Dr Olivier Saurel

Diane Jan

Daniela Tobias

Ladislav

Adrian Agnès

Anne Audrey

Bénédicte

Cécile Gozlan

Elodie Dorothea

Georges

Gilles

Houda, Kristaps, Loïc, Marcel, Manhal, Monu, Moreno, Olivier Henry

Sachin, Samuel, Sami, Sylvie, Torsten, Valérie Vanessa Je

Dans ce manuscrit, nous verrons que si l'on étudie parfois des protéines isolées de leur contexte biologique, il est indispensable de les penser dans le cadre des interactions avec les partenaires présents dans leur environnement naturel. De même, si j'écris parfois à la première personne du singulier, c'est un nous collectif (ou plutôt un we) que je devrais employer car chaque personne que je cite ci-dessous (et encore, seulement en première approximation) a contribué plus ou moins directement à l'édification de cette thèse.

Je voudrais en premier lieu remercier les Prs. Tatyana Polenova et Bernd Reif pour avoir eu la patience de relire et d'évaluer ces travaux, ainsi que les Drs. Carine Van Heijenoort et Luca Monticelli qui ont accepté de faire partie de mon jury.

Je souhaite remercier mon directeur de thèse, Dr Guido Pintacuda, pour m'avoir guidé sur les chemins sinueux de la recherche. Merci d'abord pour m'avoir permis de travailler sur tant d'aspects différents de la recherche et sur des sujets si variés. Dans le désordre créatif dont tu t'entoures, tu as toujours su me dédier du temps quand j'en avais besoin. