduced within NTAIL and variants were picked at random in the absence of any selection pressure, we showed that variants bearing substitutions within the MoRE tend to display a lower interaction strength towards XD, thus supporting the conclusion that the sequence of the MoRE is poorly evolvable in terms of XD binding [START_REF] Gruet | Dissecting partner recognition by an intrinsically disordered protein using descriptive random mutagenesis[END_REF]. The implications of this finding is that the sequence of NTAIL has been selected during evolution to achieve optimal binding with XD, reflecting the need for the NTAIL/XD interaction strength to be kept in a precise range so as to ensure dynamic anchoring of the polymerase complex (i.e., L-P) [START_REF] Brunel | Sequence of events in measles virus replication: role of phosphoprotein-nucleocapsid interactions[END_REF] and efficient transcription re-initiation at each intergenic junction of the MeV genome [START_REF] Bloyet | Modulation of reinitiation of measles virus transcription at intergenic regions by PXD to NTAIL binding strength[END_REF]. Thus, it seems that in addition to relying on a different pattern of MoRE residues, the NTAIL/hsp70 and NTAIL/XD interactions do not have the same evolution potential. One possible hypothesis for the higher evolvability of the NTAIL/hsp70 interaction might reside in the fact that the two binding partners have not been subjected to an as tight co-evolution as that of the NTAIL/XD pair due to the multiple functional roles that hsp70 plays in the cell and that are not uniquely related to MeV infection. In addition, taking into account the fact that the intracellular concentration of hsp70 has been estimated at 85 nM in Vero cells following transient hyperthermic treatment [START_REF] Oglesbee | The highly inducible member of the 70 kDa family of heat shock proteins increases canine distemper virus polymerase activity[END_REF] and that the MeV nucleoprotein is the most abundant viral protein, it is conceivable that a high affinity between the two proteins is not requested for the NTAIL/hsp70 interaction to occur and elicit the known effects on viral transcription and replication [START_REF] Zhang | Hsp72 recognizes a P binding motif in the measles virus N protein C-terminus[END_REF][START_REF] Shu | Plasticity in structural and functional interactions between the phosphoprotein and nucleoprotein of measles virus[END_REF] and on the innate immune response [START_REF] Kim | hsp70 and a novel axis of type I interferon-dependent antiviral immunity in the measles virus-infected brain[END_REF].

Impact of varying the α-helicity of the MoRE on XD and hsp70 binding

Since the MoRE was shown to fold into an α-helix upon binding to XD [START_REF] Bourhis | The C-terminal domain of measles virus nucleoprotein belongs to the class of intrinsically disordered proteins that fold upon binding to their physiological partner[END_REF][START_REF] Bourhis | The intrinsically disordered C-terminal domain of the measles virus nucleoprotein interacts with the C-terminal domain of the phosphoprotein via two distinct sites and remains predominantly unfolded[END_REF][START_REF] Johansson | Crystal structure of the measles virus phosphoprotein domain responsible for the induced folding of the C-terminal domain of the nucleoprotein[END_REF][START_REF] Kingston | Structural basis for the attachment of a paramyxoviral polymerase to its template[END_REF][START_REF] Belle | Mapping alphahelical induced folding within the intrinsically disordered C-terminal domain of the measles virus nucleoprotein by site-directed spin-labeling EPR spectroscopy[END_REF][START_REF] Bernard | Interaction between the Cterminal domains of N and P proteins of measles virus investigated by NMR[END_REF][START_REF] Gely | Solution structure of the C-terminal X domain of the measles virus phosphoprotein and interaction with the intrinsically disordered C-terminal domain of the nucleoprotein[END_REF][START_REF] Kavalenka | Conformational analysis of the partially disordered measles virus NTAIL-XD complex by SDSL EPR spectroscopy[END_REF][START_REF] Jensen | Intrinsic disorder in measles virus nucleocapsids[END_REF] and since the results of the alanine scanning mutagenesis reported in Figures 2A and2B suggest that NTAIL binds XD and hsp70 through different mechanisms, we investigated the impact of varying the helicity of the MoRE on XD and hsp70 binding. To that end, we conceived and generated a variant 471 bearing a mutated MoRE in which all the residues shown to be critical for XD binding by the alanine scanning mutagenesis (Figure 2A) were kept unchanged, and all the other residues were replaced with an alanine yielding a variant called "Ala471" (see Table 1). Concomitantly, we also generated another variant, called "Gly471" (Table 1), in which all the MoRE residues that had been swapped with alanine in Ala471 were replaced with a glycine. The rationale for the generation of these two variants was the following. Because of its high alanine content, Ala471 was expected to exhibit an increased α-helicity, whereas Gly471 was expected to be unable to fold into an α-helix even upon XD binding because of its high glycine content and of the well-known α-helix-breaking ability of this residue [START_REF] Scott | Conformational entropy of alanine versus glycine in protein denatured states[END_REF][START_REF] Borcherds | Disorder and residual helicity alter p53-Mdm2 binding affinity and signaling in cells[END_REF][START_REF] Otieno | Probing the role of nascent helicity in p27 function as a cell cycle regulator[END_REF][START_REF] Rogers | Coupled folding and binding of the disordered protein PUMA does not require particular residual structure[END_REF]. These expectations were reinforced by convergent results obtained using disorder predictions (Figure 3A), hydrophobic cluster analysis [START_REF] Callebaut | Deciphering protein sequence information through hydrophobic cluster analysis (HCA): current status and perspectives[END_REF], secondary structure predictions [START_REF] Chandonia | New methods for accurate prediction of protein secondary structure[END_REF] (supplementary Figure S1), and modeling (Figure 3B). Most importantly, they were experimentally confirmed by direct spectroscopic data. Indeed, far-UV circular dichroism (CD) spectroscopy analysis of AlaMoRE and GlyMoRE peptides (each encompassing residues 485-506, and bearing the above described substitutions) confirmed their markedly increased and decreased helicity, respectively, with respect to wtMoRE (Figure 3C). The Ala471 and Gly471 variants were assessed for their ability to bind to XD and hsp70 by split-GFP reassembly. Results are reported in Figure 3D. In this experiment, XD interaction with fulllength NTAIL (aa 401-525) was used as the reference and its fluorescence set to 100%. As already reported [START_REF] Gruet | Fuzzy regions in an intrinsically disordered protein impair protein-protein interactions[END_REF], wt471 (aa 471-525) exhibits a higher affinity for XD than full-length NTAIL (compare 401/XD and wt471/XD in Figure 3D). Notably, when the native MoRE sequence in wt471 was replaced with the sequence of AlaMoRE, the binding was found to increase even further (compare wt471/XD and Ala471/XD in Figure 3D) suggesting that a preformed α-helix leads to an increased NTAIL/XD interaction strength. This finding is in agreement with our previous kinetics studies that showed that the folding of the MoRE is the rate-limiting step of the NTAIL/XD binding reaction [START_REF] Gruet | Fuzzy regions in an intrinsically disordered protein impair protein-protein interactions[END_REF]. By contrast, when the wtMoRE sequence within variant 471 was replaced with the sequence of GlyMoRE, the binding to XD drops down to the negative control level (compare Gly471/XD and Stop/XD in Figure 3D). This latter result definitely indicates that if the MoRE cannot fold into an α-helix, then NTAIL/XD binding is impaired in spite of the presence within the MoRE of all the residues that alanine scanning mutagenesis identified as critical for the interaction. Taken together, the results obtained with the Ala471 and Gly471 variants support a scenario where the lower the helicity of the MoRE the lower the interaction strength. Importantly, results also indicate that XD is able to bind to a pre-formed αhelix arguing for a mixed mechanism involving both conformational selection and induced folding. When the same experiment was performed using hsp70 instead of XD, drastically different results were observed (Figure 3D). As expected, wt471 displays a much lower interaction strength towards hsp70 compared to XD (compare wt471/hsp and wt471/XD in Figure 3D). However, with Ala471 and Gly471, the binding to hsp70 decreases only marginally in both cases (compare wt471/hsp with Ala471/hsp or Gly471/hsp in Figure 3D). The possibility that different protein expression levels could be responsible for the observed differences in fluorescence was checked and ruled out (Figure 3E). Thus, while increasing or decreasing the α-helicity of the MoRE has tremendously opposite effects on XD binding, the effects on hsp70 binding are both small and similar. In an effort to further characterize the effect of helical propensity of NTAIL on its binding properties, in analogy to our previous studies [START_REF] Dosnon | Demonstration of a folding after binding mechanism in the recognition between the measles virus NTAIL and X domains[END_REF][START_REF] Gruet | Fuzzy regions in an intrinsically disordered protein impair protein-protein interactions[END_REF], we resorted to perform temperature-jump kinetic experiments on wtMoRE, AlaMoRE and GlyMoRE peptides. Unfortunately, due to the complexity of these experiments, we could not perform binding studies on hsp70. In fact, the relatively low affinity between hsp70 and NTAIL together with the small fluorescence changes induced by binding prevent an accurate determination of the (un)binding rate constants. The binding of XD to wtMoRE, AlaMoRE and GlyMoRE was measured by incubating a constant concentration of 5 µM XD with varying concentrations of either wtMoRE or AlaMoRE or GlyMoRE (typically ranging from 2 to 50 µM). In analogy to what we described previously, in all cases (un)binding was induced by a rapid discharge of 35 kV on a quartz cell, corresponding to a rapid increase in temperature of 9 K. Supplementary Figure S2 shows three typical fluorescence traces observed in these binding experiments. All curves conformed to a single exponential transition (Supplementary Figure S2 and data not shown). Inspection of kinetic data (Figure 3F) reveals that whilst the behavior of AlaMoRE is very similar to that of wtMoRE (wtMoRE: 𝑘 !"" #$$ = 1460 ± 80 𝑠 %& , 𝑘 !' #$$ = 30 ± 4 𝑠 %& 𝜇𝑀 %& ; AlaMoRE: 𝑘 !"" #$$ = 1400 ± 100 𝑠 %& , 𝑘 !' #$$ = 34 ± 5 𝑠 %& 𝜇𝑀 %& ), there is a detectable destabilization of the complex in the case of GlyMoRE (𝑘 !"" #$$ = 1610 ± 60 𝑠 %& , 𝑘 !' #$$ = 5 ± 2 𝑠 %& 𝜇𝑀 %& ). This conclusion is mirrored by a decrease of the association rate constant, which is represented by the slope of the observed rate constant as a function of reactant concentration. Thus, whilst an increase in helicity of the MoRE compared to the wt sequence does not appear to contribute to a stabilization of the complex in vitro, it is evident that a destabilization of the secondary structure of the MoRE corresponds to a weaker binding to XD. A possible reason for the discrepancy between split-GFP and kinetics data, as far as results with AlaMoRE are concerned, may lie in the differences in the experimental set up between the two approaches.

