David Vignolles

Alain Audouard

Rustem B Lyubovskii

Marc Nardone

Enric Canadell

Elena I Zhilayeva

Rimma N Lyubovskaya

Elena I Zhilyaeva

Shubnikov-de Haas oscillations spectrum of the strongly correlated quasi-2D organic metal (ET)8[Hg4Cl12(C6H5Br)]2 under pressure

Keywords: 71, 18, +y Fermi surface: calculations and measurements; effective mass, g factor -71, 20, Rv Polymers and organic compounds -72, 15, Gd Galvanomagnetic and other magnetotransport effects

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

Introduction

Quantum oscillations spectra of quasi-two-dimensional (q-2D) multiband metals are known to contain many Fourier components. In addition to those that can be attributed to closed orbits, eventually induced by magnetic breakdown (MB), linear combinations of few basic frequencies that are not predicted by the semiclassical model [1,[START_REF] Shoenberg | Magnetic Oscillations in Metals[END_REF] can nevertheless be observed. This frequency mixing phenomenon can be linked to both the q-2D nature of the dispersion relation which is liable to give rise to a field-dependent oscillation of the chemical potential [START_REF] Alexandrov | [END_REF] and to the Fermi surface (FS) topology that can be regarded as a network of orbits coupled by MB [4]. Despite both theoretical and experimental efforts, the latter being mainly focused on organic metals based on the ET molecule (where ET stands for bisethylenedithia-tetrathiofulvalene) with the FS originating from the hybridization of one orbit with an area equal to that of the first Brillouin zone (FBZ), quantitative interpretation of the data is still an open problem.

Networks of orbits can also result from the hybridization of two or more pairs of q-1D sheets as it is the case of the room temperature FS of numerous oxide bronzes.

However hidden nesting properties [5] lead to the condensation of a charge density wave which strongly modifies the FS at low temperature in most cases. As a result, quantum oscillations spectra observed e. g. in monophosphate tungsten bronzes [6] can hardly be reconciled with band a UMR 5147: Unité Mixte de Recherche CNRS -Université Paul Sabatier -INSA de Toulouse Correspondence to: audouard@lncmp.org structure calculations [7]. Analogous problematic can be observed in the q-2D organic metal β"-(ET)(TCNQ) [8].

On the contrary, although the FS of the q-2D charge transfer salt (ET) 8 Hg 4 Cl 12 (C 6 H 5 Cl), noted hereafter as (Cl, Cl), originates from the hybridization of two pairs of q-1D sheets as well, it remains metallic down to the lowest temperatures. According to band structure calculations [9], its FS is composed of two compensated orbits labelled a in the following with an area of 13 % of the FBZ area. Shubnikov-de Haas (SdH) oscillations spectra of this compound [10], which otherwise exhibit many frequency combinations, as discussed below, are in good agreement with this picture since the main Fourier component has a frequency F a = 241.5 ± 2 T that corresponds to 11 % of the FBZ area. This is also the case of the isostructural compound (ET) 8 [Hg 4 Cl 12 (C 6 H 5 Br)] 2 , noted hereafter as (Cl, Br), which has been even more extensively studied at high magnetic field [11,12] although its FS has not been reported up to now.

With respect to the interpretation of frequency combinations, applied pressure can be useful in order to tune both the transverse interactions (which are involved in the chemical potential oscillations) [13] and the MB gaps.

In a first step, band structure calculations of the (Cl, Br) compound at ambient pressure are reported which confirm that its FS can be regarded as a network of compensated orbits. In a second step, the temperature dependence of the interlayer resistance under pressure is considered. Finally, the pressure-induced change of the SdH oscillations spectrum, MB field, effective masses and scattering rate is investigated up to 1.1 GPa.

Experimental

The studied crystals were prepared by the electrochemical method reported in [14]. Tight binding band structure calculations, based on X-ray data collected at room temperature and ambient pressure, were performed using the same method as reported in [9]. The calculations use an extended Hückel effective one-electron Hamiltonian [15].

The off-diagonal matrix elements of the Hamiltonian were calculated according to the modified Wolfsberg-Helmholz formula [16]. All valence electrons were explicitly taken into account in the calculations and the basis set consisted of double-ζ Slater-type orbitals for all atoms, except hydrogen.

Interlayer zero-field resistance and magnetoresistance measurements were performed on crystals with approximate dimensions 0.5×0.5×0.1 mm 3 . Electrical contacts to the crystal were made using annealed platinum wires of 20 µm in diameter glued with graphite paste. Alternating current (1 µA, 77 Hz) and (5 to 10 µA, 20 kHz) was injected parallel to the a* direction for measurements of the interlayer zero-field resistance and magnetoresistance, respectively. Hydrostatic pressure was applied up to 1.1 GPa in an anvil cell designed for isothermal measurements in pulsed magnetic fields [17]. In the following, the pressure applied at room temperature is considered although a slight crystal size-dependent pressure loss on cooling cannot be excluded [18]. Magnetoresistance ex-periments were performed up to 54 T in pulsed magnetic field with pulse decay duration of 0.36 s, in the temperature range from 1.5 K to 4.2 K. Magnetic field was applied normal to the conducting (bc) plane. A lock-in amplifier with a time constant of 30 µs was used to detect the signal across the potential contacts. Analysis of the oscillatory magnetoresistance is based on discrete Fourier transforms of the data, calculated with a Blackman window.

Results and discussion

The band structure calculations and FS of the (Cl, Br) compound displayed in Fig. 1 are based on crystallographic data recorded at ambient pressure and room temperature. The repeat unit of the donor layer contains 8 ET molecules so that the band structure of Fig. 1a contains 8 bands mainly based on the highest occupied molecular orbitals of ET. With the usual oxidation states of Cl -and Hg 2+ , the average charge of the ET donors is +1/2. Consequently, two bands in Fig. 1a should be formally empty.

Since the second and third bands from the top overlap, the system must be metallic and its FS must contain both electron and hole contributions with the same area. More precisely, as it is the case for (Cl, Cl) [9], the FS (see Fig. 1b) originates from two pairs of crossing q-1D sheets and constitutes a network of compensated electron (around M) and hole (around Z) closed orbits elongated along the (b * + c * ) and c * directions, respectively. These orbits are both labelled a hereafter since they have the same area.

Both the band structure and FS of Fig. 1 are very similar to those previously reported for the (Cl, Cl) compound [9]. For instance whereas the area of the closed orbits of the (Cl, Br) compound is 16.1 % of the FBZ area, according to data in Fig. 1, calculations for (Cl, Cl) using the same computational details as in the present case lead to an area of 16.7 % of the FBZ area. Although this is not surprising since the two compounds are isostructural, these results justify a posteriori the previous data analysis based on a FS topology analogous to that of the (Cl, Cl) compound [11,12,19]. Let us note that similar calculations led to a more substantial variation of the orbits area for the (Br, Cl) compound (9.6 % of the FBZ area). This suggests that replacement of Br by Cl in the anion layer has a considerably stronger effect on the FS than the same replacement in the solvent molecules. The small difference in orbits area between the (Cl, Br) and (Cl, Cl) compounds is in agreement with the experimental results. Nevertheless, a slight discrepancy between calculations and quantum oscillations spectra can be observed since the closed orbits area of (Cl, Br) amounts to 16.1 % and 11 % of the FBZ area in the former and latter cases, respectively.

Zero-field interlayer resistance and magnetotransport were studied on three crystals. Since they yield consistent results, we concentrate on the most extensively studied one in the following. It should be noted that all the features reported hereafter are reversible as the pressure is released.

Temperature dependence of the zero-field interlayer resistance at various applied pressures is displayed in Fig. It should be also mentioned that a coherent-incoherent crossover of small polarons, driven by the strength of the electron-phonon coupling, can also lead to interlayer resistance maximum [26,27]. Alternatively, electron correlations have been invoked in order to account for such resistance behaviour [START_REF] Limelette | [END_REF]28]. As a matter of fact, a T 2 variation of the resistance (R = R 0 + AT 2 ) is observed at low temperature (see the inset of Fig. 2). Owing to the crystal dimensions, A ranges from ∼ 0.2 ΩcmK -2 at ambient pressure to ∼ 0.05 ΩcmK -2 at 1.1 GPa. Such a behaviour which is commonly observed both in charge transfer salts [20,[START_REF] Limelette | [END_REF] and inorganic low dimensional compounds such as Sr 2 RuO 4 [29] is currently regarded as the signature of a strongly correlated Fermi liquid. This latter point and the above statement regarding the pressure dependence of the scattering rate are further discussed at the light of the magnetoresistance data reported hereafter.

Fourier analysis of interlayer magnetoresistance data

(few examples of which are given in Fig. 3) are displayed in Fig. 4. As previously reported for ambient pressure data [11], many frequencies are observed in all the explored pressure range. Most of them are linear combinations of that linked to the compensated electron-and hole-type orbits (labelled a in Fig. 1) and of the FS pieces located in-between (labelled δ and ∆ in Fig. 1). These two latter FS pieces correspond to so called "forbidden orbits" 1 as it is the case of most of the observed frequency combinations. Oppositely, the Fourier component 2a + δ corre- 1 As reported in Ref. [11], these FS pieces could correspond to MB-induced closed orbits or QI paths. However, the corresponding effective mass and MB damping factor would be much too large and small, respectively, to account for the data Although more data is needed in order to discuss the origin of these latter features, it can be inferred that they could arise from slight change of the FS. Nevertheless, most of the observed Fourier components remain linear combinations of the three basic frequencies reported above which allows for the study of their pressure dependence. ) is in good agreement with data of the FBZ area of numerous ET salts [30,31,32]. While the pressure dependence of F a is the same as that of the FBZ area, F δ strongly decreases as the pressure increases. This behaviour, which lead to the non-monotonous pressure dependence of F 2a+δ reported in the inset of Fig. 5, could be understood assuming that the long and small axis of the a orbits elongates and shrinks, respectively, as the pressure increases. However, although this is actually the case of the β orbits of κ-salts [30], such a behavior should lead to an increase of the ∆ piece area steeper than observed in Fig. 5. The actual scenario is therefore certainly more complex.

The MB field (B 0 ), effective mass (m * ) and Dingle temperature (T D = h/2πk B τ D , where τ D is the relaxation time) can be extracted from the field and (or) temperature dependence of the oscillations amplitude. In the framework of the Lifshits-Kosevich model [START_REF] Shoenberg | Magnetic Oscillations in Metals[END_REF], the amplitude of the Fourier component with the frequency F i is given by

A i ∝ R T i R Di R MBi R Si ,
R T i = αT m * i Bsinh[αT m * i /B] (1)
R Di = exp[-αT Di m * i /B] (2)
R MBi = exp(- t i B MB 2B )[1 -exp(- B MB B )] bi/2 (3)
where α = 2π 2 m e k B /eh (≃ 14.69 T/K). Integers t i and b i are respectively the number of tunnelling and Bragg reflections encountered along the path of the quasiparticle.

As discussed in Refs. [11,12], only Fourier components a, 2a + δ (SdH) and b (QI) can be analyzed on the basis of closed orbits or QI paths, the other being due to, or strongly affected by, the frequency mixing phenomenon.

Therefore, we will mainly focus on these oscillations in the following.

In the field range below ∼ 30 T, the amplitude of the Fourier component linked to F b remains temperatureindependent up to 0.7 GPa (at 1.1 GPa, the signal-to-noise ratio is too small to derive a reliable temperature dependence). This feature is in agreement with the zero-effective mass value predicted for a symmetric quantum interferometer [33], as reported for ambient pressure data [11].

Oppositely, m * (a) and m * (2a + δ) decreases by roughly a factor of 2 between ambient pressure and 1.1 GPa (see Fig. 6a). Remarkably, the ratio m * (2a + δ) / m * (a) remains pressure-independent and equal to 1.8. This value is close to 2 as expected from the semiclassical model of Falicov-Stachowiak [1] for the considered coupled orbits networks [11]. The decrease of the effective mass as the applied pressure increases can be considered in the light of the T 2 dependence of the zero-field resistance observed at low-temperature (see the inset of Fig. 2). Indeed, in the case of a strongly correlated Fermi liquid, the Kadowaki-Woods ratio (A / γ 2 , where A and γ are the T 2 coefficient of the resistivity and the electronic specific heat coeffi-cient, respectively) should be proportional to the lattice parameter [34]. Assuming, as a rough approximation, that γ ∝ 1/ε F ∝ m * (a) / F a and, since the FBZ area varies as F a under pressure, the relevant lattice parameter is proportional to 1 / (F a ) 1/2 , yield:

A ∝ m * a (P ) m * a (P = 0) 2 F a (P ) F * a (P = 0) 5/2 (4)
As can be observed in Fig. 6a, Eq. 4 accounts for the data within the error bars. Such a behaviour could suggest a reduction of electron correlations under pressure, in line with a Brinkman-Rice scenario [28]. However, it should be mentioned that the product A×T 2 0 (where T 0 is the coherence temperature, above which the T 2 law is no more valid) should be roughly pressure-independent within this picture which is not the case. Indeed, according to the data of Fig. 2, it decreases by about a factor of two from ambient pressure to 1.1 GPa which suggests that electron correlations may not be the only contribution to the observed behaviour. A similar conclusion is derived from the data of κ-(ET) 2 Cu[N(CN) 2 ]Br [20].

Besides the effective mass, the two main ingredients entering the field dependence of the oscillation amplitude are the MB field and the Dingle temperature. Since the a orbits only involve Bragg reflections, both T D and B 0 2 can be derived from this component [11,12]. The MB field value, consistently deduced from both the a and b components within the error bars, strongly decreases as the 2 Two different MB gaps are observed in Fig. 1. As discussed in Ref. [11], only a value very close to their arithmetic mean, can be derived from the field-dependent data.

applied pressure increases as reported in Fig. 6c. This behaviour is consistent with the data relevant to the MB orbit β reported for few charge transfer salts [30,31]. T D has been derived from the a and 2a+δ components, adopting the B 0 value derived from a in the latter case (see Fig. 6b). As already reported [11], a slightly negative value is obtained at ambient pressure for 2a + δ. This can be due to a contribution of the frequency mixing phenomenon to this component. A lack of accuracy in the determination of B 0 (large error bars are observed in Fig. 6c) can also contribute to this result since the 2a + δ orbits involves four tunnellings (t 2a+δ = 4 in Eq. 3) which makes the deduced T D value very sensitive to B 0 . A sizeable increase of the scattering rate, deduced from the field dependence of both the a and 2a+δ components, is observed under pressure. Indeed, T D which is close to 0 at ambient pressure increases up to about 4 K at 1.1 GPa which corresponds to τ D = 0.3 ps. Pressure-induced defects seems the most plausible explanation for this behaviour. However, it must be recalled that this high τ D value decreases back as the pressure is released which rules out any sample degradation due to the pressurization process. The increase of the scattering rate is in line with the pressure-induced resistance maximum observed in Fig. 2. However, as above discussed, the decrease of its amplitude above 0.5 GPa cannot be interpreted on this basis. An interplay between the pressure-induced decrease strength of the electron correlations and the increase of scattering rate, the latter being in this case assumed to be mainly controlled by electronphonon interaction, could therefore be considered. Within this picture, it should also be assumed that the variations of the interlayer scattering rate, involved in the interlayer resistance, reflect that of the in-plane scattering rate which is probed by SdH oscillations. In any case, the observed pressure dependence of the Dingle temperature is at variance with the data of most ET-based salts for which T D remains unchanged under pressure [31,35,36].

Finally, let us examine the pressure dependence of the frequency mixing that correspond to so called "forbidden orbits". They could arise from both chemical potential oscillation [START_REF] Alexandrov | [END_REF] and MB-induced Landau level broadening [4]. As mentioned in Ref. [13], applied pressure leads to an increase of the transverse interactions and therefore to a damping of the chemical potential oscillation. Oppositely, as can be observed in Fig. 4, the amplitude of some components such as ∆ + a or b + a remains large, or even increases, as the applied pressure increases. This behaviour suggests that the field-dependent chemical potential oscillation is not the dominant contribution to the development of the observed frequency combinations. This point is in line with recent calculations [37] which indicate that, contrary to the case of numerous ET salts which are uncompensated metals, chemical potential oscillation in compensated orbits systems is strongly damped. As a consequence, Landau level broadening is likely the main source of the observed frequency combinations in compensated orbits networks.

Summary and conclusion

The SdH spectra of the charge transfer salt (ET) 8 [Hg 4 Cl 12 (C 6 H 5 Br)] exhibits many Fourier components corresponding to linear combinations of three basic frequencies (F a , F δ and F ∆ ), in agreement with band structure calculations (see Fig. 1).

Most of these frequency combinations, which correspond to "forbidden orbits" in the framework of the semiclassical model, keep a large amplitude in all the studied pressure range. This feature likely rules out chemical potential oscillation as a dominant contribution to their origin.

The pressure sensitivity of the FBZ area is similar to that of most of the ET salts. The area of the ∆ piece and of the compensated orbits (a) follow the same pressure dependence as that of the FBZ area while the area of the δ piece decreases under pressure. The measured electronic properties are significantly modified as the applied pressure increases. Either additional frequencies or shifts in few of the frequency combinations are observed in the high pressure range. Therefore, some change of the FS topology as the pressure increases cannot be excluded.

Higher pressures are needed to check this hypothesis.

The zero-field interlayer resistance follows a T 2 behaviour at low temperature with a pressure-dependent prefactor A, in agreement with the predictions for a strongly correlated Fermi liquid [34]. Since it strongly decreases under pressure, it could be inferred that the effect of electron correlations, which are important at ambient pressure decreases, under pressure. This latter point is in agreement with data relevant to e.g. κ-(ET) 2 Cu[N(CN) 2 ]Cl [START_REF] Limelette | [END_REF]. However, within this picture, the amplitude of the resistance maximum should also continuously decrease as the applied pressure increases, which is not the case. Oppositely, it increases in the pressure range below about 0.5 GPa. In line with SdH data reported for κ-(ET) 2 Cu[N(CN) 2 ]Br [23],

the increase of the amplitude of the resistance maximum in the low pressure range could be attributed to the sizeable increase of the scattering rate, deduced from the SdH data, as the applied pressure increases. Within this picture, the pressure dependence of the interlayer resistance would mainly result from the interplay between the pressure sensitivity of the scattering rate and of the strength of electron-correlations.

Fig. 1 .

 1 Fig. 1. (a) Dispersion relations at ambient pressure of (ET)8[Hg4Cl12(C6H5Br)]2. The dashed lines marks the Fermi level. (b) Fermi surface (FS) corresponding to (a). Γ , Y, Z, S and M refer to (0, 0), (b * /2, 0), (0, c * /2), (-b * /2, c * /2) and (b * /2, c * /2), respectively. Labels a, δ and ∆ correspond to the FS pieces discussed in the text.

2 .

 2 The room temperature resistance decreases as the pressure increases (dlnR/dP = -1 GPa -1 ) while the low temperature value remains almost unaffected. As a result, the residual resistivity ratio (RRR) decreases monotonously by about a factor of 3 as the pressure increases up to 1.1 GPa. Whereas the resistance continuously decreases as the temperature decreases at ambient pressure, a resistance maximum is observed under pressure, the amplitude of which is maximum around 0.5 GPa. Data under pres-

Fig. 2 .

 2 Fig. 2. (color on line) Temperature dependence of the zerofield interlayer resistance at various applied pressures. The low temperature part of the data is plotted as a function of T 2 in the inset.

Fig. 3 .

 3 Fig. 3. (color on line) Field-dependent interlayer resistance at various applied pressures. Curves have been shifted down by an arbitrary amount for clarity. Black and blue solid lines correspond to data measured at 4.2 K and 1.7 K, respectively.

Fig. 4 .

 4 Fig. 4. (color on line) Fourier analysis of the oscillatory part of the magnetoresistance at various applied pressures and field ranges. Blue, red, green and black solid lines correspond to data at 1.7, 2.5, 3.4 and 4.2 K, respectively. Black vertical lines are marks calculated with the set of frequency values (Fa, F δ and F∆) that best fits to the Fourier spectra.

Fig. 5 .

 5 Fig. 5. (color on line) Pressure dependence of the normalized frequency of the Fourier components δ, a, ∆ and b deduced from data in Fig. 4). The ambient pressure values are F δ = 135 T, Fa = 235.5 T and F∆ = 1577 T [11]. The inset displays the pressure dependence of F 2a+δ . Open squares stand for values calculated as F 2a+δ = 2Fa + F δ .

Fig. 6 .

 6 Fig. 6. Pressure dependence of the (a) effective mass, (b) Dingle temperature and (c) magnetic breakdown field, deduced from the field or temperature dependence of the Fourier components a, 2a + δ and b. Lines are guides to the eye. The inset displays the A coefficient deduced from data in Fig. 2, plotted against [m * a / m * 0 ] 2 / [Fa / F * 0 ] 5/2 , where m * 0 and F * 0 are the ambient pressure values. The straight line is the best fit of Eq. 4 to the data.

  where the spin damping factor (R Si ) depends only on the direction of the magnetic field with respect to the conducting plane. The thermal (for a 2D FS), Dingle and MB damping factors are respectively given by:

This work was supported by Euromagnet under the European Union contract R113-CT-2004-506239, MEC-Spain (Project FIS2006-12117-C04-01, CSD2007-00041), Generalitat de Catalunya (Project 2005 SGR 683), the French-Russian exchange program between CNRS and RAS (number 21451) and by RFBR grants 09-02-00899 and 08-03-00480. We thank Jean-Yves Fortin, Sergei I.

Pesotskii and Geert Rikken for interesting discussions.


Jordi Haubrich

Joan Font

Robert B Quast

Anne Goupil-Lamy

Pauline Scholler

Damien Nevoltris

Francine Acher

Patrick Chames

Philippe Rondard

Laurent Prézeau
email: laurent.prezeau@igf.cnrs.fr

Jean-Philippe Pin
email: jean-philippe.pin@igf.cnrs.fr

J.-P P Wrote

A nanobody activating metabotropic glutamate receptor 4 discriminates between homo-and heterodimers

Keywords: Biological Sciences G protein-coupled receptor, single domain antibody, activation mechanism

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

Introduction

There is more and more interest in developing antibodies as possible pharmacological and even therapeutic agents (1,2). The single domain antibodies, also named nanobodies, are prone for such activities as their short variable loops can interact in surface cavities that can vary between the conformational states of a protein (1). As such, nanobodies regulating drug targets have already been reported (3,4). Among the main drug targets are the G protein-coupled receptors (GPCRs) that play important roles in cell-cell communication (5).

Among the large GPCR family, class C receptors are those activated by the main neurotransmitters, glutamate and -amino butyric acid (GABA), as well as by Ca 2+ ions, sweet and umami compounds (6). There are eight genes encoding metabotropic glutamate (mGlu) receptor subunits. Although only considered as disulfide linked homodimers, the complexity of the mGlu receptor family increased with the description of heterodimeric entities made of two different mGlu subunits (7).

Indeed, the post-synaptic mGlu1 and 5 on one side, and pre-synaptic mGlu2, 3, 4, 7 and 8 subunits on another side can form heterodimeric entities in recombinant cells, leading to the possible existence of 16 additional mGlu subtypes (7,8). Among these, the mGlu2-4 heterodimer retained much attention. It was found to display a specific pharmacological profile and was identified in transfected neurons, as well as in both cortico-striatal (9) and lateral perforant path (10) terminals. Identifying the various heterodimeric mGlu receptors in the brain and their possible roles is essential.

We recently reported nanobodies acting as positive allosteric modulators (PAMs) specific to the mGlu2 homodimer (11). Among the mGlu subtypes, the mGlu4 containing receptors, and especially mGlu4 homodimers, are of interest for the treatment of Parkinson disease (12)(13)(14) and pain (15,16). In the present study, we report the identification of DN45, a nanobody that specifically and fully activates human mGlu4 homodimers. We show that this nanobody acts by stabilizing the closed active state of the glutamate binding Venus flytrap domain (VFT). However, this nanobody was unable to directly activate the mGlu2-4 heterodimer, acting instead as a pure PAM. These data reveal a novel way a nanobody can activate a class C GPCR, and confirm the potential of using nanobodies to discriminate between homo-and heterodimeric receptors.

Results

DN45 is specific to the human mGlu4 receptor

To identify nanobodies targeting mGlu4, HEK293 cells transiently expressing either the human or the rat mGlu4 receptor were injected in a llama. Genes encoding V H H domains were amplified by reverse transcription polymerase chain reaction (RT-PCR) from the total RNA of peripheral blood mononuclear cells and used to create a phage display library. This library was depleted on controlled HEK293 cells and enriched by two cycles of positive selection on human mGlu4 transfected cells in the presence of an excess of anti-HEK293 nanobodies (17). Nanobody containing E. coli supernatants were screened by flow cytometry.

Among the selected clones, nanobody DN45 was further characterized. We performed a binding assay based on time-resolved fluorescence energy transfer (TR-FRET) between the SNAP-tagged receptor and an anti-c-Myc-antibody-d2 interacting with the c-Myc-tag inserted at the C terminal end of the DN45 sequence (Fig. 1A).

The nanobody binds to human mGlu4 exclusively when tested at 100 nM and no binding was observed on any other human mGlu receptor (Fig. 1B), nor on the eight rat mGlu receptors (Fig. 1C). To avoid constitutive receptor activation by ambient glutamate produced by the cultured HEK293 cells, excitatory amino acid transporter 3 (EAAC1) was co-transfected with the indicated receptors, such that receptors were essentially in an inactive state under basal condition. Activating any of the mGlu receptors with an agonist did not allow binding of DN45 either, except on the human mGlu4 where a higher signal was observed (Fig. 1B,C). Indeed, DN45 had a preferred high binding affinity (K D = 3.1 nM) for the active conformation of the human mGlu4 receptor and a low binding affinity for the basal and antagonist-induced conformation (Fig. 1D and SI Appendix, Table S1).

DN45 is a full agonist of the mGlu4 receptor

In agreement with the higher binding affinity of DN45 for the active form of hmGlu4 receptor, DN45 alone stabilizes the active conformation of the VFT dimer as revealed using a TR-FRET based conformational biosensor (Fig. 2A) (7,18). This biosensor measures the distance variation between the N-terminally inserted SNAP tags of the subunits, such that a high TR-FRET is measured in the basal state (after random labelling with the SNAP substrates O 6 -benzylguanine (BG)-Lumi4-Tb and BG-Green), while a lower TR-FRET is observed upon the reorientation of the VFTs upon activation (18). DN45 displays a potency, and efficacy not significantly different from the group-III mGlu receptor agonist L-AP4 (Fig. 2A and SI Appendix, Table S2).

Subsequently, we setup a single-molecule FRET (smFRET) approach to compare the agonist effect of L-AP4 and DN45 at the single-molecule level in an environment without ambient glutamate (Fig. 2B and SI Appendix, Fig. S1, Table S3). There was no difference between the apo-state and the antagonist-stabilized state. However, both L-AP4 and DN45 acted as full agonists as illustrated by the similar increase in the proportion of molecules in the low FRET, active conformation. DN45 also activated the natural Gi protein of the mGlu4 receptor with the same potency as L-AP4, as revealed with a Gi bioluminescence resonance energy transfer (BRET) sensor (Fig. 2C and SI Appendix, Fig. S2, Table S2). The agonist activity of DN45 was further confirmed by measuring the accumulation of IP-1 upon activation of the chimeric G protein Gqi9 (Fig. 2D). DN45 potency and efficacy were similar to those obtained with L-AP4, both being more potent than glutamate (Fig. 2D and SI Appendix, Table S2). In presence of high concentrations of LY341495, the potency of DN45 was decreased (Fig. 2E). Of note, the L-AP4 potency for accumulation of IP-1 was not significantly increased in the presence of non-saturating DN45 concentrations (Fig. 2F and SI Appendix, Table S2).

Residues in lobe 2 of the human mGlu4 VFT confer subtype selectivity to DN45

To explain the subtype selectivity of DN45, we substituted any individual residue at the surface of the human mGlu4 VFT into its rat equivalent. We identified twelve residues on the surface of the VFT that are different in the human and rat proteins (Fig. 3A). Among the twelve mutations tested, three (i.e. I318S, H323R and D485G) affected DN45 binding, with D485G suppressing binding completely (Fig. 3B).

Consistent with these binding data, the DN45 agonist effect was largely affected or even suppressed in these human mGlu4 mutants (Fig. 3C), while L-AP4 could still activate them (SI Appendix, Fig. S3A).

Molecular modelling identifies DN45-mGlu4 interaction site on both lobe 2 and lobe 1

We built 3D models of both human mGlu4 (in an active closed conformation) and DN45, and performed docking experiments using ZDOCK (19) without any specific constraint. Our best scored model using ZRANK (20), that keeps the main interactions with or without L-AP4 within the VFT binding site after a 10 ns dynamics simulation, revealed possible interaction sites of DN45 on the VFT (Fig. 3A). In the presence of bound L-AP4, more interactions between the partners were detected (SI Appendix, Fig. S4A,B) in agreement with the higher binding affinity in the active state. A time-dependent analysis of the model revealed the mGlu4 residues maintaining interaction with DN45 during the 10 ns molecular dynamics simulations (SI Appendix, Fig. S5). These include the residues that are human-specific: I318, H323, D485 (lobe 2) and also residues on lobe 1 (SI Appendix, Fig. S5). The virtual mutagenesis analysis performed on our 3D model of the mGlu4-DN45 complex is perfectly in line with the evaluated impact of individually substituting the twelve human-specific residues for its rat equivalents (SI Appendix, Fig. S5 and Table S4).

In addition, the model revealed other key residues such as L322 (SI Appendix, Fig. S6).

DN45 binding was restored on rat mGlu4 mutant bearing at least four human residues

The proposed model was further confirmed by introducing human-specific residues into the rat sequence, restoring DN45 binding and agonist activity on mutated rat mGlu4 receptor. We considered five human-specific residues based on their proximity to the apparent epitope of DN45 (i.e. I318, H323, D485, V385 and H507 (Fig. 3A)). When the five rat residues were mutated to their human equivalent (rmG4-5M), DN45 bound (Fig. 3D) and activated the receptor (Fig. 3E) with an affinity (K D = 22.8 nM) and potency not different from those measured on human mGlu4 receptor (SI Appendix, Table S1, S2). The substitutions G485D (rmG4-1M), G485D+R323H (rmG4-2M) and G485D+R323H+S318I (rmG4-3M) on the rat mGlu4 receptor was however not sufficient to recover binding of DN45 (Fig. 3D). Indeed, the model of rmG4-3M reveals that DN45 is not able to interact with the three mutated residues (i.e. S318I, R323H, G485D) (Fig. S4A) because a loop is closing the entrance of the epitope cavity (SI Appendix, Fig. S7). The additional mutation of I385V or Q507H restores the opening of the cavity and allows for interactions between DN45 and the three key human residues (SI Appendix, Fig. S7). This was further supported when binding of DN45 to the rat mGlu4 bearing the three main aforementioned substitutions plus I385V (rmG4-4MB) or Q507H (rmG4-4MA) was recovered (Fig. 3D) and both mutants were activated by DN45 (Fig. 3E). The observed affinities (K D = 13.6 nM and 10.4 nM, respectively) and potencies were not significantly different from those measured for human mGlu4 (Fig. 3D,E and SI Appendix, Table S1, S2).

All mutants were functional as we successfully activated them with L-AP4 (SI Appendix, Fig. S3B).

Taken together, these modelling and mutagenesis data provide a reliable explanation for the species selectivity of DN45 by interacting with residues mainly located on lobe 2 of the VFT.

DN45 agonist activity needs interaction with both lobes of the VFT

The predictions of interactions between DN45 and lobe 1 of the VFT in a closed active state are especially comprising residues H371 to E401 (Fig. 4A, B and SI Appendix, Fig. S4B). Interactions with this region are expected to stabilize the closed conformation of the VFT, providing an explanation for the agonist activity of DN45 and its increased affinity for the active mGlu4 (Fig. 1D). In a first attempt to prevent the interaction between DN45 and lobe 1 of the mGlu4 VFT, an N-glycosylation site A399N/E401S was introduced. This resulted in a large decrease in DN45 agonist potency, but did not suppress its agonist activity (Fig. 4C).

Single-point mutations H371A, K386A, H392A, D397A, E401A and E403A did not lead to antagonist effects of DN45 on these mutants (SI Appendix, Fig. S8).

Therefore, a virtual saturation mutagenesis was performed to predict the impact of multiple mutations of lobe 1 residues on the binding of DN45 but also the impact of these mutations on the overall stability of mGlu4. The best predicted mutations were H371E, R391M, R393W and A399K in a loop of lobe 1 (Fig. 4A,B). The calculated binding energy revealed that the highest destabilizing values were found for the triple mutant R391M+R393W+A399K (4.82 kcal/mol) and quadruple mutant H371E+R391M+R393W+A399K (4.81 kcal/mol).

Experimental data support that this loop is involved in the agonist effect of DN45 as DN45 no longer activated neither the triple nor the quadruple mutant (Fig. 4D,E). Of note, the quadruple mutant displays constitutive activity that is reduced by DN45, revealing that DN45 acts as an inverse agonist on this receptor (Fig. 4E). This is confirmed by the significant antagonistic effect of DN45 on this mutant upon activation by L-AP4 (Fig. 4F and SI Appendix, Table S2).

Taken together, these data reveal that DN45 primarily binds to lobe 2, but also with lobe 1 in the active state, an interaction increasing its affinity, and required for its agonist activity.

DN45 acts as a positive allosteric modulator on heterodimeric mGlu2-4

It is now recognized that the mGlu4 subunit can associate with other mGlu subunits to form heterodimeric mGlu receptors (7,8). Among these, the mGlu2-4 heterodimer has retained much attention and has specific pharmacological properties that were used to illustrate its existence in the brain (9,10).

We first examined the binding of DN45 to the mGlu2-4 heterodimer using a combination of mGlu2 N-terminally labelled with CLIP-tag and C-terminal GABA B1 endoplasmic reticulum retention sequence C1KKXX (21) (CLIP-mGlu2-C1KKXX) and mGlu4 N-terminally tagged with SNAP-tag and C-terminal GABA B2 endoplasmic reticulum sequence C2KKXX (21) (SNAP-mGlu4-C2KKXX). After labelling CLIP-mGlu2-C1KKXX with Lumi4-Tb, binding of DN45 to the mGlu4 subunit should generate a TR-FRET signal between the Lumi4-Tb on the mGlu2 subunit and the d2 acceptor on an anti-c-Myc-antibody that binds to DN45 (Fig. 5A). A very low signal could be detected under basal condition, that almost disappeared in the presence of the competitive antagonist LY341495 (Fig. 5B). However, a large signal could be measured in the presence of the mGlu4 agonist L-AP4, the general mGlu agonist glutamate, or the mGlu2 selective agonist LY379268 (Fig. 5B), revealing binding affinities of 5.0 nM, 3.1 nM and 7.4 nM, respectively, close to the K D measured on the agonist occupied mGlu4 homodimers (3.1 nM).

The effect of DN45 on the conformation of the mGlu2-4 heterodimer was then examined. Using CLIP-mGlu2-C1KKXX labelled with O 6 -benzylcytosine (BC)-Lumi4Tb and SNAP-mGlu4-C2KKXX labelled with BG-Green, only the heterodimer generates a TR-FRET signal that is largely decreased upon activation (10,18). As previously reported, the mGlu2 agonist LY379268 activated the heterodimer, while the mGlu4 agonist L-AP4 remains very partial (10,18). DN45 on the other hand could not activate the mGlu2-4 by itself (Fig. 5C). However, in the presence of a minimal concentration (EC 5 ) of LY379268 (50 nM), DN45 could activate the receptor, revealing a clear PAM effect (Fig. 5C). Indeed, DN45 largely increased the potency of the mGlu2 agonist LY379268 (Fig. 5D), demonstrating its potent pure PAM effect on the mGlu2-4 heterodimer.

Discussion

The mGlu4 receptor has been proposed as a potential drug target for the treatment of Parkinson's disease (12,22,23) or pain (15). In the brain, mGlu4 can be found in a homodimeric form, or associated with other mGlu subunits, like mGlu2, in heteromeric complexes. Using ligands with different properties at mGlu4 homo-and heterodimers, evidence has been provided supporting homodimers as the best targets for Parkinson disease's treatment (13). However, the respective role of both types of mGlu4 containing receptors remains to be clarified (8)(9)(10). In the present study we report an mGlu4 nanobody with selective agonist activity at mGlu4 homodimers, and with a pure PAM action on the mGlu2-4 heterodimers.

Nanobodies have become popular in pharmaceutical research. They have a high specificity and affinity and can stabilize specific conformations of their targets. Due to their small size compared to traditional IgG (~15 vs. ~150 kDa), nanobodies have several advantages like lower immunogenicity, better pharmacokinetics and the possibility to reach smaller cavities in proteins (3,4). As such, nanobodies can stabilize specific conformations of their target, being interesting tools for structural studies (1) and innovative pharmacological agents (24). Moreover, some nanobodies can cross the blood-brain-barrier and can then be used to target the central nervous system (25). Of course, nanobodies may have a much better subtype selectivity than orthosteric ligands acting at a binding site often conserved in various receptor subtypes activated by the same natural ligand. Most recently described nanobodies with pharmacological action at GPCRs act as antagonists (26,27) as is the first FDAapproved nanobody, caplacizumab (24).

Recently nanobodies acting as selective PAMs for mGlu2 (11) and mGlu5 (28) homodimers were reported, the formers having in vivo activities. Both nanobodies were found to stabilize the agonist bound active orientation of the two VFTs of the dimers, with the mGlu2 nanobodies acting at the active interface of the VFTs (11), while the mGlu5 one acts on a loop on the top of the mGlu5 VFT (28). Here we identified the mode of action of DN45 by combining state of the art modelling and docking approaches and mutagenesis. We show that DN45 can stabilize the closed active state of each VFT in the mGlu4 homodimer, as the binding epitope includes residues from both lobe 1 and lobe 2. DN45, like L-AP4, increases the same proportion of mGlu4 in the active conformation, as shown by smFRET in the total absence of glutamate, demonstrating an agonist activity equivalent to that of L-AP4.

While the primary binding epitope is located on lobe 2, further interaction between the VFT and DN45 occurs upon spontaneous closure of the VFT. These additional contacts increase the affinity of DN45 and lead to stabilization of the active state of the receptor, as does the agonist. This model is well supported by the observation that preventing the contact between lobe 1 and the nanobody through mutations, converts the nanobody into an antagonist or an inverse agonist, stabilizing the inactive receptor by preventing VFT closure. Taken together, our data confirm that stabilizing the closed VFTs is sufficient for receptor activation, even in the absence of glutamate. Such a mode of action is then different from what has been proposed for the mGlu2 and mGlu5 PAM nanobodies, and reveals novel ways for the development of selective agents acting at a specific mGlu subtype.

When tested on the mGlu2-4 heterodimer, DN45 was found unable to activate the receptor on its own. In contrast to the mGlu4 homodimer, on which two nanobodies are likely acting (one per VFT), only one is expected to bind to the heterodimer, on the mGlu4 VFT only. The fact that the nanobody cannot activate such a heterodimer suggests that the mGlu4 VFT may have less tendency to spontaneously close when associated with mGlu2. One may also consider the positive allosteric effect between each VFTs (21). The transient closure of one VFT may facilitate the closure of the other, such that binding of a first nanobody may facilitate the binding of the second one in the homodimer, leading to the stabilization of the active state without agonist, and then to the full agonist activity. Such an allosteric process is unlikely to occur in the heterodimer in the presence of DN45 alone, since DN45 only acts on one VFT. In agreement with this model, agonist binding in the mGlu2 VFT, largely favored DN45 binding on the mGlu4 subunit. Such a model is consistent with the pure PAM effect of DN45 observed on the mGlu2-4 heterodimer. Such a PAM action of DN45 on mGlu2-4 is also consistent with the effects of group-III agonists acting at the mGlu4 VFT. Indeed, L-AP4 could barely activate the heterodimer, while it largely potentiates the effect of an mGlu2 agonist (10,29). This supports the idea that the closure of the mGlu2 VFT is the driving force for the activation of the mGlu2-4 heterodimer, that is then further stabilized in the active state through the closure of the mGlu4 VFT by DN45.

This model points to a transactivation mechanism within the mGlu2-4 receptor, as the mGlu4 TMD is mainly responsible for G protein activation in this heterodimer (30). This is then similar to what is well established for the GABA B receptor in which GABA stabilizes the closed state of the GB1 VFT, leading to the G protein activation by the GB2 7TM (31) .

Overall, we describe the first nanobody with a full agonist action at a GPCR. Such a tool will be useful to solve the active structure of human mGlu4. We show DN45 acts by stabilizing the closed state of the mGlu4 VFT, then through a novel mode of action compared to other nanobodies enhancing the activity of other class C GPCRs. Our study then provides a new insight in the activation mechanism of mGlu4 and mGlu2-4 receptors, and illustrates how powerful nanobodies can be to decipher the function of these different receptor subtypes.

Materials and Methods

Information on materials, llama immunization, selection, production and purification of DN45, mutagenesis, cell culture and transfection, labelling of CLIP and SNAP-tag, DN45 binding and selectivity assay, IP-1 accumulation assay, mGlu4 and mGlu2-4 biosensor assay, BRET assay, statistical analysis, single-molecule FRET approach and molecular modeling is provided in the SI Appendix. S2). S1, S2).

Methods

Llama immunization, selection, production and purification of DN45

The llama immunization, selection, production and purification of DN45 were performed as described previously (11). In brief, two llamas were immunized by four subcutaneous injections with 5 10 7 HEK293T cells, transfected with rat or human mGlu4 receptors. The V H H library constructions were performed in E. Coli TG1 strains and the library diversity was above 10 9 transformants.

Bacteria were then infected by KM13 helper phage and phage-containing pellets were purified by two selection rounds on human mGlu4 receptor transfected in HEK293T (2 10 7 ) cells. Each round was preceded by a depletion step on cells that were not transfected and positive selections were performed in the presence of an excess of anti-HEK293 cells (17). E. Coli TG1 bacteria were infected with eluted phages and could be used for production of the nanobody.

The production of DN45 was done by transforming E. Coli BL21DE3 strain bacteria.

They were grown overnight at 37 °C while agitating. Protein production was induced the day after by addition of 1 mM IPTG and bacteria were grown overnight at 28 °C while agitating. Bacteria were then collected and lysed in TES-buffer containing Tris, EDTA and sucrose. After centrifugation, the periplasmic extract was recovered and the His-tagged nanobodies were purified by Ni-NTA purification (Qiagen).

Mutagenesis

For the generation of rmG4-2M, rmG4-3M, rmG4-4MA, rmG4-4MB and rmG4-5M, synthethic genes encoding for the rat mGlu4 with corresponding mutations were ordered at GeneCust. The synthetic gene was inserted in a N-terminally SNAPtagged rat mGlu4 receptor with the DNA ligation kit from Agilent Technologies. 10566016, Gibco™) 2 hours before the start of the experiment.

Labelling of CLIP and SNAP-tag

Labelling of the CLIP and SNAP-tag with fluorescent molecules was done for 1 hour and 30 minutes at 37 °C in serum-free DMEM Glutamax. For expression and binding experiments for the mGlu4, the cells were incubated with 100 nM BG-Lumi4Tb. For measuring expression levels of the mGlu2-4, the cells were incubated with 1 µM BC-Lumi4Tb and 100 nM BG-Lumi4Tb. For binding on the mGlu2-4, the cells were incubated with 1 µM BC-Lumi4Tb. For the TR-FRET biosensor assay of the mGlu4, the cells were incubated with 100 nM BG-Lumi4Tb and 60 nM BG-Green. For the TR-FRET biosensor assay of the mGlu2-4, the cells were incubated with 100 nM BG-Lumi4Tb and 1 µM BC-Green. Unbound substrate was removed by washing each well four times with Tag-lite®-buffer.

DN45 binding and selectivity assay

Eight rat and eight human mGlu receptors were overexpressed in HEK293 cells as described in cell culture and transfection and incubated in a black 96-wells plate.

Expression levels of the receptors were measured by the PHERAstar FS as the signal at 620 nm after excitation at 337 nm by a UV-pulsed nitrogen laser. Next, cells were pre-incubated for 30 minutes at 37 °C and 5% CO 2 with 10 µM L-AP4, 100 nM LY379268, 1 µM quisqualic acid or 100 µM LY341495 in Tag-lite®-buffer depending on the overexpressed receptor. Then,105 nM of DN45 and 200 nM anti-c-Myc-d2 were added at the same time and cells were incubated for 3 hours at 20 °C. As a negative control, a nanobody against β-arrestin 2 containing the c-Myc sequence was added instead of DN45. TR-FRET was measured with the PHERAstar FS after excitation at 337 nm by a UV-pulsed nitrogen laser. Signals at 620 nm and 665 nm were integrated and the HTRF®-ratio was calculated as follows:

Values were normalized as the signal over the signal of the negative control.

IP-1 accumulation assay

Inositol phosphate (IP-1) accumulation was determined with the IP-One assay kit from Cisbio Bioassays (ref. 62IPAPEC). Cells had been prepared in a black 96-wells plate. Dilution ranges of DN45 and L-AP4 were prepared in stimulation buffer, added to the cells and incubated for 30 minutes at 37 °C and 5% CO 2 . Then, cells were lysed by addition of IP1 labelled with d2 and anti-IP1 antibody labelled with Lumi4Tb in lysis buffer. The lysate was incubated for 1 hour at 20 °C and TR-FRET was measured with the PHERAstar FS and the HTRF®-ratio was determined. Values were normalized as the percentage of response to L-AP4.

mGlu2-4 and mGlu4 biosensor assay

Compounds were added to each well of a black 96-wells plate containing cells and incubated for 30 minutes at 37 °C and 5% CO 2 in Tag-lite®-buffer. In case of preincubation steps, compounds were pre-incubated 30 minutes at 37 °C and 5% CO 2 prior to addition of the other compounds. TR-FRET was measured with the PHERAstar FS after excitation at 337 nm by a UV-pulsed nitrogen laser. The signal at 520 nm was integrated and an acceptor ratio was calculated as described before (33):

Values were normalized as the percentage of response to L-AP4.

BRET assay

Cells had been prepared in a black 96-wells plate.

