Elisa Pieri
email: elisa.pieri@univ-amu.fr

Vincent Ledentu

Michael Sahlin

François Dehez

Massimo Olivucci

Nicolas Ferré
email: nicolas.ferre@univ-amu.fr

CpHMD-then-QM/MM identification of the amino-acids responsible for the Anabaena Sensory Rhodopsin pH-dependent electronic absorption spectrum

Introduction

The pH of the water phase is of crucial importance for the activity and stability of proteins, influencing a considerable amount of chemical properties, such as the tertiary structure, 1 the enzymatic activity 2 or the solubility. [START_REF] Franco | On the solubility of proteins as a function of pH: Mathematical development and application[END_REF] It relies on the fact that, on average, 29% of protein sequence is composed by titratable (also often termed as ionizable) amino acids (i.e. ASP, GLU, HIS, LYS, CYS, TYR and ARG), [START_REF] Jordan | A universal trend of amino acid gain and loss in protein evolution[END_REF] which can dynamically change their protonation state with the pH value. The loss/binding of a proton modifies the atomic charges on the side chains, triggering adjustments in the electrostatic interactions and conformational rearrangements. Consequently, the variation of the pH can induce major changes in the protein function. The understanding of the origin of such a pH-dependent behavior faces a great challenge: at a given pH, several protein residues can be in different protonation states (ie. they titrate) at the same time, making very difficult the identification of a single aminoacid as the main responsible for the pH-dependent property of interest. In other words, the protein has to be considered as a poly-acid macromolecule featuring tens of titrating sites which individual behaviors and their possible interactions (also allowed by water networks) shape the protein pH-dependency. [START_REF] Ullmann | Protein Modelling[END_REF] Nevertheless, the interpretation of the pH-dependency in terms of titrating sites is always appealing. As a matter of fact, such understanding opens the way to site-directed mutagenesis designed for altering the protein function in a controlled manner. [START_REF] Banci | pH-dependent properties of SOD studied through mutants on Lys-136[END_REF][START_REF] O'keefe | pH-dependent insertion of proteins into membranes: B-chain mutation of diphtheria toxin that inhibits membrane translocation, Glu-349--Lys[END_REF][START_REF] Digiammarino | A novel mechanism of tumorigenesis involving pHdependent destabilization of a mutant p53 tetramer[END_REF][START_REF] Horng | Analysis of the pH-dependent Folding and Stability of Histidine Point Mutants Allows Characterization of the Denatured State and Transition State for Protein Folding[END_REF][START_REF] Ormazabal | Histidine Residues in the Na+-coupled Ascorbic Acid Transporter-2 (SVCT2) Are Central Regulators of SVCT2 Function, Modulating pH Sensitivity, Transporter Kinetics, Na+ Cooperativity, Conformational Stability, and Subcellular Localization[END_REF][START_REF] Del Carmen Marín | Fluorescence Enhancement of a Microbial Rhodopsin via Electronic Reprogramming[END_REF] In this work, we identify the titrating residues responsible for the pH-dependence of the visible-light absorption spectrum of Anabaena Sensory Rhodopsin (ASR), a transmem-brane photochromic microbial protein whose photoactivity is carried out by the retinal chromophore. The retinal, present in the protonated Schiff base form, is linked to the lysine K210 and counterbalanced by the aspartate counterion D75 (Figure 1). Its predominant configuration in the ASR dark-adapted state is the all-trans (AT), while the 13-cis (13C) configuration is the major one in the light-adapted state. [START_REF] Sineshchekov | Photochromicity of Anabaena Sensory Rhodopsin, an Atypical Microbial Receptor with a cis-Retinal Lightadapted Form[END_REF] In this protein, the pH has the capacity of influencing the entire photochemistry, including absorption spectra, lifetimes and reaction dynamics. [START_REF] Tahara | PH-Dependent Photoreaction Pathway of the All-Trans Form of Anabaena Sensory Rhodopsin[END_REF][START_REF] Rozin | pH Dependence of Anabaena Sensory Rhodopsin: Retinal Isomer Composition, Rate of Dark Adaptation, and Photochemistry[END_REF] In particular, a small red-shift (2 nm in both isomers [START_REF] Rozin | pH Dependence of Anabaena Sensory Rhodopsin: Retinal Isomer Composition, Rate of Dark Adaptation, and Photochemistry[END_REF] ) in the maximum absorption wavelength (λ max ) has been experimentally found from pH 3.6 to 5.0, while a more consistent (6 nm for the all-trans isomer and 10 nm for the 13-cis one) blueshift is evidenced when the pH changes from 5.0 to 7.6 (Figure 2). The latter shift has been experimentally attributed putatively to the D217 deprotonation. While it is located about 15 Å from retinal, this aspartic acid is known as the retinal proton acceptor during the ASR photocycle, thanks to an extended hydrogen bonding network present in ASR. [START_REF] Shi | Cytoplasmic Shuttling of Protons in Anabaena Sensory Rhodopsin: Implications for Signaling Mechanism[END_REF] Accordingly, D217's change of protonation probably perturbs this network, modifying the electrostatic potential experienced by retinal and ultimately the ASR λ max . The cause of the shift occurring at the more acidic pH remains unknown.