In conclusion, the results reported in Figures 3D andF indicate that α-helical folding is a strict requirement for binding to XD, whereas favoring of disfavoring it only partially reduces the binding to hsp70. These results reinforce the idea that XD and hsp70 bind NTAIL through different molecular mechanisms.

Mapping the reciprocal binding regions in the NTAIL/hsp70 interaction

The results reported in Figures 2 and3 suggest that the NTAIL/hsp70 interaction does not rely on few specific MoRE residues and is also relatively insensitive to the folding state of the MoRE. These observations raised the question as to which are the minimal requirements for the NTAIL/hsp70 interaction to occur, an interaction that has been much less investigated compared to the NTAIL/XD interaction and for which molecular and structural data are still lacking. In order to fill this gap and to contribute to a better understanding of this interaction, we first used a deletion approach to map the reciprocal binding sites and then targeted for site-directed mutagenesis the hsp70-binding site of NTAIL. Hsp70 consists of three domains: a nucleotide binding domain (NBD, 1-384), a peptide binding domain (PBD, 381-543) and a lid (543-641) (see [START_REF] Couturier | High Affinity Binding between Hsp70 and the C-Terminal Domain of the Measles Virus Nucleoprotein Requires an Hsp40 Co-Chaperone[END_REF] and references therein cited). To map the minimal region involved in NTAIL binding, we generated different hsp70 constructs containing one or more of these domains (Figure 4A, lower panel) and tested their ability to interact with wt471 or with its Ala471 and Gly471 derivatives. Results are shown in Figure 4A, upper panel. The three NTAIL variants provided very similar interaction profiles, with the highest interaction strength being obtained with the hsp70 construct devoid of the C-terminal lid (NBD-PBD, 1-543), and the lowest interaction strength being observed with the PBD and Lid domains. The finding that hsp70 deletion variant devoid of the C-terminal lid (NBD-PBD, 1-543) exhibits the highest interaction strength is in agreement with our previous studies where surface plasmon resonance experiments yielded a two-fold increase in the KD (68 versus 34 μM) of the NTAILbinding reaction when full-length hsp70 was used instead of NBD-PBD [START_REF] Couturier | High Affinity Binding between Hsp70 and the C-Terminal Domain of the Measles Virus Nucleoprotein Requires an Hsp40 Co-Chaperone[END_REF]. They are also in agreement with previous reports by the group of Lila Gierasch that showed that the hsp70 Cterminal region is dispensable for binding of a peptide substrate, inter-domain allostery and cochaperon interaction [START_REF] Smock | Conserved, disordered C terminus of DnaK enhances cellular survival upon stress and DnaK in vitro chaperone activity[END_REF].