Statistical analysis

Curve fitting and statistical analysis was done with Graphpad Prism software (version 9.0). Data are expressed as mean  SEM of three or more individually performed experiments. The performed statistical analysis is explained in the corresponding figure legend. P-values<0.05 were considered significant.

Single-molecule FRET approach Sample preparation and smFRET measurements

Sample preparation and smFRET measurements were done as previously described (34) with a few modifications as specified in the following.

Cell culture, transfection and membrane fraction preparation

Detergent solubilization

Receptors were solubilized on ice by resuspension of crude membranes in acquisition buffer (20 mM Tris-HCl pH7.

smFRET measurements

The pulsed interleaved excitation (PIE)multiparameter fluorescence detection (MFD) setup, data acquisition and analysis were previously described. (34) Measurements were performed at a 4 times dilution of the sample in acquisition

Refinement of Docked Structures

In order to refine the best pose generated with ZDOCK, we solvated the mGlu4 dimer-DN45 complex and the mGlu4 dimer-DN45-L-AP-4 complex in an orthorhombic periodic box of water with a minimum distance from boundary of 15.0 Å and performed with NAMD (38) a molecular dynamics simulation for 10 ns for each system. Analysis of trajectories and non-bonded interactions have been performed

with Discovery Studio.

Virtual mutations

Using the last frame of the molecular dynamics simulation, we performed a virtual mutagenesis to identify residues in mGlu4 that are critical for binding to DN45. We used the Discovery Studio Calculate Mutation Energy Binding protocol (39) that evaluates the effect of mutations on the binding affinity of molecular partners in protein-protein and protein-ligand complexes. It performs combinatorial amino-acid scanning mutagenesis on a set of selected amino-acid residues by mutating them to one or more specified amino-acid types. The energy effect of each mutation on the binding affinity (ΔΔG mut ) is calculated as the difference between the binding free energy (ΔΔG bind ) in the mutated structure and wild type protein: ΔΔG mut = ΔΔG bind (mutant) -ΔΔG bind (wild type). The ΔΔG bind is defined as the difference between the free energy of the complex and unbound state. All energy terms are calculated by CHARMm and the electrostatics energy is calculated using a Generalized Born implicit solvent model. The total energy is calculated as an empirical weighted sum of van der Waals interactions (E vdW ), electrostatic interactions (ΔG elec ), an entropy contribution (-TS sc ) related to the changes in sidechain mobility, and a non-polar, surface dependent, contribution to solvation energy (ΔG np ).

This protocol was used to predict the impact of single or a combination of mutations of human residues to their rat counterpart. Finally, the full interaction surface between DN45 and human mGlu4 was scanned.

We also calculated the impact of a single point mutation of each of these interface residues to the overall stability of mGlu4 using the Calculate Mutation Energy Stability protocol (40). It performs combinatorial amino-acid scanning mutagenesis on a set of selected amino-acid residues by mutating each of them to one or more specified amino-acid types. The energy effect of each mutation on the protein stability (mutation energy, ΔΔG mut-stab ) is calculated as the difference of the free energy of folding (ΔΔG folding ) between the mutated structure and the wild type protein:

ΔΔG mut-stab = ΔΔG folding (mutant) -ΔΔG folding (wild type). The ΔΔG folding is defined as the free energy difference between the folded (ΔG fld ) and unfolded (denaturated) states (ΔG unf ) of the protein: ΔΔG folding = ΔG fld -ΔG unf . Note that a negative value of ΔΔG mut indicates the mutation has a stabilizing effect, and conversely, a positive value indicates the mutation has a destabilizing effect. The energy of the folded state of the wild type is derived from the input structure. The structures of the mutants in the folded state are modeled by keeping the backbone rigid while optimizing the sidechains using the ChiRotor algorithm (41). Before evaluating the folding energy terms, the structures of the wild type and mutants are energy minimized using CHARMm.

The unfolded state is modeled as a relaxed penta-peptide in an extended conformation with the mutated residue in the center. The peptide model is applied only to the short range interactions involving the mutated residues and it is extended with a Gaussian chain model to account for the long-range electrostatic interactions. This is based on the hypothesis that most of the contacts between amino acid residues in the unfolded protein are only sporadic if the residues are not immediate neighbors along the sequence. Considering that van der Waals interactions decline sharply with distance and contribute only at very close contacts; the method neglects the non-polar interactions between the residues that are separated by more than two peptide bonds in sequence.

All energy terms are calculated by CHARMm, and the electrostatics energy is calculated using a Generalized Born implicit solvent model. The total energy is calculated as an empirical weighted sum of van der Waals (E vdW ) interaction, electrostatic interaction (ΔG elec ), an entropy contribution (-TS sc ) related to side-chain mobility, and a non-polar, surface dependent, contribution to solvation energy (ΔG np ).

The calculations were performed in pH-dependent mode.

Fig. 1 . 1 Fig. 2 .Fig. 3 .-Fig. 4 .-

 11234 Fig. 1.DN45 is selective for the human mGlu4 receptor and preferentially binds

Fig. 5 .

 5 Fig. 5.DN45 acts as a positive allosteric modulator on heterodimeric mGlu2-4

  SNAP-rmG4-5M-C2KKXX was generated by introducing the synthetic gene encoding for rmG4-5M into a N-terminally SNAP-tagged rat mGlu4 receptor with a C-terminal endoplasmic reticulum sequence C2KKXX. All other mutations were done by sitedirected mutagenesis following the QuickChange mutagenesis protocol from Agilent Technologies.Cell culture and transfectionHEK-293 (ATCC® CRL-1573™) cells were cultured in Dulbecco's Modified Eagle Medium (DMEM) (ref. 41965, Gibco™) containing 10% fetal bovine serum (FBS) (ref. F2442, Sigma-Aldrich). For each experiment we used 100,000 cells/well of a black 96-wells plate (ref. 655086, Greiner bio-one) coated with poly-L-ornithine (ref. P4957, Sigma-Aldrich). Cells were detached from the petridish (~80% confluency) with trypsin-EDTA (0.05%) (ref. 25300054, Gibco™) and after removal of the trypsin-EDTA by centrifuging, diluted in DMEM supplemented with 10% FBS.Depending on the experiment, HEK-293 cells were transfected by electroporation(32) or reverse lipofectamine transfection following the manufacturer's protocol (Invitrogen™ Lipofectamine 2000™). To prevent toxic concentrations of glutamate in the medium, excitatory amino acid transporter 3 (EAAC1) was co-transfected. For binding experiments 9 µg of plasmid of the receptor and 1 µg EAAC1 were cotransfected by electroporation. For IP-1 accumulation experiments, 8 µg of plasmid of the receptor, 1 µg Gqi9 and 1 µg EAAC1 were co-transfected by electroporation. For the Gi-BRET assay, 4 µg of plasmid of the receptor, 0.8 µg Gi-Rluc, 0.8 µg G, 1.6 µg Gγ-Venus and 1 µg EAAC1 were co-transfected by electroporation. For the mGlu2-4 biosensor experiment, 80 ng CLIP-mGlu2-C1KKXX, 40 ng SNAP-rmG4-5M-C2KKXX, 20 ng EAAC1 and 10 ng pRK6 empty vector per well were transfected by reverse lipofectamine transfection. Cells were incubated for 24 hours at 37 °C and 5% CO 2 and medium was changed for pre-heated serum-free DMEM Glutamax (ref.

  Coelenterazine and dilution ranges of DN45, L-AP4 were prepared in PBS buffer. The signals at 530 nm and 480 nm were measured by the Mithras LB 940 (Berthold). Coelenterazine was added and signals were measured for 10 minutes. Then, coelenterazine and compounds were added directly after each other. Signals were measured for 25 minutes and BRETvalues were calculated by dividing the signals at 530 nm and 480 nm. Values were normalized as percentage of response to L-AP4.

  HEK293T cells (ATCC CRL-3216, LGC Standards S.a.r.l., France) were grown in Gibco™ DMEM, high glucose, GlutaMAX™ Supplement, pyruvate (Thermo Fisher Scientific, France) supplemented with 10% (vol/vol) FBS (Sigma-Aldrich, France) at 37°C, 5% CO 2 in 25cm 2 flasks to approximately 80 % confluence. Transfection was carried out by mixing 4 µg SNAP-mGlu4 plasmid DNA and 8 µl Lipofectamine 2000 (Thermo Fisher Scientific, France) in 500 µl Gibco™ Opti-MEM™ I reduced serum medium (Thermo Fisher Scientific, France) and incubation at room temperature for 25 minutes. The mixture was then added to the cells in DMEM GlutaMax supplemented with10 % FBS and expression was carried out for 72h at 37°C and 5% CO 2 . The medium was replaced once with fresh DMEM GlutaMax with 10 % FBS after 48h. SNAP-tag labeling was achieved by addition of 2 mL DMEM GlutaMax supplemented with 900 nM BG-Cy3b and 300 nM BG-d2 and carried out for 1.5 hours at 37°C and 5 % CO 2 , followed by three washes with 5 mL PBS DPBS w/o Ca 2+ and Mg 2+ (Thermo Fischer Scientific, France) at ambient temperature. Then cells were detached mechanically using a cell scraper in DPBS and collected at 1000x g and 22°C for 5 minutes. Cells were resuspended in cold hypotonic lysis buffer (10 mM HEPES pH 7.4, cOmplete™ protease inhibitor (Sigma-Aldrich, France)), frozen and stored at -80°C. For the preparation of membrane fractions cells were thawed on ice and passed 30-times through a 200 µL pipette tip. After two rounds of centrifugation at 500 x g and 4°C for 5 min, the supernatant was aliquoted and centrifuged at 21,000 x g and 4°C for 30 min to collect crude membranes. The pellets were washed once with 20 mM HEPES pH 7.4, 118 mM NaCl, flash frozen in liquid N 2 and stored at -80°C.

  buffer without detergent and further dilution in acquisition buffer containing 0.0025 % LMNG, 0.0005 CHS tris, 0.0025 % GDN to achieve single molecule-compatible concentrations of labeled receptors (approximately 30-50 pM) in the absence of ligand or in presence of 100 µM LY34, 10 µM DN45 or 10 µM L-AP4. Apparent FRET efficiencies (E PR ) corrected for direct acceptor excitation and donor leakage into the acceptor channel were determined using the Software Package for Multiparameter Fluorescence Spectroscopy, Full Correlation and Multiparameter Fluorescence The docking of DN45 to mGlu4 VFT was performed with ZDOCK (19) as implemented in Discovery Studio (BIOVIA, Dassault Systèmes, Discovery Studio Modeling Environment, Release 2020, San Diego: Dassault Systèmes, 2020). After generating 54000 poses, a filtering was then used to keep only poses where CDRs from the nanobody DN45 were interacting with mGlu4. The distance criterion was 10 Å. The two first poses with the best ZRANK (20) score were in the same cluster, at the interface of the two lobes.

  Fig. S1. Agonist action of DN45 nanobody on VFT reorientation analyzed

Fig. S5 .

 S5 Fig. S5.Non-boned interactions as function of time for 10 ns molecular

  4, 118 mM NaCl, 1.2 mM KH 2 PO 4 , 1.2 mM

	MgSO 4 , 4.7 mM KCl, 1.8 mM CaCl 2 ) supplemented with 1 % (w/v) lauryl maltose
	neopentyl glycol (LMNG, Anatrace purchased through CliniSciences, France) + 0.2
	% (w/v) cholesteryl hemisuccinate tris salt (CHS Tris, Anatrace purchased through
	CliniSciences, France) for 5 min. Then Glyco-diosgenin (GDN, Avanti Polar Lipids
	purchased through Merck) was added (final detergent concentration 0.83 % LMNG +
	0.167 % CHS Tris + 0.83 % GDN) and the solution was centrifuged for 10 min at
	21,000 x g and 4°C. The supernatant was diluted 8.33-times in acquisition buffer and
	applied to a Zeba Spin Desalting Column (7 kDa cut-off, Thermo Fisher Scientific,
	France) equilibrated in acquisition buffer containing 0.005 % LMNG + 0.001 % CHS
	Tris + 0.005 % GDN and centrifuged 2 min at 1,500 x g and 4°C. The flow-through
	was then immediately diluted 10-times in cold acquisition buffer and kept on ice in the
	dark until use.