The elucidation of the molecular origin of such small λ max shifts is not always easy from the experimental point of view, and can take benefit from multiscale atomistic models rooted in theoretical chemistry: quantum mechanical (QM), molecular mechanical (MM) or hybrid QM/MM). [START_REF] Hayashi | Structural Determinants of Spectral Tuning in Retinal Pro-teinsBacteriorhodopsin vs Sensory Rhodopsin II[END_REF][START_REF] Bucher | The Protonation State of the Glu-71/Asp-80 Residues in the KcsA Potassium Channel: A First-Principles QM/MM Molecular Dynamics Study[END_REF][START_REF] Rangarajan | Mechanism of Spectral Tuning in Green-Absorbing Proteorhodopsin[END_REF][START_REF] Frähmcke | The Protonation State of Glu181 in Rhodopsin Revisited: Interpretation of Experimental Data on the Basis of QM/MM Calculations[END_REF][START_REF] Vanni | A Conserved Protonation-Induced Switch can Trigger Ionic-Lock Formation in Adrenergic Receptors[END_REF][START_REF] Grante | Protonation effects on the UV/Vis absorption spectra of imatinib: A theoretical and experimental study[END_REF][START_REF] Driant | On the Influence of the Protonation States of Active Site Residues on AChE Reactivation: A QM/MM Approach[END_REF][START_REF] Molakarimi | QM/MM simulations provide insight into the mechanism of bioluminescence triggering in ctenophore photoproteins[END_REF][START_REF] Gozem | Theory and Simulation of the Ultrafast Double-Bond Isomerization of Biological Chromophores[END_REF][START_REF] Iijima | QM/MM Investigation for Protonation States in a Bilin Reductase PcyA-Biliverdin IXα Complex[END_REF][START_REF] Ling | Protonation state and fine structure of the active site determine the reactivity of dehydratase: hydration and isomerization of β-myrcene catalyzed by linalool dehydratase/isomerase from Castellaniella defragrans[END_REF] Most of the time, the pH effect is introduced thanks to very specific change of individual protonation states, without considering the relevance (probability) of such a chemical process at a given pH. In the present work, we will demonstrate that, indeed, the number of populated protonation states combinations (microstates) is always large. Accordingly, the corresponding ensemble has to be fully considered when small λ max shifts have to be reproduced and analyzed, as it is the case in the ASR absorption spectrum. Therefore, we have derived and used a molecular model which has been especially targeted towards the investigation of the ASR protonation microstates. For that purpose, we have chosen the Constant-pH Molecular Dynamics (CpHMD) approach which consists in sampling both the conformation and protonation state spaces. [START_REF] Swails | Enhancing Conformation and Protonation State Sampling of Hen Egg White Lysozyme Using pH Replica Exchange Molecular Dynamics[END_REF][START_REF] Huang | All-Atom Continuous Constant pH Molecular Dynamics With Particle Mesh Ewald and Titratable Water[END_REF] With respect to usual molecular dynamics simulations, CpHMD ones include Monte Carlo-based sampling of the system Figure 2: ASR experimental λ max values (in nm) between pH=3 and pH=8. Black (grey): AT retinal (13C retinal). Data taken from Rozin et al. [START_REF] Rozin | pH Dependence of Anabaena Sensory Rhodopsin: Retinal Isomer Composition, Rate of Dark Adaptation, and Photochemistry[END_REF] The red (blue) arrow indicates the pH region characterized by a λ max red-shift (blue-shift). Note the lack of data between pH=5 and pH=7. protonation states, weighted for the semigrand canonical ensemble (constant temperature, volume and hydronium chemical potential). The nature of the opsin-retinal interactions responsible for the tuning of the chromophore photochemical properties like λ max has been studied for many years. [START_REF] Hayashi | Structural Determinants of Spectral Tuning in Retinal Pro-teinsBacteriorhodopsin vs Sensory Rhodopsin II[END_REF][START_REF] Wanko | Calculating Absorption Shifts for Retinal Proteins: Computational Challenges[END_REF][START_REF] Fujimoto | Mechanism of color tuning in retinal protein: SAC-CI and QM/MM study[END_REF][START_REF] Altun | Spectral Tuning in Visual Pigments: An ONIOM(QM:MM) Study on Bovine Rhodopsin and its Mutants[END_REF][START_REF] Tomasello | Electrostatic Control of the Photoisomerization Efficiency and Optical Properties in Visual Pigments: On the Role of Counterion Quenching[END_REF][START_REF] Rajamani | The opsin shift and mechanism of spectral tuning in rhodopsin[END_REF][START_REF] Hasegawa | A Configuration Interaction Picture for a Molecular Environment Using Localized Molecular Orbitals: The Excited States of Retinal Proteins[END_REF][START_REF] Campomanes | Origin of the Spectral Shifts among the Early Intermediates of the Rhodopsin Photocycle[END_REF] Within recent years, new models have emerged in which the direct electrostatic interactions are found to be less important than inductive, dispersive, [START_REF] Guareschi | Electrostatic versus Resonance Interactions in Photoreceptor Proteins: The Case of Rhodopsin[END_REF] excitonic or charge transfer ones. [START_REF] Yanai | First-Order Interacting Space Approach to Excited-State Molecular Interaction: Solvatochromic Shift of p-Coumaric Acid and Retinal Schiff Base[END_REF] Nevertheless, in the context of the present study regarding pH-effects in ASR, in which most of the titratable amino-acids are not in the immediate vicinity of its chromophore (see Figure 1), we have based our model on a very qualitative, yet simple, QM/MM with electrostatic embedding approach. Such an approach has proven to be effective when one is looking for reproducing trends, eg due to rhodopsin mutations [START_REF] Pedraza-González | Automatic Rhodopsin Modeling with Chromophore Cavity Generation, Ionization State Selection, and External Counterion Placement[END_REF] or in other photoactive proteins. [START_REF] Udvarhelyi | Role of the Molecular Environment in Flavoprotein Color and Redox Tuning: QM Cluster versus QM/MM Modeling[END_REF] The successful reproduction of the ASR λ max variations with the pH validates such an effective model.

Methods and computational details

From the microscopic point of view, the large number of interacting titratable sites translates to the existence of an ensemble of protonation microstates whose respective populations are pH-dependent. [START_REF] Bombarda | pH-Dependent pKa Values in Proteins -A Theoretical Analysis of Protonation Energies with Practical Consequences for Enzymatic Reactions[END_REF][START_REF] Ullmann | pKa values and redox potentials of proteins. What do they mean?[END_REF] Consequently, macroscopic pK a values characterizing the acid-base properties of a macromolecule cannot be strictly attributed to independent and well-identified titrating amino-acid. Only the pK 1/2 , ie the pH value at which a given titrating site is equally populated in its protonated and deprotonated forms, keeps a physically-ground meaning.

Accordingly, a pH-dependent macroscopic property A can be evaluated as the average of individual contributions a i for each member of the ensemble containing N possible microstates, weighted by their respective probabilities w i :

A (pH) = N i=1 w i (pH) a i (1)
Because the size of the ensemble grows exponentially with the number of titratable sites, it is very often assumed that a unique microstate dominates the sum in Equation 1, being then used as a template for further atomistic modeling, and eventually allowing for the identification of the microscopic origin of the protein pH-dependency. However, the sum in Equation 1 can be rewritten in a different way, grouping in a cluster all the microstates sharing the same a i value (M being the number of clusters, M ≤ N ):

A (pH) = M j W j (pH) a j (2)
in which W j denotes the cumulated weight of a particular cluster, W j = i w i , ∀a i = a j .

There is no mathematical evidence that the most probable cluster (ie the largest W j ) coincides with the most probable microstate (ie the largest w i ). As a consequence, A may significantly deviate from the value a i characterizing the most probable protonation microstate.

In order to mitigate the bias introduced by the selection of a single microstate, some of the authors have recently designed a multiscale modeling protocol, denoted as CpHMD-then-QM/MM, which heavily relies on the statistical sampling of both the protonation state and conformational spaces, together with large scale (thousands) of QM/MM λ max evaluations.

This approach, successfully applied to the modeling of a polypeptide dyad, [START_REF] Pieri | Sampling the protonation states: the pH-dependent UV absorption spectrum of a polypeptide dyad[END_REF] is summarized hereafter (all details are given in SuppInfo).

System setup and CpHMD calculations. The ASR initial structure has been retrieved from the 1XIO PDB entry, 43 a dimeric system in which both the retinal AT and the 13C forms are present. Having cleaned the structure of a monomer (see SuppInfo for the details),

we have built two ASR models: (i) one solvated in water; (ii) one embedded in a membrane, modeled as a 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphoethanolamine (POPE) lipid bilayer.

Using the Amber16 package, 44 explicit solvent CpHMD simulations [START_REF] Swails | Enhancing Conformation and Protonation State Sampling of Hen Egg White Lysozyme Using pH Replica Exchange Molecular Dynamics[END_REF][START_REF] Swails | Constant pH Replica Exchange Molecular Dynamics in Explicit Solvent Using Discrete Protonation States: Implementation, Testing, and Validation[END_REF] have been carried out using the ff14SB Amber and TIP3P forcefields, [START_REF] Maier | Improving the Accuracy of Protein Side Chain and Backbone Parameters from ff99SB[END_REF] complemented with parameters for retinal. [START_REF] Hayashi | Structural Changes during the Formation of Early Intermediates in the Bacteriorhodopsin Photocycle[END_REF] Enhanced sampling has been achieved using the Replica-Exchange technique for pH values between 3.5 and 8.5, for a total simulation length of 20 to 30 ns (reduced sets of titrated residues) or 40 ns (full set of titrated residues) per pH value and per retinal isomer.

For the sake of simplicity, individual pK 1/2 values are simply called pK a in the following and are fitted to the calculated deprotonated fraction x i using the Hill equation:

pK a (i) = pH -n log x i 1 -x i (3)
in which the fitted n value represents cooperative effects between residue i and the other titrating residues.

Semi-empirical QM/MM calculations. Using 20000 to 40000 equally-spaced snapshots (geometries and protonation microstate) extracted from CpHMD trajectories obtained at pH values between 3.5 and 8.5, we have carried out QM/MM vertical excitation energy calculations of retinal, in electrostatic interaction with the rest of the system. In order to keep reasonable the computational cost, we have used the PM7 semi-empirical Hamiltonian, [START_REF] Stewart | Optimization of parameters for semiempirical methods VI: more modifications to the NDDO approximations and re-optimization of parameters[END_REF] together with the Multi-Electron Configuration Interaction 49 approach implemented in Mopac. [START_REF] Stewart | Stewart Computational Chemistry[END_REF] The absorption spectra at different pH values have been generated using the Newton-X 2.0 package. [START_REF] Barbatti | Newton-X: a surface-hopping program for nonadiabatic molecular dynamics[END_REF][START_REF] Barbatti | NEWTON-X: a package for Newtonian dynamics close to the crossing seam[END_REF] ARM calculations. Automatic Rhodopsin Modeling (ARM) [START_REF] Del Carmen Marín | Fluorescence Enhancement of a Microbial Rhodopsin via Electronic Reprogramming[END_REF][START_REF] Melaccio | Toward Automatic Rhodopsin Modeling as a Tool for High-Throughput Computational Photobiology[END_REF] calculations have been used to obtain accurate vertical excitation energies on a selected subset of protonation microstates. In short, ARM is a semi-automated protocol which starts from a PDB structure, builds a minimal rhodopsin model, samples the conformational space of the retinal cavity, before calculating vertical excitation energies using the high-level CASPT2 method. [START_REF] Finley | The multi-state CASPT2 method[END_REF] In its current stage of development, it only deals with a single protonation microstate.