In the case of wt471, the second best binder is full-length hsp70 (1-641). PBD-Lid and NBD each show a significant binding to wt471, although lower than full-length hsp70. However, when the Ala471 and Gly471 variants were used instead of wt471, the binding of NBD proved to be slightly more efficient than full-length hsp70. Since the NBD and the PBD-lid together cover the full-length hsp70 these results suggest that wt471 might form weak contacts throughout more than one hsp70 domains. By contrast, the PBD and the Lid on their own are unable to bind to wt471 and its two derivatives. However, we cannot exclude the possibility that results obtained with the PBD and the Lid alone could arise from a low solubility/expression of these constructs, but due to the lack of affinity tag, the various hsp70 constructs fused to CGFP could not be purified as was the case of NGFP fusions. To better visualize/appreciate the binding behavior of full-length hsp70 and of its lid-free derivative, their binding data, as obtained with wt471 and with the Ala471 and Gly471 variants, were compared (Figure 4B). This comparison confirms the slightly reduced binding of full-length hsp70 to Ala471 and Gly471 variants with respect to wt471 as already observed in the experiment reported in Figure 3D. By contrast, the lid-free hsp70 binds all the three NTAIL variants in a similar manner. In conclusion, results indicate that removing the lid not only increases binding towards NTAIL but also renders it less sensitive to substitutions within the MoRE that have profound effects on XD binding.

To further dissect the NTAIL/hsp70 interaction, we generated a series of NTAIL truncation variants (Figure 5A) and assessed their binding properties towards hsp70 by split-GFP reassembly. The rationale for doing that was to map the minimal binding region and to assess the contribution of NTAIL Box3 (aa 517-525) to binding. Indeed, previous studies showed that although hsp70 binds with a weak affinity a Box3 peptide [START_REF] Zhang | Identification and characterization of a regulatory domain on the carboxyl terminus of the measles virus nucleocapsid protein[END_REF], the presence of this motif stabilizes NTAIL/hsp70 complexes [START_REF] Couturier | High Affinity Binding between Hsp70 and the C-Terminal Domain of the Measles Virus Nucleoprotein Requires an Hsp40 Co-Chaperone[END_REF] that are otherwise mediated by high affinity binding to Box2 [START_REF] Zhang | Hsp72 recognizes a P binding motif in the measles virus N protein C-terminus[END_REF]. As shown in Figure 5B, the NTAIL/hsp70 interaction strength increases with increasing N-and C-terminal shortening of NTAIL, with the wtMoRE on its own providing the highest binding. Note that, once again, the increase in fluorescence is not accounted for by an increased expression level (Figure 5C). These results therefore indicate that as in the case of XD, the MoRE is the only NTAIL region responsible for binding to full-length hsp70, and that Box3 behaves as a weak binding dampener.

Although the NTAIL/hsp70 interaction relies only on the MoRE, the latter is seemingly capable of binding, albeit weakly, to both NBD and PBD-lid, thus raising the question as to how such a short peptide can stretch over to cover the whole hsp70. One possibility, that remains to be assessed, is that NTAIL binds to hsp70 with a 2:1 stoichiometry. That hsp70 may form substoichiometric chaperone-substrate complexes is supported by recent studies that showed that DnaK binds to hTRF-1 with a 1:3 stoichiometry [START_REF] Sekhar | Conformational heterogeneity in the Hsp70 chaperone-substrate ensemble identified from analysis of NMR-detected titration data[END_REF]. Alternatively, it is conceivable that the MoRE binds to multiple hsp70 sites in a dynamic and promiscuous manner, as already observed in the case of hsp33 (Dana Reichmann, personal communication).

Rational design of an NTAIL variant with enhanced interaction strength towards hsp70

Since alanine scanning mutagenesis showed that several single substitutions lead to an increase in NTAIL/hsp70 binding, we reasoned that this could be the basis for devising an hsp70 "super binder" (hsb). The latter (see Table 1) was obtained by collectively introducing all the substitutions that individually increase the binding to hsp70 in the context of 471 (hsb471).

Swapping the sequence of wtMoRE with that of hsb is associated with a slightly lower disorder probability (Figure 5D) reflecting the higher alanine content of the latter (Table 1). In line with expectations, this rationally designed 471 variant displays a much higher binding (2.35 times) towards full-length hsp70 than wt471 in a split-GFP reassembly assay (compare wt471 and hsb471 in Figure 5B). Since results reported in Figure 5B definitely identified the MoRE as the only interacting unit between NTAIL and hsp70, and since substitution of wt471 with hsb471 resulted in a significant increase in NTAIL/hsp70 binding, we reasoned that hsbMoRE should be the strongest hsp70 binder. This hypothesis was experimentally confirmed: hsbMoRE indeed binds hsp70 almost three times better than wtMoRE (Figure 5B). In other words, the difference between wtMoRE and hsbMoRE in terms of binding to hsp70 is higher when the MoRE is used alone than when it is used in the context of 471 (2.35 times, see above). In the results reported in Figure 4, hsp70 truncation variant NBD-PBD (hsp1-543) proved to be the best NTAIL binder. Results reported in Figure 5E show that when wtMoRE was substituted with hsbMoRE in the context of 471, the binding to hsp1-543 increases (compare wt471 and hsb471) and reaches maximal level when hsbMoRE was used alone.

The finding that replacement of as much as 70% (13 residues out of 19, Table 1) of the sequence of the wtMoRE with alanine or glycine residues results in an almost 3-fold increase in binding to hsp70 (Figure 5B), raises the question as to how can hsp70 binding to NTAIL be specific of the MoRE while being relatively independent of the sequence of the latter. It is conceivable that hsp70 recognizes not a precise amino acid sequence or motif but rather a pattern of few residues with specific chemical features with no strict positional conservation. Which could be these chemical features? The sequence of hsbMoRE is highly enriched in Ala, Gly and Leu residues (in this order) and has a null net charge (Table 1). Of note, these features were also found to favor binding of peptides to hsp33, an ATP-independent, redox-regulated chaperone [START_REF] Reichmann | Order out of disorder: working cycle of an intrinsically unfolded chaperone[END_REF]. This study also identified Asp/Glu residues as strongly disfavoring binding. Noteworthy, in hsbMoRE the two naturally occurring Asp residues were removed, thereby providing a conceptual framework to explain increased binding by hsbMoRE. Interestingly, the study by Reichmann and co-workers also unveiled that increased hydrophobicity is correlated to increased binding strength [START_REF] Reichmann | Order out of disorder: working cycle of an intrinsically unfolded chaperone[END_REF]. Although previous data identified exposed patches of hydrophobic residues in native proteins [START_REF] De Crouy-Chanel | Interaction of DnaK with native proteins and membrane proteins correlates with their accessible hydrophobicity[END_REF] and hydrophobic residues in peptides as targets of hsp70 binding [START_REF] Clerico | How hsp70 molecular machines interact with their substrates to mediate diverse physiological functions[END_REF], it should be emphasized that hydropathy per se is not sufficient to explain increased binding strength by hsbMoRE. Indeed, Ala471 is a weaker hsp70 binder than wt471 (Figure 3D) and wt471 is a weaker hsp70 binder than hsb471 (Figure 5B) although the hydropathy of AlaMoRE (1.6), as obtained using the Kyte-Doolittle scale and the expasy server (https://www.expasy.org/), is higher than that of wtMoRE (-0.24) and the hydropathy of hsbMoRE (0.96) is higher than that of wtMoRE. Finally, since the generation of hsbMoRE implied introducing all the Ala/Gly substitutions that individually led to increased binding strength while keeping unchanged all MoRE positions that once substituted led to diminished binding, it follows that the latter can be regarded as determinants of NTAIL/hsp70 binding. Among these residues, a stretch of three Leu (495, 496 and 498) is reminiscent of the sequence of the NRLLLTG heptapeptide co-crystallized with DnaK, the bacterial counterpart of hsp70 [START_REF] Zhu | Structural analysis of substrate binding by the molecular chaperone DnaK[END_REF].