Table S4 .

 S4 Impact on the binding of DN45 to hmGlu4 and overall stability of folding of the complex upon virtual mutation of twelve human residues to corresponding rat residues at the end of 10 ns simulation.

				Mutation energy		Mutation energy
	# Human Rat		Effect		Effect
				binding (kcal/mol)		stability (kcal/mol)
	66	P	A	0.07	Neutral	0.54	Destabilizing
	224	V	L	-0.04	Neutral	1.22	Destabilizing
	245	D	N	0.12	Neutral	0.53	Destabilizing
	262	A	T	0.11	Neutral	-0.42	Neutral
	270	R	K	0.02	Neutral	0.06	Neutral
	280	A	G	0.09	Neutral	1.97	Destabilizing
	281	V	I	0.01	Neutral	-0.06	Neutral
	385	V	I	0.01	Neutral	-1.23	Stabilizing
	507	H	Q	0.01	Neutral	-0.3	Neutral
	318	I	S	1.55	Destabilizing	3.14	Destabilizing
	323	H	R	2.96	Destabilizing	2.92	Destabilizing
	485	D	G	0.69	Destabilizing	0.89	Destabilizing

Table S5 .

 S5 Predicted difference in binding energy of DN45 to rat mGlu4 upon mutation of 5 selected rat residues to corresponding human residues.

	# Rat Human	Mutation energy binding (kcal/mol)	Effect
	385 I	V	0.02	Neutral
	507 Q	H	0.02	Neutral
	318 S	I	-1.04	Stabilizing
	323 R	H	-2.52	Stabilizing
	485 G	D	-0.81	Stabilizing

Acknowledgements and funding sources

We thank the ARPEGE platform facilities at the Institut de Génomique Fonctionnelle for all fluorescence-based assays. J.-P.P. was supported by la Fondation pour la Recherche Médicale (ref. DEQ20170336747), Cisbio Bioassays (EIDOS collaborative team IGF-CISBIO, ref. 039293) and LabEx MAbImprove (ref. NR-10-LABX-5301). P.R. was supported by the Agence Nationale de la Recherche (ref. ANR-15-CE18-0020-01). J.-P.P., L.P. and P.R. were further supported by the Centre National de la Recherche Scientifique and the Institut National de la Santé et de la Recherche Médicale. P.C. and D.N. were supported by the Fond Unique Interministériel of the French government (FUI, Cell2Lead project). F.A. was supported by the Centre National de la Recherche Scientifique and the Science Ambassador Program from Dassault Systèmes BIOVIA. J.H. was supported by a fellowship from la Région Occitanie and Cisbio Bioassays (TransACT, ref. 156544) and la Ligue contre le cancer (PhD grant, ref. IP/SC-16487). J.F. was supported by la Fondation pour la Recherche Médicale (ref. DEQ20170336747). R.Q. was supported by a grant from the Agence Nationale pour la Recherche (ANR 18-CE11-0004-02). The CBS belongs to the France-BioImaging national infrastructure supported by the French National Research Agency (ANR-10-INBS-04, "Investments for the future").

Data availability

All data and associated protocols are available in the main paper and the SI Appendix and materials and methods upon request to J.-P.P.

Abbreviations

 (18), rat mGlu2-C1KKXX (7) and rat SNAP-mGlu4-C2KKXX (30)

Molecular modelling

Homology model of mGlu4 VFT

The model for the mGlu4 homodimer was built using MODELER (36) as implemented in Discovery Studio. Two PDB structures 6BSZ (Human mGlu8

Receptor complexed with glutamate at 2.65 Å resolution) and 6BT5 (Human mGlu8

Receptor complexed with L-AP4 at 2.92 Å resolution) were used as templates. A disulfide bridge was defined between the Cys 136 of the two loops. The ligand L-AP4 from 6BT5 was used when constructing the model. We generated 100 models and the one with the lowest probability density factor (PDF) and lowest discrete optimized protein energy (DOPE) (37) score was selected.

Homology model of DN45

We then built the homology model of DN45 with the Model Antibody Cascade protocol from Discovery Studio 2020. This protocol automatically identified most similar PDB structures to model the framework and used 5GXB, 5IMM, 4LGS and 5M2W. The protocol also identifies the best templates for each CDR loop. For loop H1, it used loop H1 from PDB structures 5LMJ and EJ1. For loop H2, the templates were the H2 loop from 6EHG, 4TVS. Finally, for loop H3, the templates loops where from 5LWF and 1U0Q.

Docking of DN45 to mGlu4 VFT


Eduardo Fancello

Luca Goglio
email: luca.goglio@polito.it

Laurent Stainier

Jakson Manfredini Vassoler

Identification of the strain rate parameters for structural adhesives

Keywords: Adhesives, elasto-viscoplasticity, strain rate

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

Introduction

In many applications of the adhesives, e.g. in the automotive and transportation fields, impact conditions are common or, however, they must be considered for design. Relevant applications are in the production of structural members for which the traditional joining techniques (welding, riveting) are replaced by adhesive bonding. In such cases, the adhesive itself must not only ensure a strength level no lower than that offered by the traditional techniques, but it can also be required to act as an additional medium for dissipating energy. An example of this is represented by the structural foams, recently available on the market, used to fill car body beams.

The virtual validation of the structure against crash conditions is carried out by Finite Element Method (FEM) simulations, in which the behavior of new materials such as adhesives must be correctly reproduced by means of models which must include their typical features, i.e. non-linear elastic and viscous behavior, and moreover the strain rate dependence should be accounted for. Clearly, in addition to their mathematical development, the models must be supported by experiments that should, on the one hand, allow for assessing the quantitative values of the related parameters, and, on the other hand, validate their capability to reproduce the reality. About this aspect, it must be remarked that, unlike the case of metals and other structural materials (for which many results are available), not many studies have been published regarding the strain rate dependence of adhesives under impact.

Regarding the simulation of the (non-linear) material behavior, considerable advance has been made in the last decade by the development of models under a variational formulation. The main advantages of such approaches are that: (i) they can be regarded under a unifying minimum principle, (ii) they are very flexible in the sense that they can accommodate a wide collection of material behaviors by varying the potentials assumed and (iii) they are suitable for error estimation. However, their application is not so widespread yet; thus there is a considerable interest in testing their capability to model practical cases. At the same time, there is a practical need to show how the material parameters appearing in the adopted formulae (potentials, etc.) can be identified from an experimental data base.

This paper is structured as follows. First, the theoretical formulation is briefly described in terms of mathematical quantities and principles involved (this is done also with the aim to give an essential outline to the reader not familiar with the variational formulation; clearly, the specialized literature must be consulted for a deeper insight). Second, the experimental tests and results are described. Then the choice of the models and the identification of the parameters are presented and the final conclusions are drawn.

Symbols

Mathematical Background

Principles of the Continuum Mechanics Formulation

To account for the non-linear behavior, the formulation is based on the typical quantities of continuum mechanics. First, the conservative aspect of the phenomenon is considered by assuming an energy potential [1], in the form W = W(F, F i , Q).In W,F = ∂x/∂X accounts for the total deformation, whilst F i represents the inelastic deformation only; they are related by the multiplicative decomposition F = F e F i in which the elastic part F e also appears [2]. Q contains other internal variables to account for hardening. The whole set of internal variables formally includes both F i and Q, aprioriconsidered as independent. Later, in the process of adapting the formulation to specific constitutive behaviors (viscoplasticity), we will introduce relations linking F i and Q (e.g. a flow rule in plasticity -Section 2.2.2). These relations should be seen as non-holonomic constraints between F i and Q, which have to be taken into account in the minimization process, and which in practice render F i dependent on the subset of internal variables Q. Thus, for the sake of clarity, we make the distinction between F i and Q from the onset. By taking the time derivative of W , which gives its rate of change, one obtains:

Ẇ = ∂W ∂F : Ḟ + ∂W ∂F i : Ḟi + ∂W ∂Q : Q. (1)
Second, the non-conservative aspects are considered by means of two pseudopotentials φ = φ( Ḟ), that accounts for the dependence of the material behavior on the total rate of deformation, and ψ = ψ( Ḟi , Q), that accounts for the dependence on the rate of inelastic deformation as well as on the rate of the internal variables.

It can be proven [3] that the following relationships hold:

P = ∂W ∂F + ∂φ ∂ Ḟ , (2)
∂W ∂F i + ∂ψ ∂ Ḟi = 0, ∂W ∂Q + ∂ψ ∂ Q = 0. (3a, b)
Equation (2) shows how the first Piola-Kirchhoff tensor can be expressed from appropriate potentials and equation (3a, b) define the restrictions on the evolution of the internal variables (originated from the self-equilibrium of the stresses conjugated to F i and Q).

On the basis of the previous mathematical properties, the essence of the variational formulation is the definition of the potential , based on the work performed by the Piola-Kirchhoff stress (which is energetically conjugated to the rate of the deformation gradient) in the time interval (0, t f ):

= t f 0 P : Ḟ dt. ( 4
)
In case of (hyper-)elastic material, the value of would depend on the final state only. Due to the inelastic behavior, depends also on the path followed during the evolution of the system. By minimizing the work, after several mathematical manipulations, it can be found that [4]:

δ = P(t f ) : δF(t f ). (5)
This allows for obtaining an expression for P:

P(t f ) = ∂ (F(t f ); F(0), F i (0), Q(0)) ∂F(t f ) , (6)
in which it is worth noting the dependence on all variables in the initial state (t = 0), and on F only in the final state (t = t f ). In incremental terms, from the state n to the state n + 1, equation ( 6) becomes:

P n+1 = ∂ (F n+1 ; F n , F i n , Q n ) ∂F n+1 . (7)
The incremental potential (F n+1 ; F n , F i n , Q n ) can be written in the form [3]:

F n+1 ; F n , F i n , Q n = tφ( • F ) + min F i n+1 ,Q n+1 W n+1 -W n + tψ • F i , • Q , (8)
where

• F(F n+1 ), • F i (F i n+1 ) and • Q(Q n+1 )
are, respectively, approximations of the rates Ḟ, Ḟi and Q. Equation (8) restates the constitutive equation ( 2) (from which P can be obtained) and, at the same time, satisfies the conditions (3) on the internal variables by means of the included minimization. The functions W,φ,ψ contained in (8) may assume different forms, according to the models adopted. The energy W is, in general terms, additively expressed as:

W = ϕ(F) + ϕ e (F e ) + ϕ i (F i , Q), (9)
where ϕ,ϕ e ,ϕ i are potentials related to the particular models adopted, as described in the following section.

Models and Parameters

In this work the material is described as elasto-viscoplastic, by means of the rheological model shown in Fig. 1. The outline of this particular elasto-viscoplastic model is also shown in [START_REF] Fancello | Proceedings of the IX International Conference on Computational Plasticity[END_REF]. The upper branch is purely elastic, and the associated potential is a function of the total deformation gradient ϕ = ϕ(F).T h el o we r (Maxwell) branch accounts for the inelastic behavior, since the inelasticity is viscoplastic, and the superscript i of the inelastic deformation gradient F i in the previous formulae is replaced in the following by p to represent the viscoplastic deformation gradient F p . The internal variables Q account for the modifications of the material due to plasticity. The energy potential of the model is written as: (10) where ϕ(F) and ϕ e (F e ) correspond to the elastic behavior of the springs, and ϕ p (F p , Q) accounts for the material strain hardening. In addition, the viscous behavior is accounted for by a dissipative potential ψ = ψ( Ḟp , Q) which explicitly depends on the rate of change of the internal variables. In this model no dependence is assumed on the rate of the total deformation gradient Ḟ,i.e.φ = φ( Ḟ) = 0.