Analysis tools. In order to decipher the relations between calculated λ max and the pH, we have mainly used two analysis approaches.

The first one is an improved version of the Minimal Electrostatic Model derived by some of us and already applied to the experimental ASR absorption spectrum. [START_REF] Stenrup | pH-Dependent absorption spectrum of a protein: a minimal electrostatic model of Anabaena sensory rhodopsin[END_REF] It is based on the sounded assumption that the λ max changes with the pH are governed by modified electrostatic interactions between the retinal chromophore and the protein protonation state changes. By fitting the λ max variations (actually, the related ∆E max variations) to the computed changes of the deprotonated fraction x i (which depends on the pH) for each of the N tit titrating residue i experiencing the pH-independent retinal transition electrostatic potential φ S 0 →S 1 ret (only the 3 terms E f ull , int and surf are optimized, all the other ones are constants or parameters, c i , φ S 0 →S 1 ret and r i which are derived independently, see SuppInfo):

∆E max (pH) = E f ull + N tit i=1 x i (pH) e 4π 0 [c i int + (1 -c i ) surf ] φ S 0 →S 1 ret (r i ) (4)
the contribution ∆E i (corresponding to the term in between the braces) of each individual titrating residue i can be obtained at all the considered pH values.

The second analysis has been done using multilinear regression analysis of λ max as a function of the protonation states for all the considered independent titratable residues. 56 Ultimately, this analysis provides correlations between λ max and individual (de)protonation events.

Results

Before explicitly considering the very large ensemble of the ASR protonation microstates, we first report accurate λ max values for a selected set of microstates, in order to illustrate how it can be difficult to pick one or few ASR models as candidates for explaining the origin of its pH-dependency.

ARM calculations of some individual microstate λ max

When one builds an atomistic model of a protein, one of the main assumptions concerns the protonation state of each and every titratable residue. This is usually based on the comparison of each microscopic pK a value with the targeted pH. It is then possible to roughly mimic a change of pH by modifying the protonation state of one or several titratable residues with respect to a reference microstate. We have applied this procedure to ASR λ max by means of ARM calculations (Figure 3). As expected, the largest λ max shift (> 120 nm) is due to protonation of retinal counterion, D75, keeping retinal as a protonated Schiff base.

Because this shift is out of the range reported experimentally (max 10 nm), we will keep D75 as deprotonated in the following.

The deprotonation of D217 (15.4 Å from retinal) induces a blue-shift of 13 nm, in quali-tative agreement with the experimental assignment. [START_REF] Rozin | pH Dependence of Anabaena Sensory Rhodopsin: Retinal Isomer Composition, Rate of Dark Adaptation, and Photochemistry[END_REF] Closer to retinal (14.2 Å), the deprotonation of D198 causes a much larger 41 nm λ max blue-shift. Simultaneous deprotonation of both D198 and D217 results in a blue-shift which is 32 nm with respect to the reference value, hence less than the one obtained by the single deprotonation of D198.

The smallest red-shift (2 nm) is due to protonation of H219. Accordingly, H219 could be a good candidate for explaining the ASR red-shift at acidic pH. This tiny effect is probably related to the large distance between H219 and retinal, more than 19 Å. The smaller distance between H21 and retinal (18 Å) results in a slightly larger red-shift (8 nm) upon H21 protonation. However, protonation of both H21 and H219 results in a 22 nm red-shift, larger than the ones resulting from each protonation taken separately.

In this configuration, H21 and H219 form an interacting pair in the ASR structure.

Because of the large electrostatic repulsion between two close positively charged moieties, this doubly-protonated situation has very little chance to occur in the considered pH range.

In other words, the corresponding protonation microstate is unlikely to be populated.

These ARM calculations result in accurate λ max values for each microstate. Accordingly, they can be used for predicting how λ max changes between two different microstates. However, they need to be complemented with the probability of such a process to take place at a given pH. For instance, it may occur that the large blue-shift caused by D198 deprotonation actually does not matter if this reaction is not likely to happen, because D198 pK a value could be significantly far from the pH range under study. Rephrased in a probabilistic language, they don't take into account the relative populations of D198 protonated and deprotonated forms at a given pH. The same remark applies to all the above-considered changes of protonation and this is the very reason why a correct ASR modeling protocol needs to include the determination of microstate populations. The latter are evaluated by means of CpHMD simulations.

3.2 Preliminary CpHMD-then-QM/MM study: effect of the membrane.

We have first applied the CpHMD-then-QM/MM protocol to an ASR model featuring 13C

retinal at three pH values: 3.5, 5.5 and 7.5, using 6 equally-spaced 40 ns trajectories obtained between pH=3.5 and pH=8.5. Because the pK a values of the 18 selected titrating residues (all aspartic and glutamic acids, all histidines) obtained from CpHMD simulations are known to be slowly converging with the system size and the number of titrating residues, [START_REF] Swails | Constant pH Replica Exchange Molecular Dynamics in Explicit Solvent Using Discrete Protonation States: Implementation, Testing, and Validation[END_REF] we have considered two models of ASR: in water and in a membrane solvated in water, using several 10, 20 or 40 ns windows (see the discussion in SuppInfo). As mentioned above, the retinal counterion D75 is always deprotonated. In all, our study demonstrates that most of the pK a values are converged after 20 ns, with the exception of D57, H69, D98, D120 and D198 in water, but only D57 and D98 in the membrane (see Figure 4). Indeed, the titration curve of the latter residue shows a chaotic behavior which can be attributed to its peculiar location.

It allows both a complete exposure to water and embedding in the membrane, so that the side chain can easily flip between the two opposite environments and experience completely different dielectric constants. As a matter of fact, the distance between the centers of mass of D98 and the closest phospholipid head changes sharply between 6 Å and 18 Å at all pH values (see SuppInfo). This behavior is responsible for the oscillating titration curve of D98, which would require prohibitively long simulations to achieve a good convergence. D57 belongs to a very flexible part of the protein, which was actually not resolved in the original 1XIO PDB structure. Accordingly, it is likely that its pK a value would converge using a larger sampling (see SuppInfo for the time evolution of its pK a value).

Even if the 40 ns trajectory length is relatively short to observe any major secondary or tertiary rearrangement, the ASR structural fluctuations are indeed smaller when the membrane is included into the model, as exemplified by the RMSD values reported in Table 1. The cavity in which the retinal chromophore lies keeps the same shape whatever the ASR environment. Its fluctuations, as well as those characterizing the retinal structure are reduced and independent of the pH value. The average distance between the centers of mass of the titrating residues is also independent of the pH (see SuppInfo). Moreover, the distances between titrating residues and retinal only change slightly, the maximum variations being observed for residues close or belonging to the reconstructed Q58 -Q66 missing loop (see SuppInfo). Accordingly, it is likely that the pH-dependence of ASR λ max is not primarily related to structural rearrangements involving the retinal cavity.

The importance of the membrane is striking when water-to-membrane pK a shifts are considered (Table 2). Besides D57 and D98 which are not discussed for the reasons explained above, 2 pK a differences are much larger than 1 pH unit, corresponding to titrating D125 and D198. Actually, it turns out that D198 is always deprotonated in the membrane, while it can exist in water within its protonated form at acidic pH. The cause of this behavior can be found in the distances between D198 and the closest titratable sites (E4, H8, H69).