Conclusions

Using alanine scanning mutagenesis, truncation variants and rational site-directed mutagenesis in combination with a protein complementation assay based on split-GFP reassembly, we have shed light on the mechanisms by which NTAIL recognizes XD and hsp70, two structured partners differing in size and fold. Results indicate that although NTAIL uses the same binding region (i.e., the MoRE) to bind the two partners, the binding mechanisms are not the same. Hsp70 is much more tolerant of NTAIL substitutions than XD, with this higher tolerance likely coming at the cost of a lower affinity compared to the NTAIL/XD pair. The NTAIL/hsp70 interaction is more evolvable compared to the NTAIL/XD interaction, i.e., the majority of substitutions lead to increased interaction strength. Notably, while an increased and a decreased α-helicity of the MoRE lead to enhanced and reduced interaction strength with XD, respectively, the extent of helicity of the free form of the MoRE has seemingly a negligible impact on hsp70 binding, suggesting that the MoRE does not adopt an α-helical conformation once bound to hsp70. Shortening hsp70 by removing the lid results in increased binding towards NTAIL, as does the shortening of the latter to encompass only the MoRE. Finally, NTAIL/hsp70 binding can be further increased by introducing within the MoRE all the substitutions that individually result in increased interaction strength. The enhanced interaction strength of hsbMoRE towards hsp70 NBD-PBD may be instrumental for obtaining crystals of a complex in view of structure determination by X-ray crystallography.

The availability of the structure of the complex will shed light on the binding mode of this mutated MoRE to hsp70. In particular, the high-resolution structure of the complex will enable ascertaining whether the NBD and the PBD bind each a MoRE molecule, and will unveil the actual conformation that the substrate adopts once bound to hsp70. Works are in progress in our laboratory to reach this goal. Finally, and from a more general perspective, the present results contribute to shed light on the molecular mechanisms by which IDPs recognize their partners. Indeed, in spite of the increasing interest being paid to IDPs, the molecular features that encode their binding efficiency remain elusive, and the variety of binding modes and mechanisms exhibited by IDPs (see [START_REF] Habchi | Introducing Protein Intrinsic Disorder[END_REF] and references therein cited) suggests that there is no general mechanism for their binding-induced folding. In addition, Here, by showing that the conformation sampled by the free form of the MoRE of NTAIL does not necessarily commit NTAIL to adopt an α-helical conformation in the bound form, we bring an additional "brick in the wall", i.e. we provide an additional example of partner-mediated polymorphism and of the relative insensitiveness of the bound structure to the pre-recognition state. These results are in line with recent findings that revealed that NTAIL folds as anti-parallel β sheet at the air/water interface [START_REF] Bénarouche | Interfacial properties of NTAIL, an intrinsically disordered protein[END_REF] thereby further underscoring the extreme polymorphism of this IDP.

Materials and Methods

Alanine scanning mutagenesis and generation of NGFP-NTAIL constructs

Two complementary mutagenic primers were used in two separate PCR amplification experiments using the wtNTAIL471 construct in the pNGG vector [START_REF] Gruet | Dissecting partner recognition by an intrinsically disordered protein using descriptive random mutagenesis[END_REF][START_REF] Gruet | One-step generation of error-prone PCR libraries using Gateway(R) technology[END_REF] as template. The first one used attB1 and the reverse mutagenic primer, and the second one used attB2 and the forward mutagenic primer. After DpnI treatment to remove methylated parental DNA, the two products of the first PCR amplification were mixed and then used as overlapping megaprimers for a second PCR amplification with primers attL1a and attL2a [START_REF] Gruet | One-step generation of error-prone PCR libraries using Gateway(R) technology[END_REF]. The second PCR product was directly used in an LR reaction (Invitrogen) with expression vector pNGG as previously described [START_REF] Gruet | One-step generation of error-prone PCR libraries using Gateway(R) technology[END_REF]. The mutagenic primers used to generate all the single-site alanine variants are listed in supplementary Table S1.

Full-length wtNTAIL401 (aa 401-525), wtNTAIL471 (aa 471-525) and wtNTAIL481 (aa 481-525) have been already described [START_REF] Gruet | Fuzzy regions in an intrinsically disordered protein impair protein-protein interactions[END_REF]. Truncation variant wtNTAIL401ΔBox3 (aa 401-516) was obtained by PCR amplification using the wtNTAIL401 construct in pNGG as template and primers attB1 and D3-attB2. After DpnI treatment, the PCR product was used as template for a second PCR amplification using attL1a and attL2a primers. The second PCR product was directly used as template in an LR reaction with expression vector pNGG as described above.

Truncation variant wtMoRE (aa 485-506) coding sequence was obtained by PCR amplification using primers B1MoRE , B2MoRE and the wtNTAIL401 construct in pNGG as template. After DpnI treatment, the PCR product was used as template in a second PCR amplification using attL1a and attL2a primers. The second PCR product was directly used in an LR reaction with expression vector pNGG as above. The hsb471 variant coding sequence (aa 471-525 with mutated MoRE) was obtained by PCR amplification using the wtNTAIL471 construct in pNGG as template and primers hsbNTailF1 and hsbNTailR1 in two separate PCR amplifications. The first one used primers attB1 and hsbNTailR1, and the second one used attB2 and hsbNTailF1. After DpnI treatment, the two products of the first PCR amplification were mixed and then used as overlapping megaprimers for a second PCR amplification with primers attL1a and attL2a. The second PCR product was directly used in an LR reaction with expression vector pNGG as described above.

The hsbMoRE variant coding sequence was obtained by PCR amplification using the wt471 in pNGG as template. In the first PCR amplification, the two halves of the coding sequence were amplified separately using primers 4hsb5a and 4hsb5b in one tube, and primers 4hsb5c and 4hsb5d in a second tube. After DpnI treatment, a third PCR amplification was performed using 1 µl of each of the first two tubes and primers attL1a and attL2a. The PCR product of the third PCR amplification was then directly used in an LR reaction with expression vector pNGG as described above. The Ala471 and Gly471 NTAIL variants coding sequences were obtained as that of hsbMoRE471 except that primer pairs AlaNTailR1 and AlaNTailF1, and GlyNTailR1 and GlyNTailF1 were respectively used in the first PCR experiment.

The sequences of all the primers used to generate the NGFP-NTAIL constructs are shown in Supplementary Table S2.

CaCl2 competent E. coli TAM1 cells (Active Motif) were transformed with the LR mixtures, and selected on ampicillin-containing plates. All the sequences were checked by sequencing (GATC Biotech) and found to conform to expectations.