W = ϕ(F) + ϕ e (F e ) + ϕ p (F p , Q) = ϕ(F) + ϕ e (FF p-1 ) + ϕ p (F p , Q),

Elastic Branch

In this branch, the potential ϕ(F) is split into an isochoric part, i.e., a function of the isochoric Cauchy-Green strain tensor Ĉ, and a volumetric part, i.e., a function of J , the Jacobian of F:

ϕ(F) = ϕ( Ĉ) + U(J). ( 11
)
Thus it is assumed that the whole volumetric contribution of the model is contained in U .

The explicit dependence on the Cauchy-Green strain holds because (as it is obvious) the energy is only related to material stretching (whilst F is also influenced by rigid body rotation), and thus it is possible to write:

ϕ( Ĉ) = ϕ(c 1 ,c 2 ,c 3 ), (12)
where c 1 ,c 2 ,c 3 are the eigenvalues of Ĉ.

The volumetric part is a function of the Jacobian only, and is written as:

U(J) = K 2 [ln(J )] 2 , ( 13
)
where K is a volumetric material constant. It is evident from equation ( 13) that in case of an isochoric deformation J = 1andU = 0.

Viscoplastic Branch

For this branch, the isochoric deformation gradient F is decomposed as follows:

F = Fe F p , (14)
in which Fe is the elastic deformation gradient, corresponding to the spring, and F p is the viscoplastic deformation gradient, corresponding to the friction element, which is intrinsically isochoric (det F p = 1). The viscoplastic rate of deformation tensor is assumed to coincide with the viscoplastic spatial gradient of velocity (i.e. the plastic spin is null), and so

D p = 1 2 (L p + L pT ) = L p = Ḟp F p-1 . (15)
From equation (15) the flow rule follows:

Ḟp = D p F p . (16)
The viscoplastic rate of deformation tensor is expressed by means of a spectral decomposition, with the additional assumptions of a von Mises-type plasticity model:

D p =q 3 j =1 q j m j ⊗ m j =q 3 j =1 q j M j , (17)
q j ∈ K Q = p j ∈ℜ, 3 j =1 p j = 0; 3 j =1 p 2 j = 3/2 , (18)
M j ∈ K M ={N j ∈ Sym; N j : N j = 1, N i : N j = 0,i = j }, ( 19
)
where q is the rate of accumulated plastic deformation, q j are linear combination coefficients, m j and M j are, respectively, the eigenvalues and the eigenprojections of D p . The conditions of isochoric deformation and the radius of the von Mises yield surface (set of scalars K Q ) are enforced by equation (18), while equation ( 19) enforces the eigenprojections to be symmetric orthogonal (set of second-order tensors K M ). The viscoplastic deformations are incrementally updated by means of an exponential mapping. The increments of viscoplastic deformation gradient and viscoplastic Cauchy-Green strain are respectively:

F p = F p n+1 F p-1 n = R exp[ tD p ], (20)
C p = ( F p ) T F p = F p-T n C p n+1 F p-1 n = exp[ tD p ] 2 , (21)
where the rotation increment R comes from the polar decomposition of F p . From equation ( 21) and considering equation ( 17) it follows that

1 2 ln( C p ) = tD p = t q 3 j =1 q j M j = q 3 j =1 q j M j , (22)
where q = t q. Substituting equation ( 22) in equation ( 20) it is found that

F p n+1 = exp q 3 j =1 q j M j F p n . (23)
Here the rotation R appearing in equation ( 20) has been disregarded since the constitutive expressions depend on the Cauchy-Green tensor in which the rotation terms are eliminated. The potentials of the viscoplastic branch can now be defined by means of the internal variables q,q j and M j only.

The elastic potential of the branch is written as a function of the eigenvalues of Ĉe , similarly to the case of equation (12):

ϕ e ( Ĉe ) = ϕ e c e 1 ,c e 2 ,c e 3 . (24)
The plastic potential ϕ p accounts for the strain hardening of the material. In case of isotropic hardening the potential is defined as

ϕ p = ϕ p (q), where q = t 0 q dt. ( 25
)
In incremental terms the accumulated plastic deformation can be written as:

q n+1 = q n + t q = q n + q. ( 26
)
The dissipative viscoplastic potential ψ depends on D p and, under the adopted hypotheses, on q only:

ψ(D p ) = ψ q t = ψ(q) = ψ(q), if q 0, +∞, if q<0, (27)
where in the first case ψ is a convenient formula for the potential, whilst the second case has the meaning of a penalization that constraints the rate q to be non-negative.

Adopted Models

Within this work the following models have been used in the potentials.

Elastic Potential

Two different models have been tested to account for the elastic behavior. The first is the Hencky model

ϕ e = μ e 3 j =1 ε e j 2 , (28)
in which μ e is a material constant, and ε e j the j th component of elastic logarithmic strain. Alternatively, also Ogden-type models have been tested:

ϕ e = 3 j =1 N e k=1 μ e k α e k exp ε e j α e k -1 , (29)
where μ e k and α e k are material constants, and N e is the number of terms included.

Plastic Potential

A general model has been tested in the form

ϕ p = 0 q + 1 2 Hq 2 + μ q + 1 α exp(-αq) + N p k=1 μ p k α p k + 1 (q) α p k +1 , ( 30
)
in which 0 is the yield stress and H is the plastic modulus (linear hardening), μ and α are material constants of the exponential law (saturation), and μ p k and α p k are material constants of the power law that includes N p terms. According to the coefficients included, equation (30) can account for many material models adopted in plasticity.

Viscous Potential

The Perzyna [5] potential

ψ(q) = ⎧ ⎨ ⎩ mY 0 q0 m + 1 q q0 (m+1)/m , if q 0, +∞, if q<0, (31)
has been adopted where m and Y 0 are material constants, q0 is a reference rate. In this work it is assumed that Y 0 coincides with the yield stress 0 .

Minimization at Each Increment

It is assumed that the potential ϕ( F) is null, thus only the volumetric contribution U(J) can be present in the elastic branch. The minimization problem (8) can be rewritten as

(F n+1 ) = (C n+1 ) = U (J n+1 ) + min q,M j ,q j ϕ e Ĉe n+1 + ϕ p (q n+1 ) + tψ q t , (32)
where:

U (J n+1 ) = U(J n+1 ) -U(J n ), (33)
ϕ e Ĉe n+1 = ϕ e Ĉe n+1 -ϕ e Ĉe n , (34)
ϕ p (q n+1 ) = ϕ p (q n+1 )ϕ p (q n ).

During each increment, the internal variables q j and M j must satisfy the conditions (18) and (19), respectively, and q 0. Following a procedure similar to that shown in [3], it is possible to show that the minimization with respect to M j implies co-linearity of Ĉe n+1 , Ĉpr = F p-T n Ĉn+1 F p-1 n and D p . The minimum conditions with respect to q, q j give the system of simultaneous non-linear equations:

- ∂ ϕ e ∂ε e i q + β 1 + 2β 2 q i = 0,i = 1, 2, 3, (36a,b,c) - 3 j =1 ∂ ϕ e ∂ε e i q j + ∂ ϕ p ∂ q + ∂ψ ∂ q = 0, ( 37
) 3 j =1 q j = 0, ( 38
) 3 j =1 q 2 j - 3 2 = 0, (39)
in which, besides the already mentioned variables q, q 1 ,q 2 , q 3 , also appear the Lagrange multipliers β 1 and β 2 which enforce the conditions of isochoric deformation and the radius of the von Mises yield surface. The system of equations is solved numerically. The particular case of uniaxial stress is treated without loss of generality. After minimization, the Piola-Kirchhof stress is evaluated as

P n+1 = 2F n+1 ∂ (C n+1 ) ∂C n+1 = F n+1 2J -2/3 n+1 DEV ∂ϕ ∂ Ĉn+1 + ∂ϕ e ∂ Ĉn+1 + t ∂φ ∂ Ĉn+1 + ∂U ∂J n+1 J n+1 C -1 n+1 , (40)
where DEV(A) = A -1 3 (A : C)C -1 . Assuming isochoric behavior (J = 1) the deformation gradient F and the Cauchy-Green strain C tensors become:

F = F = λ 00 01 / √ λ 0 001 / √ λ (41)
and

C = Ĉ = F 2 , ( 42
)
where λ is the stretch in the uniaxial loading direction.

Denoting σ = σ 11 , the only non-zero component of the Cauchy stress, the volumetric incremental relationship becomes

p n+1 = ∂U ∂J n+1 = σ 3 . ( 43
)
Substituting expressions (41), ( 42) and ( 43) in (40), and recalling the relationship σ = J -1 PF T between Cauchy and Piola-Kirchhoff stress tensors, it is found that

σ n+1 = C n+1 2DEV ∂ϕ ∂ Ĉn+1 + ∂ϕ e ∂ Ĉn+1 + t ∂φ ∂ Ĉn+1 + p n+1 C -1 n+1 . ( 44
)

Experimental

Test Configuration

The experimental tests, described in more detail in [START_REF] Goglio | [END_REF], were designed to investigate the influence of the strain rate on the material response over a wide range, from 1 × 10 -3 s -1 to 3 × 10 3 s -1 . The stress-strain curves were measured from specimens made entirely of the adhesive, to avoid the influence of the adherends and joint geometry on the results. Due to the very large range of strain rates considered, two different test facilities were needed: the tests at strain rates lower than 20 s -1 were carried out on a universal servo-hydraulic testing machine, while the tests at higher strain rates, up to 3.0 × 10 3 s -1 , were carried out on a Split Hopkinson Pressure Bar (SHPB). Traditionally, the use of the SHPB has been typical for compression tests on metals [START_REF] Kolsky | Stress Waves in Solids[END_REF], but in recent years several studies [START_REF] Hou | [END_REF][10][11][12] have considered other kinds of materials like polymers, foams, ceramics, etc. and have also tested under different stress states (tension, torsion, bending) by means of special set-ups. However, studies specifically concerning adhesives are relatively rare [13,14].

The standard SHPB configuration performs dynamic compression tests, usually on cylindrical specimens. Figure 2 shows the simplified sketch of a compression SHPB. The test starts when the impact of the striker bar against the input bar generates a compression pulse. The pulse propagates along the input bar, then it is partially reflected and partially transmitted through the specimen and the output bar. By measuring the reflected pulse on the input bar and the transmitted pulse on the output bar it is possible to reconstruct the dynamic stress-strain curve of the tested material by means of the following equations [START_REF] Kolsky | Stress Waves in Solids[END_REF]:

σ specimen (t) = E 0 A 0 A ε transmitted (t), (45)
ε specimen (t) =- 2c 0 L ε reflected (t) • dt, ( 46
)
εspecimen (t) =- 2c 0 L ε reflected (t), (47)
where A and L are, respectively, the cross-section and length of the specimen; A 0 ,c 0 and E 0 are cross-section, wave velocity and elastic modulus of the bar, respectively.