For instance, the E4 -D198 distance is almost constant when ASR is embedded in the membrane (9.4 Å(±0.3)) while it fluctuates between 9.8 Å and 13.6 Å in its absence (see SuppInfo). The same behavior is found for H8, while it is reduced in the case of H69. In their protonated forms, these residues favor the deprotonation of D198, especially when they remain close to it. Eventually, D198 pK a becomes negative. Regarding D125, this residue is very close to the membrane in the acidic pH range (the distance between its carboxylate and the closest phospholipid head is 5 Å as an average); the low dielectric constant of the membrane environment favors the neutral form of D125. Hill factors are also reported in Table 2. Ideally, for a mono-acid without any interaction with other titrating sites, the Hill factor n = 1. [START_REF] Weiss | The Hill equation revisited: uses and misuses[END_REF] If n is very far from 1, it probably means that the associated pK a value has to be considered carefully, probably due to convergence issues (for instance D98 or D120). Besides this technical issue, it is expected that acidic sites are interacting with each other in a protein like ASR, eventually resulting in n values different from 1, as it is the case for the triad E4-H8-H69. This is also true for other titrating residues like H21-E36-E217-H219 which are close in space. Actually, in the latter case, H21 seems to behave as an independent titrating residue when ASR is not embedded in the membrane, resulting in a pK a value closer to the reference one for histidine (6.5).

These pK a changes with the nature of ASR surroundings have an impact on λ max values.

In Figure 5 we have reported the QM/MM λ max values computed in both water and the membrane environments. When calculated in the membrane, ASR λ max values are closer to the experimental ones. This result suggests an improved accuracy of our model when the protein is embedded in a lipid bilayer that resembles the physiological conditions. Between pH=3.5 and pH=5.5, we obtain a 0.6 nm red-shift, in agreement with the experimental 2 nm (note that the latter shift comes out of λ max values obtained at pH=5.0, not pH=5.5). [START_REF] Rozin | pH Dependence of Anabaena Sensory Rhodopsin: Retinal Isomer Composition, Rate of Dark Adaptation, and Photochemistry[END_REF] Between pH=5.5 and pH=7.5, our protocol predicts a 12.2 nm blue-shift, in excellent agreement with the experimentally reported 10 nm value. These two excellent results demonstrate that our CpHMD-then-QM/MM protocol is particularly well adapted to the study of ASR λ max pH-dependency. However, since we are titrating 18 residues at the same time, the analysis of its molecular origin is rather complex and cumbersome. Therefore, we decided to reduce the 3.3 Refinements using three pH windows.

CpHMD-then-QM/MM in each pH window. In order to select a minimal number of titrating residues in each pH window, we have applied a two-step filtering procedure to our 13C ASR model:

1. A titratable residue is not titrated in a window if its deprotonated fraction within the pH range is lower than 5% or larger than 95%, according to our preliminary 40 ns CpHMD calculations.

2. Each list of titrating residues is then complemented with titratable residues whose protonation states show a large interaction with at least one of the members of the list.

All details are given in SuppInfo. The number and the list of titrating residues in each pH window are reported in Figure 6. Four residues (E4, D57, D98, E123) are included in all the three windows, due to their irregular or very flat titration curves (see SuppInfo). Some other ones belong to two consecutive pH windows (eg H69 which is titrating between pH=4.5 and 7.5).

Figure 6: Selection of the titrated residues in the three narrow pH windows 3.0-4.5, 4.5-6.0 and 6.0-7.5. In a given window, the protonation state of the other titratable residues is fixed and decided according to their deprotonated fractions.

Each pH window is treated independently of the other ones. As a consequence of their different numbers of titrating residues, and the resultant fixed protonation state of the nontitrated amino acids, the 3 different potential energy surfaces yield two λ max discontinuities at pH=4.5 and pH=6.0 (Figure 7). At first glance, our CphMD-then-QM/MM protocol is able to catch the qualitative features of λ max variations induced by the considered pH variations, even using a reduced number of titrating residues in each window. As a matter of fact, our results show a redshift between pH=3 and pH=5, and then a blue-shift between pH=5 and pH=7.5. As shown in Figure 7 and confirmed by a bootstrapping analysis (see SuppInfo forthe details), the λ max convergence in each window is improved by increasing the CpHMD sampling, while keeping reasonable the consumed computational resources.

Global electrostatic analysis of the ASR λ max pH-dependency. The retinal chromophore and its counterion (D75) keeping the same protonation states, the overall ASR structure being conserved at all pH values, the pH-dependency of the ASR absorption spec-trum is probably due to pH-dependent electrostatic interactions between titrating residues and retinal. In order to test this hypothesis, we have calculated the average external potential felt by the retinal atoms for each considered pH value, as reported in Figure 8a. Two main trends can be highlighted. The external potential felt by the Schiff base is very small at acidic pH and becomes more negative when the pH increases. Conversely, the external potential felt by the β-ionone ring is very positive at acidic pH and decreases to reach small negative intensity at pH=7.5. Remembering that the retinal positive charge, mainly located in the Schiff base region in the electronic ground state (S 0 ), is translocated towards the β-ionone ring when retinal is excited to S 1 , [START_REF] Demoulin | Intramolecular photo-induced charge transfer in visual retinal chromophore mimics: elec-tron density-based indices at the TD-DFT and post-HF levels[END_REF] we can immediately infer that both S 0 and S 1 are more and more stabilized when the pH increases from acidic to neutral (Figure 8b).

Of course, the pH-induced stabilizations of each electronic state have no reason to be equal.

Accordingly, we can rationalize the acidic λ max red-shift by a slightly larger stabilization of S 1 than S 0 . Conversely, the λ max blue-shift between pH=5 and 7.5 can be attributed to a larger stabilization of S 0 than S 1 .

As a final check of the electrostatic origin of the pH-dependence of ASR 13C absorption spectrum, the λ max values at pH=3, 5 and 7 have been computed in the absence of the external potential in the PM7 calculations, keeping frozen the retinal structures extracted from the corresponding CpHMD trajectories. Whatever the pH, retinal λ max is 632 nm (±0.4), ie red-shifted by almost 100 nm, in agreement with experimental maximum absorption wavelength of retinal in vacuo. [START_REF] Knudsen | The UV-visible action-absorption spectrum of all-trans and 11-cis protonated Schiff base retinal in the gas Phase[END_REF] Deciphering the molecular origin of the ASR 13C λ max pH-dependence. Correlations between computed QM/MM vertical excitation energies and the individual protonation states of all the titrating residues have been determined for each pH window, using MLR analysis. First, it must be remembered that using three different pH windows with different sets of titrating residues results in exploring three different potential energy surfaces. As a consequence, some MLR coefficients are significantly different in different windows. For instance, D57 deprotonation induces a red-shift in both the acidic and intermediate windows and a tiny blue-shift in the neutral one. However, in the latter window, its deprotonated fraction is larger than 90%, casting doubts on the statistical relevance of the blue-shift. In H69 in the acidic one, despite the spatial proximity of these three amino-acids.

In the acidic window (pH=3.0-4.5), deprotonated fractions of D120 and D57 are 47% and 38% respectively. Both titrating residues contribute to λ max variations with a red-shift.

Three blue-shifting contributions are due to E4, D226 and E160, all with a deprotonated fraction comprised between 15 and 20%. Accordingly, we propose D120 as the main molecular origin of λ max red-shift. This residue is located at about 8 Å from the retinal β-ionone ring.

Accordingly, upon photo-excitation, the negative charge of the D120 deprotonated form will stabilize the retinal excited state, whose positive charge moves away from the Schiff base in S 1 with respect to S 0 , ultimately inducing the reported λ max red-shift.

In the middle window (pH=4.5-6.0), many important contributions are found, either red-shifting (D57, H69, E123, D125) or blue-shifting (D217, D226) ASR 13C retinal λ max .

This complicated situation reflects the large number of titrating amino-acids in this window, i.e. many carboxylic acids and histidines showing deprotonated fractions comprised between 20% and 70%. Accordingly, the presence of the λ max peak can be roughly rationalized using the different stabilizations of both the retinal ground and excited state energies, as already presented above.