Generation of hsp70-CGFP constructs

Hsp70-CGFP constructs were PCR amplified using the full-length hsp70 coding sequence as template [START_REF] Couturier | High Affinity Binding between Hsp70 and the C-Terminal Domain of the Measles Virus Nucleoprotein Requires an Hsp40 Co-Chaperone[END_REF] and the following primers: Hsp70BspHI and 384AatII for NBD, Hsp70BspHI and Hsp70_543 for NBD-PBD, Hsp70BspHI and Hsp70AatII for NBD-PBD-lid, Hsp70_384 and Hsp70_543 for PBD, Hsp70_384 and Hsp70AatII for PBD-lid, BspHI543 and Hsp70AatII for the lid. The sequences of all the primers used to generate the hsp70-CGFP constructs are shown in Supplementary Table S2.

After DpnI treatment, the PCR product was purified and then digested with BspHI and AatII. After purification, digested PCR products were ligated to BspHI and AatII digested pMRBAD [START_REF] Wilson | Detecting protein-protein interactions with GFPfragment reassembly[END_REF]. CaCl2 competent E. coli TAM1 cells (Active Motif) were transformed with the ligation mixture, and selected on kanamycin plates. All the sequences were checked by sequencing (GATC Biotech) and found to conform to expectations.

The XD-CGFP construct and the split-GFP reassembly assay have been described already [START_REF] Gruet | Fuzzy regions in an intrinsically disordered protein impair protein-protein interactions[END_REF].

Purification of NGFP fusions under denaturing conditions

The expression level of the NGFP constructs was assessed as follows. After fluorescence and OD600 has been measured, the whole volume of triplicate cultures (12 ml) was spun for 5' at 4000g in a single well of a 24-wells deep-well. Cell pellets were re-suspended in 1 ml of 50 mM Tris/HCl pH 8, 0.3 M NaCl, 10 mM imidazole, 0.1% Triton X100, 5 mM phenylmethylsulfonyl fluoride (PMSF) and 0.25 mg/ml Lysozyme, and frozen. After thawing, cell lysates were supplemented with 20 µg/ml DNAse I and 20 mM MgSO4 and incubated at 37°C for 30' under shaking. Urea (2 g per well) was added and allowed to dissolve and to denature the proteins contained in the lysate by an additional 30' incubation at 37°C under shaking. After spinning for 5'at 4000g, supernatants were supplemented with 100 µl of a 50% suspension of IMAC sepharose high performance beads (GE Healthcare). His-tagged NGFP fusions were allowed to bind to the beads for 30' at room temperature on a rotating wheel. After washing with 5 x 1 ml of 50 mM Tris/HCl pH 8, 0.3 M NaCl, 50 mM imidazole, beads were re-suspended in 50 µl of reducing SDS-PAGE loading buffer and the proteins contained in 10 µl of this suspension were resolved by SDS-PAGE.

Modeling of the MoREs

The structural models of the free form of the MoREs were obtained using the PEP-FOLD3 server (http://mobyle.rpbs.univ-paris-diderot.fr/cgi-bin/portal.py#forms::PEP-FOLD3) [START_REF] Lamiable | PEP-FOLD3: faster de novo structure prediction for linear peptides in solution and in complex[END_REF].

Synthetic peptides and far-UV circular dichroism (CD) measurements

The synthetic peptide mimicking wtMoRE has been already described [START_REF] Gruet | Fuzzy regions in an intrinsically disordered protein impair protein-protein interactions[END_REF]. Peptides mimicking the AlaMoRE and the GlyMoRE were purchased from JPT (Berlin, Germany). All the MoRE peptides were designed to encompass residues 485-506 of NTAIL, and possess an additional nonnative tyrosine at the C-terminus to allow their concentration to be inferred from the absorbance at 280 nm.

The CD spectra of the MoRE peptides were measured using a Jasco 810 dichrograph, flushed with N2 and equipped with a Peltier thermoregulation system. One-mm thick quartz cuvettes were used. Peptide concentrations were 0.1 mg mL -1 . Peptides were dissolved in 10 mM sodium phosphate pH 7. Spectra were measured between 190 and 260 nm at 20°C. The scanning speed was 50 nm/min, with data pitch of 0.2 nm. Each spectrum is the average of three acquisitions. The spectrum of buffer was subtracted from the protein spectrum. Spectra were smoothed using the ''means-movement'' smoothing procedure implemented in the Spectra Manager package.

The BESTSEL website (http://bestsel.elte.hu/) [START_REF] Micsonai | Accurate secondary structure prediction and fold recognition for circular dichroism spectroscopy[END_REF] was used to analyze the experimental data in the 190-250 nm range.

Temperature-jump fluorescence spectroscopy

Kinetic binding experiments were performed by using a Hi-Tech PTJ-64 capacitordischarge T-jump apparatus (Hi-Tech, Salisbury, UK). Temperature was rapidly changed with a jump-size of 9 °C, from 16 °C to 25 °C. Usually 10-20 individual traces were averaged. The fluorescence change of N-acetyl-tryptophanamide (NATA) was used in control measurements. Degassed and filtered samples were slowly pumped through the 0.5 x 2 mm quartz flow cell before data acquisition. The excitation wavelength was 296 nm and the fluorescence emission was measured using a 320 nm cut-off glass filter.

Experiments made use of a previously reported XD variant Y480W with a C-terminal hexahistidine tag [START_REF] Dosnon | Demonstration of a folding after binding mechanism in the recognition between the measles virus NTAIL and X domains[END_REF][START_REF] Gruet | Fuzzy regions in an intrinsically disordered protein impair protein-protein interactions[END_REF] and the three MoRE-mimicking peptides described above. The XD variant was purified as already described [START_REF] Dosnon | Demonstration of a folding after binding mechanism in the recognition between the measles virus NTAIL and X domains[END_REF].

The experiments were carried out by mixing a constant concentration of Y480W XD (5 µM) with excess concentrations of either wtMoRE, or AlaMoRE or GlyMoRE peptide. The buffer used was 10 mM sodium phosphate and 150 mM NaCl at pH 7.0. ---------------------------------------------------------------------hsp70 ------------------------------------------------------------------------------- S1 andS2).

Supplementary Information

FiguresFigure 1 .Figure 2 .Fluo

 12 Figures

*Figure 3 .