Among the cases of structural adhesives and curing methods tested in [START_REF] Goglio | [END_REF], the results chosen for this work are those on the bi-component, cold cured Loctite Hysol 9466 epoxy (Henkel, Düsseldorf, Germany). All specimens were made by injecting the adhesive into Teflon ® moulds and curing them at room temperature for 24 h. After removal they were aged for one week before testing. The specimens for static and dynamic compression tests were cylinders with a diameter of 10 mm. For the tests carried out on the hydraulic machine (on which the crosshead speed can be adjusted), the adopted length values were 10 mm for strain rates up to 0.5 × 10 1 s -1 and 4 mm for higher strain rates. The length values adopted to obtain the desired nominal (in the sense that the actual strain rate depends also on the intensity of the impact and can be assessed only after the test) strain rate on the SHPB varied from 4mm(3.0 × 10 3 s -1 )to10mm(1.5 × 10 3 s -1 ).

In [START_REF] Goglio | [END_REF] are reported also cases of tensile tests carried out on specimens with threaded ends. To test them at high strain rates, a modified version of the SHPB was used, in which the tensile pulse is created by means of a reflection system [12]. Due to the lower precision of this system, the results obtained were not suitable for the type of processing carried out in the present work, thus only results from compression tests are considered here. Moreover, the failures in these tension tests are -unlike the case of compression -substantially brittle and thus give little information about the inelastic behavior of the adhesive.

Results

Figure 3 shows the results of the compression tests for the cold-cured samples (two replicates for each case). The progressive growth of the curves under increasing strain rate can be noticed and the dynamic stress-strain curves present approximately the same shape as the static ones. A generic curve contains: an elastic initial part (linear with good approximation); a rounded knee; and a wide plateau, initially softening (at high strain rate) then progressively hardening. The shorter extension of the curves corresponding to nominal strain rate from 1.2 × 10 3 to 2.0 × 10 3 s -1 is due to the fact that, under the same displacement applied by the bar to the specimen end, the longer the specimen, the lower the total strain. The relevant corresponding

Table 1.

Nominal strain rate, actual mean strain rate, elastic limit, 2% proof stress (for each replicate the values are separated by "," in the box)

εnominal (s -1 ) εactual,mean (s -1 ) E (GPa) R p0.2 (MPa)
Static tests 1.0 × 10 -3 1.00 × 10 -3 , 1.00 × 10 -3 1.98, 2.20 -47, -47 1.0 × 10 -2 1.10 × 10 -2 , 1.00 × 10 -2 2.00, 1.99 -52, -53 1.0 × 10 -1

1.15 × 10 -1 , 1.13 × 10 -1 numerical data are reported in Table 1. As expected, the values of elastic modulus at high strain rate are in general slightly higher than those obtained statically or at low strain rate. Much more relevant is the increase in absolute value of the elastic limit with the strain rate, that changes from about -47 MPa in static condition to about -143 MPa at 3.0 × 10 3 s -1 .

Identification of the Parameters

The uniaxial case has been implemented in the MATLAB ® code (The MathWorks Inc., Natick, MA, USA), with the aim of building a tool able to determine the values of the different parameters that appear in the models by reproducing the experimental results. The advantage of doing so is that the computation time is much shorter than it would be required using an actual FEM code; moreover, no additional problem related to discretization (meshing, element formulation) is introduced. The identification of the parameters is based on a least squares algorithm that minimizes the difference between model response and measurements. The experimental curves are the average of the two replicates. For the elastic potentials both the Hencky and Ogden (N e = 1, 2, 3) models were tried. The plastic potential cases have been tested with H forced to be zero or left unconstrained, and with the number of terms in the power law N p = 1, 2, 3. The viscous potential is in all cases that of Perzyna. Table 2 gives a complete list of the tested combinations, for a total of 24 cases. The number of parameters simultaneously identified varies from 8 (elastic potential Hencky, case 1) to 18 (elastic potential Ogden N e = 3, case 6). The computation time is of the order of some hours per case on a 64 bit PC. For the sake of brevity only a small selection of the results from the 24 cases is reported here, in terms of the corresponding strain-stress diagrams and values of the parameters.

Regarding the role of the elastic potential, the use of the Ogden vs. the simpler Hencky model was tested to account for the possible non-linear elasticity. It was found that for this material the Ogden model did not offer any advantages. This is probably caused by the fact that the "S" shape of the curves is due to plasticity and not to hyperelasticity: the elastic behavior is substantially linear and, therefore, the Hencky model is already adequate.

Regarding the plastic potential, the tests have tried to identify the contributions of the different terms, in particular that of linear strain hardening (H ) and that of the Table 2. List of the 24 tested combinations of models for the elastic and plastic potentials

Elastic potential

Plastic potential

H = 0 H unconstrained Hencky N p = 1 N p = 2 N p = 3 N p = 1 N p = 2 N p = 3 C a s e 123456 Ogden N e = 1 N p = 1 N p = 2 N p = 3 N p = 1 N p = 2 N p = 3 C a s e 123456 Ogden N e = 2 N p = 1 N p = 2 N p = 3 N p = 1 N p = 2 N p = 3 C a s e 123456
Ogden N e = 3

N p = 1 N p = 2 N p = 3 N p = 1 N p = 2 N p = 3 C a s e 123456
Viscous potential: Perzyna in all cases.

power law. The yield 0 and the exponential part of the potential (more adequate to describe the knee) have always been included. In general, all cases have proven to perform satisfactorily, in the sense that the experimental curves are reproduced, even when a limited number of parameters are used. The major improvement that can be achieved, by increasing the complexity of the model, is obtained when going from cases 1, 2 and 4 (quite similar to each other) to cases 3, 5 and 6 (also similar to each other) that are able to reproduce -at least the trend -the softening. The general conclusion is that a model of reasonable complexity can reproduce the desired behavior; for instance, the case with linear strain hardening and N p = 1 already shows post-yield softening.

A sa ne x a m p l e ,t h ec a s e sH e n c k y#1( H = 0, N p = 1), Hencky # 5 (H unconstrained, N p = 2), and the most complex Ogden N e = 3#6(H unconstrained, N p = 3) are plotted in Figs 4, 5 and 6, respectively. Table 3 lists the corresponding parameters. The main difference among them is that the simplest case of Fig. 4 does not describe the stress peak and the subsequent slight softening, which are already (even if not completely) reproduced by the case in Fig. 5. It can be noticed from Fig. 6 that no significant improvement is added by the most complex case.

Regarding the viscous potential, it is interesting to notice how the implemented model accounts for the strain rate dependence by reproducing quite well the experimental curves, in a strain rate range that spans over six orders of magnitude (10 -3 -10 3 s -1 ). This justifies the choice of the potential adopted, although the Perzyna model had been initially proposed (and usually applied) for materials with linear strain hardening. The m coefficient in (31) remains approximately in the range 8-9 in all cases. Conversely, the reference rate q0 interacts with the other parameters of the simulation and is more subjected to changes from case to case. Considering the graphs presented in Figs 4, 5 and 6, the only objection that can be raised is that if one extrapolates the curves of the model towards higher strain, they all tend to superpose each other. This behavior, physically not plausible, is due to the mathematical structure of the potential. Nevertheless, in the strain range tested the model response is correct.

Conclusions

The variational formulation of the elasto-viscoplastic material model has been applied to the case of a structural bi-component epoxy adhesive, with the aim of reproducing its behavior under compression at different test velocities. The work has been carried out within a variational framework previously developed, that makes it possible to tackle the problem at general level, accommodating different models according to the specific potentials implemented. The natural use of this formulation is the implementation in non-linear finite element codes. However, in the present work it has been utilized analytically, due to the uni-axiality of the problem.

The main aspects which have been considered in this work are the choice of the potentials and the evaluation of the capability for reproducing the experiments. As far as elasticity is concerned, since this adhesive is stiff and linear elastic, a Hencky potential shows to be adequate. Regarding plasticity, the adopted potential -which combines the typical behaviors (linear hardening, saturation, power law) -is also able to reproduce material softening with a limited number of parameters. Finally, what is perhaps more remarkable is the capability of the adopted Perzyna potential to fit the curves at different strain rates, from quasi-static conditions to 103 s -1 .

  Main Quantities c j (j th) eigenvalue C right Cauchy-Green strain tensor D rate of deformation tensor F deformation gradient tensor J Jacobian of F L spatial gradient of the velocity tensor m material constant (viscous potential) m j (j th) eigenvector M j (j th) eigenprojection matrix N e ,N p number of parameters (elastic, plastic) P first Piola-Kirchhoff stress tensor R rotation matrix q accumulated plastic deformation Q internal state variable t time X reference configuration (undeformed) x present configuration (deformed) W energy potential α e k material constants (elastic potential) α, α p k material constants (plastic potential) β 1 ,β 2 Lagrange multipliers ε e j (j th) elastic logarithmic strain λ stretch μ e ,μ e k material constants (elastic potential) μ, μ p k material constants (plastic potential) 0 yield stress σ Cauchy stress tensor φ,ψ dissipative potentials incremental potential (to be minimized) Subscripts, Superscripts, Indices and Operators

Figure 1 .

 1 Figure 1. Rheological model for the elasto-viscoplastic behavior; ϕ potential of the elastic branch, ϕ e elastic potential of the Maxwell branch, ϕ p plastic potential and ψ viscoplastic potential.

Figure 2 .

 2 Figure 2. SHPB compression test configuration.

Figure 3 .

 3 Figure 3. Experimental compression test results on the Hysol 9466 at different strain rates (a and b denote the two replicates).

Figure 4 .

 4 Figure 4. Experimental (symbols) and simulated (lines) stress-strain curves at different strain rates; model Hencky # 1.

Figure 5 .

 5 Figure 5. Experimental (symbols) and simulated (lines) stress-strain curves at different strain rates; model Hencky # 5.

Figure 6 .

 6 Figure 6. Experimental (symbols) and simulated (lines) stress-strain curves at different strain rates; model Ogden N e = 3#6.

  Data in parentheses are affected by measurement inaccuracy but are reported for completeness only.

	Dynamic tests Hydraulic machine 5.0 × 10 0 5.0 × 10 -1 1.2 × 10 1 Dynamic tests 1.5 × 10 3 SHPB 2.0 × 10 3 3.0 × 10 3	1.92, 1.93 5.23 × 10 -1 , 5.11 × 10 -1 1.85, 1.90 4.83 × 10 0 , 4.98 × 10 0 2.01, 2.25 1.21 × 10 1 , 1.21 × 10 1 1.99, 1.95 1.49 × 10 3 , 1.48 × 10 3 (2.17, 2.24) -126, -122 -60, -61 -74, -75 -86, -86 -93, -95 2.01 × 10 3 , 2.00 × 10 3 (2.30, 2.09) -131, -126 3.04 × 10 3 , 2.99 × 10 3 (2.44, 2.30) -143, -142

Table 3 .

 3 Model parameters for the cases Hencky # 1, Hencky # 5, Ogden N e = 3#6

		Hencky # 1	Hencky # 5	Ogden N e = 3#6
	μ e 1 , μ e	759.17	775.76	15171
	μ e 2	-	-	988.17
	μ e 3	---1508.2
	α e 1	-	-	0.82038
	α e 2	-	-	0.85850
	α e 3	-	-	8.1714
	0	24.994	22.525	15.507
	H μ	-5.0436	-202.74 17.970	-342.23 32.330
	α	80798	115.01	75.979
	α α α μ p 1 p 2 p 3 p 1 μ p 2 μ p 3 q0 m	3.5871 --381.44 ----575.49 10.069 5.6594 1.4307 1.5239 -3.2143 2002.4 942.59 332.53 660.63 1.0628 × 10 -2 1.2147 × 10 -2 5.0476 × 10 -4 8.9796 8.4443 8.5219

Acknowledgements

Jakson Manfredini Vassoler is presently enrolled in his PhD course at the Politecnico di Torino under the sponsorship of the ALPIP (America Latina Plemonte Politecnico) interchange program.

Eduardo Fancello also thanks the Brazilian organization CNPq that provided partial financial support for this research.

Laurent Stainier is a Research Associate at the Belgian "Fonds de la Recherche Scientifique (F.R.S.-FNRS)".