In the neutral window (pH=6.0-7.5), D217 gives the largest blue-shifting contribution, while being mainly deprotonated. Other residues significantly contribute to the blue-shift: H8, E36, H219. Residues H69, D98, E123, D125 induce a red-shift, however smaller. With the exception of H8, all the blue-shifting residues are spatially close and are possibly interacting, also with H21. Accordingly, we propose that the titrating residues H21, E36, D217 and H219 behave as a single acid-base cluster, being characterized by the experimental pK a = 6.4 value. [START_REF] Rozin | pH Dependence of Anabaena Sensory Rhodopsin: Retinal Isomer Composition, Rate of Dark Adaptation, and Photochemistry[END_REF] Confirmation of the electrostatic origin using the Minimal Electrostatic Model.

Recalling that the Minimal Electrostatic Model is based on the assumption that pH-effects are mainly electrostatic (Equation 4), we have used different structures extracted from the CpHMD trajectories in the acidic and neutral pH windows and fitted the calculated λ max to the deprotonated fractions of each titrated residue. As expected from the low structural fluc-tuations highlighted above, the results of the present analysis are qualitatively independent of the selected molecular model (see SuppInfo). In Figure 10, we present the energetic contribution of each titrating residue as a function of pH. In the acidic window, D120 is clearly the most λ max red-shifting residue. In the neutral window, D217 contributes the most to the λ max blue-shift. However, H8, H21, E36 and H219 also contribute to the blue-shift while H69, E123 and D125 are found to red-shift λ max . All these results are in line with the ones presented in the previous section and confirm the electrostatic origin of the pH effects on ASR λ max . pH=5, then blue-shift. Moreover, the ordering of 13C and AT λ max is correctly reproduced, even if the energy gap between the two retinal conformations is too small with respect to the experimental data, probably due to the qualitative QM level of theory. Nevertheless, the same convergence issues are met in both AT and 13C isomers.

Conclusions

In this article, we have reported the successful application of our CpHMD-then-QM/MM protocol to the study of the pH-dependent ASR absorption spectrum between pH=3 and pH=7.5.

The main result is the identification of the titrating residue responsible for the λ max redshift in acidic conditions, namely D120 which was never suggested in the available literature.

We have also confirmed the implication of D217 in the λ max blue-shift between pH=5 and pH=7.5. However, we have found out that other residues, close to D217, are also involved.

Accordingly, we suggest that the experimental pK a (about 6.5) has to be attributed to a cluster of titrating residues located on the ASR cytoplasmic side, instead of D217 alone.

Overall, it is demonstrated that pH-induced λ max tuning in ASR can be reproduced by means of simple opsin -retinal electrostatic interactions and achieved by residues located far from its retinal chromophore, as seen in figure 12. More involved and challenging studies would be needed in order to understand the very nature of these long-range interactions. [START_REF] Hasegawa | A Configuration Interaction Picture for a Molecular Environment Using Localized Molecular Orbitals: The Excited States of Retinal Proteins[END_REF][START_REF] Guareschi | Electrostatic versus Resonance Interactions in Photoreceptor Proteins: The Case of Rhodopsin[END_REF][START_REF] Yanai | First-Order Interacting Space Approach to Excited-State Molecular Interaction: Solvatochromic Shift of p-Coumaric Acid and Retinal Schiff Base[END_REF] This finding suggests that the ASR structure is actually designed for reducing as much as possible the effect of pH on its absorption spectrum, and maybe on other photochemical properties.

In principle, the methodology used in this work could be applied to understand the origin of the pK a =8.5 value experimentally reported for the 13C retinal isomer only. [START_REF] Rozin | pH Dependence of Anabaena Sensory Rhodopsin: Retinal Isomer Composition, Rate of Dark Adaptation, and Photochemistry[END_REF] However, major complications arise. First, at this pH value, other titratable residues can enter into the game: lysine and maybe some cysteine or tyrosine amino-acids. Second, it would be necessary to use a different starting structure (the one used in the current work, 1XIO, was obtained at pH=5.6), for instance the NMR-based 2M3G structure 60 obtained at pH=9.

On the methodological side, we have shown evidence that atomistic models based on a single protonation microstate are not relevant when pH-dependent properties are investigated. As a matter of fact, the predominant microstate represents less than 5% of the populated microstates, even at pH=7.5 which is the most favorable case in the case of ASR.

The CpHMD-then-QMMM protocol remains a very expensive tool, requiring a lot of computational resources. Accordingly, it is important to reduce the number of titrating residues, using a screening procedure based on the determination of correlations between the targeted property and the individual (de)protonation events. The minimal electrostatic model [START_REF] Stenrup | pH-Dependent absorption spectrum of a protein: a minimal electrostatic model of Anabaena sensory rhodopsin[END_REF] endorses the electrostatic origin and the identification of the microscopic sources of the ASR pH-dependent absorption spectrum.

Figure 12: ASR representation featuring its retinal chromophore (RET), its counterion (D75) and its most important titrating residues. D120 (mainly) and D57 are responsible for the λ max red-shift between pH=3 and pH=5. H8, H21, E36, D217 and H219 are collectively responsible for the λ max blue-shift between pH=5 and pH=7.

Figure 1 :

 1 Figure 1: Schematic representation of the ASR model, featuring (i) retinal chromophore bonded to lysine 210 and (ii) 18 titratable residues between pH=3 and pH=8: aspartic acids (1-letter code D) in red; glutamic acids (1-letter code E) in yellow, histidines (1-letter code H) in blue.

Figure 3 :

 3 Figure 3: λ max values (in nm) of several ASR models, using the ARM protocol. The REF model corresponds to the most likely protonation microstate at pH=7. The 3-letter code for each amino-acid has been chosen to unambiguously denote each change of protonation state on the y-axis, as follows: initial protonation state-amino-acid number-final protonation state.

Figure 4 :

 4 Figure 4: ASR structure with 13C retinal, bonded to K210. Retinal counterion is D75. Membrane phospholipid headgroups are represented as transparent surfaces. Other 4 titrating residues (D57, D98, D125, D198) are discussed in the text.

Figure 5 :

 5 Figure 5: ASR (13C retinal) QM/MM (using 40000 structures extracted from 40 ns CpHMD trajectories) and experimental (black line) λ max values in nm as a function of the pH; the red line represents the calculations of ASR embedded in a membrane, while the blue line stands for the results of ASR in water.

  complexity of the problem by studying separately three smaller pH windows, each featuring a different λ max behavior: red-shifting in the acidic 3.0 to 4.5 pH range, showing a maximum in the intermediate 4.5 to 6.0 pH window, and blue-shifting in the neutral 6.0 to 7.5 pH window.

Figure 7 :

 7 Figure 7: Experimental (dashed line) and computed (full lines) ASR 13C λ max (in nm) variation with the pH, increasing the CpHMD trajectory length: 10 ns in yellow, 20 ns in red, 30 ns in blue.

Figure 8 :

 8 Figure 8: (a): Average external electrostatic potential felt by the retinal atoms, from pH=3.0 to pH=7.5. From blue (positive potential), to red (negative potential). (b): Stabilization/destabilization of the retinal ground (S 0 ) and excited (S 1 ) states triggered by either a positive external potential (left, in blue) in the β-ionone ring region and/or a negative external potential (right, in red) in the protonated Schiff base region, with respect to the reference situation in which no external potential is applied (middle).

Figure 9 :

 9 Figure 9: Left axis: individual contributions to λ max variations (bars, in nm). Right axis: deprotonated fraction of each titrating residue (black dots). Top: acidic window, middle: intermediate window, bottom: neutral window.

Figure 10 :Figure 11 .Figure 11 :

 101111 Figure 10: Minimal Electrostatic Model energy contributions of each important residue in the acidic and neutral pH windows. Negative (resp. positive) contribution translates to λ max red-shift (resp. blue-shift).

Table 1 :

 1 Root-mean-square deviation (RMSD, in Å) with respect to the original crystallographic structure, without (Water column) and with the membrane (Membrane column), averaged over the 3 pH values 3.5, 5.5, 7.5 (individual pH RMSD values are available in SuppInfo).

	RMSD	Water Membrane
	All	3.14	2.03
	Backbone 2.36	1.27
	Cavity	1.22	1.04
	Retinal	1.03	1.03

Table 2 :

 2 pK a values calculated fitting the Hill equation (3) using the deprotonated fractions retrieved from 40 ns long CpHMD trajectories (13C isomer) in water and in membrane.