 3 Figure 3. Binding efficiency towards XD and hsp70 of NTAIL variants mutated within the MoRE as assessed by split-GFP reassembly. (A) Disorder probability, as obtained using IUPred [79] of wt471, Ala471 and Gly471. The x axis is residue number of 471. The position of the MoRE within the sequence is shown. (B) Structural models, as obtained using PEP-FOLD3, of wtMoRE, AlaMoRe and GlyMoRE peptides. (C) Far-UV CD spectra of the MoRE peptides. The inset shows the secondary structure content as derived using BESTSEL. (D) Normalized fluorescence of NTAIL variants fused to NGFP upon co-expression with XD-CGFP (XD) and hsp70-CGFP (hsp). Stop, NGFP alone; 401, full-length wtNTAIL (aa 401-525); wt471, Ala471 and Gly471, 471 variants (aa 471-525) bearing the wt sequence or multiple alanine or glycine substitutions, respectively, within the MoRE (see Table 1 for details). The black asterisk denotes a statistically significant difference (T Student's test, with p<0.01) with respect to wt471. The grey asterisk denotes a statistically significant difference (T Student's test, with p<0.01) with respect to 401. (E) SDS-PAGE analysis of the total expression of NGFP-NTAIL variants. XD and hsp indicate the CGFP fusions co-expressed, while labels on top of each lane indicate the expressed NGP-NTAIL variant (W, wt471; A, Ala471; G, Gly471). The band corresponding to NGFP alone (resulting from expression of the Stop construct) is indicated by a white star. M, molecular weight markers: 212, 158, 116, 97.2, 66.4, 55.6, 42.7, 34.6, 27, 20, 14.3 kDa. Underlined values correspond to most intense bands in the gel. Expected molecular masses of the GFP fusion proteins are: 20.4 kDa (Stop), 34 kDa (401), 25.9 kDa (wt471, Ala471, Gly471). (F) Pseudo-first-order kinetics of the binding between Y480W XD and wtMoRE, AlaMoRE and GlyMoRE peptides.

Figure 4 .Figure 5 .

 45 Figure 4. Binding efficiency of hsp70 truncation variants as assessed by split-GFP reassembly. (A) Upper panel: normalized fluorescence of hsp70 constructs schematically represented in the lower panel, fused to CGFP, as obtained upon co-expression with wt471, Ala471 and Gly471 all fused to NGFP. On the y axis are shown the fluorescence of the culture normalized as a function of culture biomass (OD600). Standard deviations of triplicate experiments are indicated for each histogram. Lower panel: Scheme of the various hsp70 constructs used in panel A. NBD, nucleotide binding domain. PBD, peptide binding domain.Numbers on the left indicate the first and last residue of the construct with respect to full-length hsp70 (aa 1-641). (B) Results obtained with full-length hsp70 (aa 1-641) or with its lid-free deletion variant (aa 1-543). The asterisk denotes a statistically significant difference (T Student's test, with p<0.01) with respect to 471. NS: non statistically significant with respect to wt471.

D

  Two supplementary figures (FiguresS1 and S2) and two supplementary tables (Tables

Acknowledgements

This work was supported by the CNRS. It was also partly supported by the European program H2020 under the EVAg Research Infrastructure (grant agreement 653316). FT is a recipient of a PhD fellowship from the Italo-French University.


Louis Moresi
email: l.moresi@ned.dem.csiro.au

Hans M Ühlhaus

Fr Éd Éric Dufour

An overview of numerical methods for Earth simulations

Geological simulation problems are distinct from engineering problems in having a strongly evolving geometry which is often developed through non-linear interaction between structure and rheology. Engineering simulation codes are therefore often, by design, unsuitable for geological applications. We summarize the important features of a small number of numerical methods which have been developed with geological and geotechnical simulations in mind, summarize the advantages and disadvantages of some of these methods and introduce the sorts of problems for which each method is best suited.

Introduction

A long-term goal of geological modeling is to achieve the degree of simulation capability currently enjoyed by the engineering community. The routine ability to recreate the evolution of geological structures during deformation and simultaneously compute stresses, temperatures, fluid flow vectors and chemical evolution would greatly enhance our ability to understand the Earth. A reliable forward modeling capability is also a prerequisite to any attempt to develop an inverse method for structural geology. Many questions in geology are formulated in terms of inverse analysis ("what produced this final state ?"). Boschetti & Moresi consider this issue explicitly in another paper at this conference. Unlike most engineering simulations, however, the geometry of the geological model is a result of the continuously evolving non-linear interaction of the structure and the rheology at extremely large strains. Initially the geometry might be relatively simple, e.g. flat layering, but during the course of the simulation very intricate patterns will develop and need to be accurately resolved by the numerical method. In many cases the complexity reflected in the geometry results from post-failure deformation of the material. To make matters worse, the rheological laws which govern particular geological materials at a given scale are often poorly determined -this is partly due to the difficulty in measuring behaviour at realistic strain rates, and partly due to the microstructural complexity of the materials which make up a given rock suite. The fact that the materials also undergo metamorphic transformations and phase changes during the evolution of a geological structure is another complicating factor which makes simulation difficult. As an example, on a global scale, the continents drift as an integral part of the surface thermal boundary layer of the convecting mantle. They have retained a distinct identity within the mantle flow for billions of years while developing a strong physical and chemical fabric along the way. From a modeling point of view, it is necessary to consider the fluid convection of the mantle and the history-dependent viscoelastic/brittle behaviour of the continental crust as a single coupled system. At the same time, the precise structure and composition of the deep continental crust is not well known. The requirements for a geological simulation code are therefore an ability to track boundaries and interfaces through extremely large deformation, including fluid convection, of non-linear history dependent materials. The wide range of physical and temporal scales, and the many coupled physical processes also impose a need for computational efficiency. The code should also be very flexible in the rheological laws which it can treat. Many different numerical methods have been devised for mechanical simulations of this kind. Some derive from standard engineering methods, while others were developed to handle specific problems in the physical sciences. We provide an introduction to a number of these methods in order to illustrate the difficulties involved in creating convincing, realistic simulations. By way of example, we demonstrate how large deformation viscoelasticity is handled in our own ELLIPSIS Lagrangian Integration Point code.

Modelling techniques

The key to all these problems is to find a way to deal efficiently with the most general case: finite strain viscoelasticity with strongly history-dependent material behaviour. This is a specialized problem: most engineering codes are optimized to study the modest strains which accumulate prior to failure. Nevertheless, an enormous amount of research has been done in developing numerical methods suited for large strain problems in different application areas.

The principal difficulty in finite strain modelling is the tendency for any computational or logical connecting mesh between material points to become arbitrarily tangled during the simulation. For example, the mesh in Figure 1 shows how relatively modest deformation in the global sense rapidly becomes tangled in regions where strain is concentrated.

Figure 1: This figure from a simulation by the Geodynamics modeling group at Dalhousie University ( shows the deformation of an initially orthogonal mesh during a transient simulation of convection in a fluid. Incedentally, the mesh in this example is used to track strain, not compute velocity vectors.

Broadly speaking there have been two approaches to dealing with this problem. One approach is to retain the mesh, but to reorganize it periodically and, perhaps, modify the method to cope with a less-than-optimal distribution of discretization points, and the other is to dispense with the mesh entirely for computing mechanical behaviour and derive a formulation where the material points interact with each other directly. The approach used in ELLIPSIS is distinct from these methods in relaxing the requirement that material points and computational points need to be the same.

Large deformation mesh-based methods

Mesh based Lagrangian methods are not particularly well suited to very large strain fluid flow applications as the mesh is subject to tangling and considerable effort is required to regrid or reconnect the mesh to prevent the computation of derivatives and shape functions from becoming inaccurate. The Arbitrary Lagrangian Eulerian method [START_REF] Huerta | Viscous flow with large free surface motion[END_REF]) avoids mesh tangling by allowing computational points to move independently from the underlying material. This strategy makes it possible to prevent mesh points from approaching one another too closely, while still preserving important interface details. However, additional advection terms are required to handle transport of quantities relative to the mesh and these terms are dispersive. The Dynamical Lagrangian Remeshing method (Braun and Sambridge 1994) reconnects all mesh points after any deformation to ensure the mesh always remains optimally configured (in DLR this means the mesh is always a Delaunay triangulation). Stress histories can be computed at integration points and therefore need to be interpolated when new element connections occur. The extension to 3D is non-trivial and the method is limited to linear triangles/tetrahedra. An alternative approach, the Natural Element Method (NEM, Braun and Sambridge 1995) uses a natural-neighbour interpolation scheme over a Delaunay triangulation to develop finite-element-like shape functions associated with each computational, material point. The Natural Element Method (NEM) does not suffer from the same difficulties as FEM when elements are highly distorted and, although it uses a triangulated mesh, has the advantage that the interpolation functions are continuously differentiable away from the computational points. NEM shares with DLR the difficulty associated with preserving tensor information at integration points after reconnection of the mesh. [START_REF] Braun | A numerical nethod for solving partial differential equations on highly irregular evolving grids[END_REF] point out that the use of a standard gaussian integration scheme in NEM is not exact for natural neighbour interpolation functions. Another complication is that the shape functions overlap element boundaries, and therefore descriptions of flux boundary conditions may not be accurate.

Meshless methods

Meshless methods include, among others, the Discrete Element Method (DEM, [START_REF] Cundall | A Discrete Numerical Model for Granular Assemblies[END_REF], Smoothed Particles Hydrodynamics (SPH, Monaghan 1992), Element Free Galerkin (EFG, [START_REF] Belytschko | Element-free galerkin methods[END_REF], and the Point Integration Method (PIM, [START_REF] Liu | A point interpolation method for two-dimensional solids[END_REF].

In the DEM, computational points are associated with a finite size and shape. Interaction between points (particles) occurs only where they are in contact -according tothe specified interaction rules. This method is very well suited to modeling of fracture since particle interactions can take the form of breakable bonds. The interaction of a particle with it's neighbours varies with angle and time, since it is a function of the neighbours' coordinates,and the interaction history. SPH and EFG are continuum methods in which the material points are associated with a radial basis function which can be used as an interpolant for field variables, and to develop an expansion for PDEs. The basis functions overlap other particles, making it more complicated to apply constraints to values at individual material points but allowing particles to pass arbitrarily close to one another during simulations of fluid motion with stagnation points. PIM is also a continuum method. In PIM, global basis functions are defined which can be combined to produce nonoverlapping shape functions. The method therefore eliminates one of the principal obstacles in efficient formulation of meshless solvers. However, very little in the way of practical applications for the method has been published to date so the promise of the method is largely unexplored. SPH, EFG and PIM are formulated in such a way that implicit solution methods or explicit methods can be used.

Figure 2: A conceptual comparison of meshless methods, the standard finite element method, and the Lagrangian Integration Point finite element method. The latter differs in having a set of computational points and a distinct set of material points. In most other methods the two sets coincide, or one is neglected completely.

Lagrangian integration point Finite Element methods

The method used in ELLIPSIS is a hybrid mesh / particle code --the idea being to retain the generality and robustness of mesh-based FEM, and capture the geometrical flexibility of a fully-Lagrangian set of particles for tracking material deformation. The computational points are a set of nodes fixed in space, connected by a mesh of finite elements. An independent set of material points which carry material properties and the solution history is embedded in the mesh. Since the material and computational points have been formally separated, a strategy for coupling the two representations is needed. The usual finite element interpolation of nodal point values to element interiors is used to update the locations of particles and history variables. The particle properties are coupled to the computational mesh through a non-standard element quadrature in which the particles which happen to be in a given element are used as integration points. Boundary conditions constrain unknowns and therefore belong to the mesh ... This formulation is very close to the material point method developed by Sulsky, Schreyer and their coworkers (e.g. [START_REF] Sulsky | Axisymmetric form of the material point method with applications to upsetting and Taylor impact problems[END_REF] who use an Eulerian mesh with Lagrangian particles. Their formulation was derived from problems where momentum dominates: impacts between elastoviscoplastic materials, suspensions in fast-moving fluids, and fast granular flows. Considerable modifications are needed to model the extreme deformations of convecting fluids with history dependent properties.

Standard Finite Element Analysis v. ELLIPSIS

Simply put, the algorithm used in ELLIPSIS is identical to the standard FEM but with a modified integration scheme. There are, however, a number of practical details involved in building such a scheme. Full implementation details are given in [START_REF] Moresi | Mantle convection modeling with viscoelastic/brittle lithosphere: Numerical methodology and plate tectonic modeling[END_REF]Muhlhaus et al (2001), the salient ingredients are summarized here.

• A scheme for updating the locations of the integration points (material points) based on the velocity solution at the computational points.

• The formulation of a weak form based on integration-point-derived material property matrices, and the storage of constitutive relationships (and the required variables and history) on the integration points.

• The ability to map arbitrary locations within distorted elements to a master-element coordinate scheme. This is the inverse mapping of points within the element onto the regular domain in which element integrals are computed. It is required because the integration scheme is derived "on the fly" rather than being optimized in advance to specific coordinates in the master-element.

• A mechanism for splitting and merging integration points based on strain history. This is required to prevent flow near stagnation points from producing distributions of particles which are poorly distributed for integration.

Example / Benchmark

Viscoelastic formulation

We use a Maxwell viscoelastic constitutive relationship which assumes that the deformation rate is the sum of viscous and elastic parts

τ 2µ + τ 2η = Dv + De = D (1)
where τ is the Jaumann corotational stress rate for an element of the continuum, µ is the shear modulus and η is shear viscosity. D is the deviatoric part of D.

τ = τ + τ W -Wτ ; W ij = 1 2 ∂V i ∂x j - ∂V j ∂x i (2)
W is the material spin tensor, The material spin tensor, W , terms account for material spin during advection which reorients the elastic stored-stress tensor.

As we are primarily interested in solutions where very large deformations may occur -such as buoyancy driven fluid convection, we prefer to work with a fluid-like system of equations from the outset.

Hence we obtain a stress / strain-rate relation from (1) by expressing the Jaumann stress-rate in a difference form:

τ ≈ τ t+∆t e -τ t ∆t e -W t τ t + τ t W t (3)
where the superscripts t, t+∆t e indicate values at the current and future timestep respectively. ∆t e is a timestep which captures the relevant timescales of the changes in elastic stresses. This timestep could, in fact, differ from that chosen for updating the particle positions.

(1) becomes

τ t+∆t e = η eff 2 Dt+∆t e + τ t µ∆t e + W t τ t -τ t W t µ ; η eff = η ∆t e ∆t e + α ( 4
)
Where α = η/µ is the shear relaxation time, and η eff is an effective viscosity.

Our system of equations is thus composed of a quasi-viscous part with modified material parameters and a right-handside term depending on values from the previous timestep. This approach makes it possible to develop a viscoelastic formulation in the context of a viscous flow code.