		Water		membrane
	Residue pK a	n	pK a	n
	H8	7.07 0.63 6.15 0.88
	H21	6.27 0.99 7.21 0.73
	H69	5.43 0.75 6.05 0.83
	H219	5.70 0.84 6.70 0.49
	E4	4.89 0.78 5.27 0.60
	E36	6.63 0.75 7.56 0.83
	E62	4.83 0.86 4.72 0.83
	E123	4.52 0.82 5.19 0.71
	E160	4.77 0.81 4.92 0.90
	D57	3.84 0.73 4.94 0.43
	D75	<0	-	<0	-
	D98	2.94 0.76 3.75 0.18
	D120	5.30 0.69 2.05 0.36
	D125	3.66 0.92 6.28 0.73
	D166	<0	-	<0	-
	D198	3.28 1.44 <0	-
	D217	5.99 0.59 6.05 0.73
	D226	4.71 0.84 4.51 0.88

Acknowledgement

EP, VL, MS and NF thank the French Agence Nationale de la Recherche for funding (grant

Supporting Information Available

Computational details, CpHMD convergence analysis (titration curves, structural modifications), details regarding the 3 pH windows (selection, distance distributions, bootstrapping analysis, multilinear regression analysis, minimal electrostatic model analysis). This material is available free of charge via the Internet at http://pubs.acs.org/.


R Darleux
email: robin.darleux@lecnam.net

B Lossouarn

J.-F De

Effect of temperature on the tuning of a piezoelectric resonant shunt composed of variable inductance or variable capacitance

The piezoelectric resonant shunt is an electromechanical device that can significantly damp mechanical vibrations if it is finely tuned. However, temperature may influence the electrical components of the shunt and the mechanical parameters of the vibrating structure, and deteriorate the damping performance as a consequence. The objective of this work is to describe how inductive and capacitive components can be chosen so that the shunt tuning is maintained in the case of temperature variations. Two solutions of fully passive adaptive shunts are developed, tested and compared. The first solution includes a variable inductance. In contrast, the second solution includes a variable capacitance and intends on keeping the shunt inductance as constant as possible. Experiments validate the concept of a resonant shunt that autonomously adapts to temperature variations.

Introduction

Mechanical vibrations can be mitigated by bonding piezoelectric patches on the surface of a vibrating structure, as it allows transferring part of the vibration energy to an electrical circuit where it can be dissipated in a resistive component. A well documented electrical circuit is the resonant shunt [START_REF] Hagood | Damping of structural vibrations with piezoelectric materials and passive electrical networks[END_REF], which is the electromechanical equivalent of a tuned mass damper (also called dynamic vibration absorber). The vibration damping induced by this device is mainly owed to a precise tuning of the electrical parameters [START_REF] Thomas | Performance of piezoelectric shunts for vibration reduction[END_REF][START_REF] Park | Sensitivity analysis of damping performances for passive shunted piezoelectrics[END_REF][START_REF] Berardengo | Vibration Control by Means of Piezoelectric Actuators Shunted with LR Impedances: Performance and Robustness Analysis[END_REF]. Besides, electrical parameters and mechanical characteristics of the system may strongly evolve with environmental parameters such as temperature. Several studies have highlighted the dependence in temperature of piezoelectric properties [START_REF] Hooker | Properties of PZT-based piezoelectric ceramics between -150 • C and 250 • C[END_REF][START_REF] Miclea | Effect of temperature on the main piezoelectric parameters of a soft PZT ceramic[END_REF]. While solutions have been proposed for an online tuning of the resonant shunt, they are either limited to a few decibels of vibration damping [START_REF] Niederberger | An autonomous shunt circuit for vibration damping[END_REF], or active solutions that require external power [START_REF] Hollkamp | A Self-Tuning Piezoelectric Vibration Absorber[END_REF]. Recently, the choice of the magnetic material of the considered inductor has been highlighted as a way to passively counterbalance the tuning variations of a resonant shunt [START_REF] Darleux | Passive self-tuning inductor for piezoelectric shunt damping considering temperature variations[END_REF].

The objective of this work is to validate the concept of an autonomous resonant shunt, whose tuning is maintained in case of temperature variations. To do so, an electromechanical model of the structure is developed in section 2. Two solutions, denoted solutions A and B, are proposed: the adapting component is a variable inductance in the solution A, while it is a variable capacitance in the solution B. Then in section 3, we describe the experimental setup of a vibrating cantilever beam covered with a pair of piezoelectric patches. From there, the design of electrical components for both solutions is performed. Finally, experimental results in section 4 show that both solutions sustain the tuning up to at least 50 • C. In the end, the concept of a reliable and fully passive resonant shunt adpating to temperature variations is validated.

-F -N eV V U e U q 1/K E C ε Z m e : 1

Piezoelectric shunt damping

Piezoelectric shunt damping consists in bonding a vibrating structure with piezoelectric transducers, and connecting them to an electrical circuit. The electrical energy transferred to this circuit is dissipated in a resistive component, and mechanical vibrations are mitigated as a consequence. In this article, we consider the case of two piezoelectric patches covering opposite faces of a cantilever beam. Three stages are needed to describe the dynamics of the system: developing an electromechanical model of the piezoelectric transducer, developing a mechanical model of the structure, and defining the shunt components.

We propose a one-dimensional model of a piezoelectric transducer:

N = K E U -eV, q = C ε V + eU, (1)
where N is the normal force applied to the transducer, q is the electric charge on an electrode, U is the displacement in the longitudinal direction, and V is the voltage between the electrodes of the piezoelectric device. The constants e, K E and C ε are the coupling coefficient, the structural stiffness when the piezoelectric transducer is short-circuited (i.e. when V = 0), and the piezoelectric capacitance under no displacement (i.e. when U = 0), respectively. The mechanical structure is then modeled as an undamped one degree-offreedom system, whose dynamics can be expressed by

m Ü + K D U = F + e C ε q, ( 2
)
where m is the mass, F is the excitation force, and

K D = K E + e 2 /C ε is the stiffness in open-circuit.
Equation (2) highlights the equivalent mechanical quantities on its left side, and the external forces applied to the structure on its right side. Finally, the structure is shunted with an impedance Z. Ohm's law applied to the shunt impedance gives

V = -Z q. ( 3
)
The use of a direct electromechanical analogy, summed up in Table 1, allows to define the equivalent electrical circuit of the electromechanical system based on equations (1), ( 2) and (3). It is represented in Figure 1. These equations can also be combined to obtain the dimensionless transfer function between the displacement and the excitation at angular frequency Ω:

U F/K D = 1 + jΩC ε Z 1 -Ω Ω O 2 (1 + jΩC ε Z) - k 2 c0 1+k 2 c0 , (4)
with j 2 = -1. In this last equation, k c0 is the coupling factor, and Ω O and Ω S are the mechanical resonance angular frequencies in open-circuit (i.e. when q = 0) and in short-circuit (i.e. when V = 0), respectively. These quantities are defined by

Ω O = K D m , Ω S = K E m , k c0 = Ω 2 O -Ω 2 S Ω 2 S . (5
Z = R + jΩL. ( 6
)
The transfer function in equation ( 4) then becomes

U F/K D = 1 + 2j ξe Ωe Ω -Ω Ωe 2 1 -Ω Ω O 2 1 + 2j ξe Ωe Ω -Ω Ωe 2 - k 2 c0 1+k 2 c0 , (7)
with Ω e the electrical resonance angular frequency and ξ e the electrical damping defined by

Ω e = 1 √ LC ε , ξ e = R 2
C ε L . (8)
The main drawback of the resonant shunt is the need for a precise tuning [START_REF] Thomas | Performance of piezoelectric shunts for vibration reduction[END_REF][START_REF] Park | Sensitivity analysis of damping performances for passive shunted piezoelectrics[END_REF][START_REF] Berardengo | Vibration Control by Means of Piezoelectric Actuators Shunted with LR Impedances: Performance and Robustness Analysis[END_REF]. The transfer function criterion detailed in [START_REF] Thomas | Performance of piezoelectric shunts for vibration reduction[END_REF] states that the optimal values of these electrical parameters are:

(Ω e ) opt = Ω O , (ξ e ) opt = 3 8 k c0 . (9)
The frequency response function (FRF) in equation ( 7) is represented in Figure 2. Variations of the electric damping ξ e are not considered since it has little influence on the vibration mitigation when compared to Ω e variations [START_REF] Thomas | Performance of piezoelectric shunts for vibration reduction[END_REF][START_REF] Park | Sensitivity analysis of damping performances for passive shunted piezoelectrics[END_REF][START_REF] Berardengo | Vibration Control by Means of Piezoelectric Actuators Shunted with LR Impedances: Performance and Robustness Analysis[END_REF]. Figure 2 shows that the damping performance drops as soon as the electrical resonance differs from its optimal value: for k c0 = 0.12, a difference of 10 % between Ω e and its optimal value adds nearly 10 dB to the transfer function peak. Hence, the tuning condition in equation ( 9) should be verified in any situation to maintain the vibration damping. However, mechanical and electrical parameters of the system may vary when temperature changes. If the evolutions with temperature of the mechanical resonance and the piezoelectric capacitance are given data, then the inductance is the control parameter. That means the value of L should be adapted so that it always verifies

Ω 2 O (T ) = 1 L (T ) C ε (T ) . (10)
In this work, we suppose that Ω O , C ε and L evolve linearly over the considered temperature range, and the associated temperature coefficients are noted α Ω O , α C ε and α L , respectively. Room temperature is noted T 0 . The evolutions with temperature of these quantities can be written as follows:

Ω O (T ) = Ω O (T 0 ) [1 + α Ω O (T -T 0 )] , C ε (T ) = C ε (T 0 ) [1 + α C ε (T -T 0 )] , L (T ) = L (T 0 ) [1 + α L (T -T 0 )] . (11)
A linear approximation of equation [START_REF] Legg | Magnetic Measurements at Low Flux Densities Using the Alternating Current Bridge[END_REF] gives the relation between the temperature coefficients that should be verified to keep the resonant shunt tuned:

α L = -(2α Ω O + α C ε ) . (12)
This solution for a passive temperature adjustment of the resonant shunt tuning, denoted solution A, requires to design the inductor according to temperature characteristics of the system. A limit of this solution would be that the inductance of the passive inductors may depend on the electrical current, as Legg reported in [START_REF] Legg | Magnetic Measurements at Low Flux Densities Using the Alternating Current Bridge[END_REF].

Since the electrical current in the shunt is generated by mechanical vibrations, the shunt inductance may strongly depend on the amplitude of the mechanical excitation. Furthermore, the temperature cal current characteristics of these inductors are because they both depend on the magnetic circuit properties. As a consequence, designing inductors with large variations of inductance with temperature may lead to large variations with the excitation amplitude as well. For this reason, another solution is developed below.

Solution B: Resonant shunt with additional variable capacitance

For this second proposed solution, the objective is first to design an inductor that is as stable as possible with respect to temperature and electrical current. Then, a capacitance C P is added in parallel with a series RL shunt. This added capacitance should be designed to maintain the damping performance when temperature evolves. In this case, the impedance Z is

Z = R + jΩL 1 + jΩRC P -LC P Ω 2 . ( 13
)
The system FRF is still expressed by equation [START_REF] Niederberger | An autonomous shunt circuit for vibration damping[END_REF]. The only differences are that the piezoelectric capacitance C ε is replaced by C ε + C P , and that the coupling factor k c0 is replaced by a lower coupling factor k c . Indeed, using the parameters defined in (5) allows to express the equivalent coupling coefficient k c :

k c k c0 = 1 1 + C P C ε . ( 14
)
The evolution of the coupling coefficient with the additional capacitance is represented in Figure 3. It shows that adding some capacitance to the shunt deteriorates the electromechanical coupling, and thus the damping performance [START_REF] Thomas | Performance of piezoelectric shunts for vibration reduction[END_REF][START_REF] Fleming | Reducing the inductance requirements of piezoelectric shunt damping systems[END_REF]. In this situation, the tuning condition derived from equation [START_REF] Legg | Magnetic Measurements at Low Flux Densities Using the Alternating Current Bridge[END_REF] becomes

Ω 2 O (T ) = 1 L (T ) [C ε (T ) + C P (T )] . ( 15
)
If the evolutions with temperature of the mechanical resonance, the piezoelectric capacitance and the inductance are given data, then the control parameters are the added capacitance and its temperature characteristics. We assume that C P evolves linearly over the considered temperature range, which means

C P (T ) = C P (T 0 ) [1 + α C P (T -T 0 )
] with α C P its temperature coefficient. A linear approximation of equation ( 15) shows that the tuning condition becomes

α C P = -(α L + 2α Ω O + α C ε ) C ε (T 0 ) C P (T 0 ) + α L + 2α Ω O . (16)
Using dimensionless quantities, the tuning condition can also be written:

Y = - a X + b , with Y = α C P α C ε , X = C P (T 0 ) C ε (T 0 ) , b = α L + 2α Ω O α C ε , a = b + 1. ( 17
)
This solution for a passive temperature adaptation of the resonant shunt is denoted solution B. Solution B may offer a more stable solution than solution A. Indeed, the electrical current should have little influence on the inductance, since it has been designed to this end. However, more stability is achieved at the expense of the damping performance. For this reason, the added capacitance should be as small as possible.

3 Design of electrical components

Experimental setup

The experimental setup consists of a cantilever beam covered with two piezoelectric patches close to the clamped end. The duralumin beam is 170 mm long, 25 mm large and 2 mm thick. The PIC 151 PZT patches are 25 mm long, 20 mm large and 0.5 mm thick. The excitation and the velocity measurement are both made at the free end of the beam. The setup is eventually heated with a heat gun inside a thermal chamber, in which measurements are made with an infrared thermometer in steady thermal state. A sketch of the setup is represented in Figure 4.

The evolutions with temperature of the mechanical resonance Ω O and of the piezoelectric capacitance C ε are plotted in Figure 5. The estimated temperature coefficients are α Ω O = -0.5 • 10 -3 K -1 and α C ε = 3.1•10 -3 K -1 . The evolution with temperature of the piezoelectric capacitance is in rough agreement with the manufacturer data. The main reason for the remaining difference is the variations of the structural properties, as it leads to piezoelectric capacitance variations because of the electromechanical coupling.

Variable inductor for solution A

Optimized inductors for piezoelectric shunt damping usually require large inductance and small resistance values. Passive inductors meeting these requirements can be made following the method described in [START_REF] Lossouarn | Design of inductors with high inductance values for resonant piezoelectric damping[END_REF]. Inductors are produced by winding N turns of conductive wire around magnetic cores of effective permeability µ e . This permeability depends on the properties of the magnetic circuit [START_REF] Snelling | Soft ferrites[END_REF]. The obtained inductance value is proportionnal to N 2 and µ e , with a geometry depending constant β:

L = βµ e N 2 . (18)
Equation [START_REF] Lossouarn | Design of inductors with high inductance values for resonant piezoelectric damping[END_REF] shows the inductance has to vary with temperature to keep the resonant shunt tuned. Since β and N do not depend on temperature, equation (18) shows the temperature coefficient of the effective permeability α µe is equal to α L . For practical reasons related to ferrite materials availability, the ungapped E 16/8/5 core geometry from core manufacturer Epcos TDK has been selected. The temperature characteristics of several available ferrite materials are compared in Figure 6. According to equation ( 12) and measurements in subsection 3.1, designing a resonant shunt adapting to temperature variations requires an inductor whose inductance decreases when temperature rises. In an other work [START_REF] Darleux | Passive self-tuning inductor for piezoelectric shunt damping considering temperature variations[END_REF], we have shown that the T46 ferrite material is the one that should be selected among the available ones.

We measure that C ε = 38.3 nF and Ω O /2π = 322 Hz, so the required inductance is estimated at L = 6.38 H using equation [START_REF] Legg | Magnetic Measurements at Low Flux Densities Using the Alternating Current Bridge[END_REF]. Since data from manufacturers give β = 0.672 nH and µ e = 1410, we calculate N = 2600 turns using equation ( 18). The produced inductor is then tested with an LCR-meter at 322 Hz. Inductance variations with temperature and electrical current are plotted in Figure 7. When available, data from manufacturers are represented as well.

In Figure 7(a), the available datasheet do not accurately predict the inductance evolution from 22 • C to 60 • C. Moreover, the predicted decrease of the inductance is only obtained at low levels of excitation and below 42 • C. In Figure 7(b), our measurements show as well a decrease of the inductance under 40 µA and 42 • C. However, as soon as the environmental parameters are outside this range of low temperature and low levels of excitation, the inductance increases. In hindsight, this practical characterization highlights the correlation that can exist between temperature and excitation effects on the inductance. For this reason, it can be difficult to design an inductor whose characteristics meet our requirements over a wide range of both temperature and excitation amplitude. This is the reason why the solution B has been developed.

Variable capacitor for solution B

In the case of solution B, the objective is first to design an whose variations with electrical current and temperature are as small as possible. Equation ( 18) is still considered. Obtaining the required inductance value while minimizing α µe is possible by adding an air gap in the magnetic circuit of the component [START_REF] Snelling | Soft ferrites[END_REF].

The N48 ferrite material with the gapped RM10 core geometry from Epcos TDK has been selected. In this case, β = 2.48 nH, µ e = 254 and N = 2870 turns.

Figure 8 shows the characteristic of the produced inductor. Data from core manufacturers do not precisely predict the inductance evolution with temperature, which may be due to a smaller air gap than anticipated. The estimated temperature coefficient for L is (α L ) B = 2.5 • 10 -4 K -1 . This value is nearly thirty times smaller than (α L ) A from subsection 3.2. Moreover, electrical current and temperature effects do not seem strongly correlated. Hence, the produced inductor meet our requirements by having a nearly constant inductance when environmental parameters evolve.

The next step is to select the added capacitance C P . Given the decrease of k c expressed in equation ( 14) and plotted in Figure 3, the goal is to add the smallest possible value of C P . Figure 9 represents the optimal tuning possibilities for solution B, as expressed in equation (17). Values of α Ω O and α C ε are estimated in subsection 3.1. The graph shows that the smaller C P is, the bigger its variations have to be to maintain the tuning. As a consequence, we selected ceramic capacitors of class 2 [START_REF] Pan | A brief introduction to ceramic capacitors[END_REF], which are characterized by large variations of capacitance with temperature over their working range. Characteristics of some of the dielectrics that vary the most with temperature are extracted from datasheets and added to Figure 9. The selected capacitor is a Y5V 4.7 nF capacitor, as it is the closest component to the optimal curve among the available ones. Its measured characteristics is rougly in agreement with its datasheet. Both are placed on Figure 9. The selected component is added to Figure 3 as well, and it shows that trading off some of the damping performance for more stability makes sense, since the coupling coefficient k c is only 6 % lower than k c0 .

4 Vibration damping of a clamped beam

Adaptive resonant shunt using solution A

The solution A is considered for passive vibration damping of the second bending mode of a clamped beam. The inductance of the resonant shunt is the T46 ferrite-based inductor designed in subsection 3.2. At room temperature, the shunt resistance is intentionally set under its theoretical optimal value [START_REF] Thomas | Performance of piezoelectric shunts for vibration reduction[END_REF][START_REF] Berardengo | Vibration Control by Means of Piezoelectric Actuators Shunted with LR Impedances: Performance and Robustness Analysis[END_REF]. The evolution of the shunt tuning is thus more easily detected. Experiments are conducted at low levels of excitation to benefit from the inductance decrease spotted on Figure 7. The experimental setup is represented in Figure 4.

Experimental FRFs are compared at different temperatures in Figure 10. The shunt tuning is preserved from room temperature to around 51 • C. It appears that the system benefits from a larger decrease of the inductance value than what had been anticipated by looking at Figure 7. Moreover, it is experimentally noticed that the shunt gets mistuned at steady thermal state if the excitation amplitude increases. This is a consequence of the strong mutual dependence electrical and temperature effects on the in Figure 7. For this reason, the solution A might keep the shunt tuned only if the mechanical excitation is stable enough over time. This could be a restrictive condition in some practical applications.

Adaptive resonant shunt using solution B

The solution B is now considered to damp bending vibrations of a clamped beam. The inductance of the resonant shunt is the N48 ferrite-based inductor designed in subsection 3.3. The shunt resistance is also set under its optimal value to spot tuning variations more easily.

Experimental FRFs are compared at different temperatures in Figure 11. The shunt seems to get slightly mistuned from room temperature to 50 • C, before coming back to a better tuning at 62 • C. This can be explained by the nonlinear variations with temperature of the added capacitance. Its variations are more important below 50 • C, which explains the initial mistuning. The reduction of the height between the initial peaks at 304 Hz and 334 Hz and the initial valley at 322 Hz may be explained by the increase of the electrical damping with temperature. Moreover, changing the excitation amplitude at steady thermal state do not get the shunt mistuned as much as with solution A. In the end, the shunt tuning is preserved from room temperature to 62 • C, so these results are promising. The electromechanical coupling is reduced when compared to solution A, which leads to a slightly lower damping performance, but stability over a wider range of temperature is achieved.

Conclusions

The objective of this work is to design a piezoelectric resonant shunt that passively adapts to temperature variations. Two solutions are developed: the first one (denoted solution A) involves a variable inductance, while the second one (denoted solution B) keeps the shunt inductance constant and uses a variable parallel capacitance. An electromechanical model of the vibrating structure connected to the resonant shunt leads to a tuning criterion for each solution. Electrical components are then designed to meet these criteria by taking into account temperature characteristics of the system parameters. When it comes to inductors, we produce them with magnetic circuits. The evolution with temperature of such components depends on the geometry and the material of the magnetic circuit. When it comes to the added parallel capacitance, the class 2 capacitors are selected since they are the ones that vary the most with temperature. All these components are tested, and then used to damp bending vibrations of a cantilever beam. Experimental results show that the shunt tuning is vaguely maintained from room temperature to around 50 • C for solution A, and more precisely maintained from room temperature to more than 60 • C for solution B.

The design of a fully passive and adaptive resonant shunt over a wider range of temperature would require predictive models of electrical components evolutions with temperature. For this matter, solution A would be quite complex to implement, because of the mutual dependence of electrical current and temperature characteristics of the produced inductors. Solution B seems easier to design and implement, but this more predictable and stable solution comes at the expense of the electromechanical coupling. If the slight induced drop of damping performance is acceptable, then solution B may be recommended.

Figure 1 :

 1 Equivalent electrical circuit of the shunted structure 2 Electromechanical model of a piezoelectric resonant shunt 2.

Figure 3 :

 3 Figure 3: Evolution of the coupling coefficient in case of an added parallel capacitance

Figure 4 :

 4 Figure 4: Sketch of the experimental setup inside the thermal chamber

Figure 5 :Figure 6 :

 56 Figure 5: Evolution with temperature of the piezoelectric capacitance and the mechanical resonance angular frequency in open-circuit divided by their respective values at room temperature

Figure 7 :

 7 Figure 7: Effect of (a) temperature and (b) electrical current on the T46 ferrite-based inductor.

Figure 8 :

 8 Figure 8: Effect of (a) temperature and (b) electrical current on the N48 ferrite-based inductor. Notice the scale of the y-axis.

Figure 9 :

 9 Figure 9: Solid line: Optimal characteristics of the added capacitance. Dashed lines: Usual temperature variations of several class 2 dielectrics. Red points: selected capacitor.

Figure 10 :Figure 11 :

 1011 Figure 10: Experimental displacement FRFs at different temperatures with solution A

Table 1 :

 1 Direct electromechanical analogy

	Mechanical quantity	Electrical quantity
	Force Velocity Compliance Mass Viscous damping	Voltage ↔ Electrical current ↔ Capacitance ↔ Inductance ↔ ↔ Resistance

opt 2.2 Adaptive tuning of a resonant shunt in case of temperature variations 2.2.1 Solution A: Resonant shunt with variable inductance A

  ) basic resonant shunt is first considered. The impedance Z consists of a resistance R and an inductance L connected in series:

	40
	30
	20
	10
	0.85	0.9	0.95	1	1.05	1.1	1.15
	Figure 2: Displacement FRF around the resonance for k c0 = 0.12 and ξ e = (ξ e )