In the finite element momentum balance equation, to solve for terms at t + ∆t e we require access to values stored at time t at the integration points of the mesh at t + ∆t e . In the ELLIPSIS code this corresponds to storing the Jaumann material derivative tensor on the Lagrangian integration points.

Yielding

In a geological context we frequently deal with situations where part of the system is subjected to stresses greater than the yield stress. Under such conditions the material fails, but, unlike many engineering simulations, we are interested in simulating in the post-failure behaviour up to very large strains. On the basis that the post yield deformation trends increasingly to dominant, simple structures with increasing strain (Ben-Zion and Sammis, 2001), our geological modeling for large strains uses very simple descriptions of yielding. Brittle behaviour is parameterized using a non-linear effective viscosity which is introduced whenever the stress would otherwise exceed the yield value τ yield . This approach ignores details of individual faults, and treats only the influence of fault systems on the large-scale convective flow.

To determine the value of the effective viscosity at any point we extend (1) by introducing a Prandtl-Reuss flow rule for the plastic part of the stretching:

τ 2µ + τ 2η + λ τ 2 |τ | = De + Dv + Dp = D (5)
where λ is a parameter to be determined such that the stress remains on the yield surface, and |τ | ≡ (τ ij τ ij /2) (1/2) . The plastic flow rule introduces a non-linearity into the constitutive law which, in general, requires iteration to determine the equilibrium state.

The implementation is as follows, starting from equation ( 5), we again express the Jaumann stress rate in first order difference form (using the Lagrangian particle reference frame):

τ t+∆t e 1 2µ∆t e + 1 2η + λ 2 |τ | + = Dt+∆t e + 1 2µ∆t e τ t + 1 2µ (W t τ t -τ t W t ) (6)
No modification to the isotropic part of the problem is required when the von Mises yield criterion is used. At yield we use the fact that |τ | = τ yield to write

τ t+∆t e = η 2 Dt+∆t e + 1 µ∆t e τ t + 1 µ (W t τ t -τ t W t )
where η = ητ yield µ∆t e ητ yield + τ yield µ∆t e + ληµ∆t e (7)

We determine λ by equating the value of |τ t+∆t e | with the yield stress in (7). Alternatively, in this particular case, we can obtain η directly as

η = τ yield / Deff where Deff = 2 Dt+∆t e + 1 µ∆t e τ t + 1 µ (W t τ t -τ t W t ) and |D| = (2D ij D ij ) 1/2 (8)
The value of λ or η is iterated to allow stress to redistribute from points which become unloaded. The iteration is repeated until the velocity solution is unchanged to within the error tolerance required for the solution as a whole.

A beam failing under extension

The yielding algorithm is benchmarked by measuring the second invariant of the stress and displacement at points within a viscoelastic beam extendedat a fixed rate, v = 5, by an imposed velocity boundary condition at one end.

The sample was 0.5 units thick, occupying the central half of a 3 x 1 mesh, and was surrounded by a low viscosity, compressible material. Three sampling points (a,b,c) for recording the stress invariant and displacement were chosen within the sample initially placed along the mid-line at x = 0.2, 0.5, 0.8. The material parameters of the sample (η = 10 8 , µ = 10 6 ) were chosen such that the relaxation time was long (α = η/µ = 100 ) compared to the duration of the experiment ( 0.25 ) so that the material behaved almost as an elastic solid.

Figure 3 shows the progress of the experiment. Initially, deformation was uniform, resulting in gradual stretching of the sample (t ≤ 0.180). The entire sample reached the yield point at the same time (t = 0.212) and initially deformed uniformly with all points yielding. However, the deformation soon localized to a number of shear bands (t = 0.220), then to two places along the sample (t = 0.227), and finally to a single location (t = 0.2359) which focussed all subsequent deformation until the sample failed entirely (t ≥ 0.2404). The frames are not uniformly spaced in time since the post-failure behaviour occurred on a much shorter timescale than the gradual loading.

Even with a material which has no strain softening, there is a tendency for deformation to localize in a particle-in-cell representation of the sample. This occurs because the sample boundary is never perfectly flat (as in real life) due to numerical fluctuations in the particle locations, and to a mild interference (moiré) effect between the array of particles and the underlying grid. These effects produce small fluctuations in the stress field which can result in early failure at certain points. Once nucleated, shear bands can propagate from these points -ultimately resulting in necking and complete separation of the two halves of the sample. It is worth discussing at this point a consequence of the fact that the yield criterion only applies to the deviatoric stress.

During the separation of the layer, the pressure becomes enormous at the constriction, which obviously could not occur in a real material. To model this situation in a more realistic manner we would need to complement the yield criterion on the deviatoric stress with a suitable tension cutoff condition.

Discussion

Geological modeling problems present an unusual set of difficulties which are every bit as challenging as problems encountered elsewhere in the simulation community. Engineering codes may not be optimal for geological modeling -whether in terms of computational efficiency or ability to handle unusual rheological descriptions in very large deformation. The algorithms we have covered are all well suited to one or other aspects of the spectrum of geological modeling. No one method can deal with the entire range. In developing a "universal solver" for geological and geomechanical one would therefore be required to build a suite of codes which can hand off elements of a problem to each Taken from [START_REF] Moresi | Mantle convection modeling with viscoelastic/brittle lithosphere: Numerical methodology and plate tectonic modeling[END_REF].

other according to the special needs inherent in a particular simulation. Some of these issues will be addressed at this meeting.

Figure 3 :

 3 Figure 3: Simulation of the extension of a viscoelastic bar with yield stress. Black shading indicates regions deforming at yield. Embedded marker points which follow the material deformation are indicated by a,b,c. Taken from Moresi et al (2001).

Figure 4a ,

 4a Figure4a, is a plot of the stress at each of the sample points in the material as a function of time for a fixed end velocity. The yield stress of the material was set at 3 × 10 5 . Stress increased within the sample at the same rate for all the sample points until the yield stress was reached. At this stage the material deformed uniformly at the yield stress. Once localization had occurred, however, points outside the necking area begin to unload, and the stress dropped dramatically. The rate at which stress drops from yield back to zero is governed by the viscous part of the rheology, and the presence of a low viscosity background material. The unloading path is clearer in plots of the displacement of the sample points through time in Figure4b. Before yielding, the displacement of each sample point increases monotonically. Once yielding occurs, and the deformation localizes, the sample points on the left of the break (a,b), under the action of stored elastic stresses, rapidly retreat towards their original locations. The sample point on the right of the break (c) moves rapidly to the right as the elastic deformation relaxes. It is worth discussing at this point a consequence of the fact that the yield criterion only applies to the deviatoric stress. During the separation of the layer, the pressure becomes enormous at the constriction, which obviously could not occur in a real material. To model this situation in a more realistic manner we would need to complement the yield criterion on the deviatoric stress with a suitable tension cutoff condition.

Figure 4 :

 4 Figure 4: Stress, and displacement at sample points a,b,c as a function of time for the extension experiment of figure 3.Taken from[START_REF] Moresi | Mantle convection modeling with viscoelastic/brittle lithosphere: Numerical methodology and plate tectonic modeling[END_REF].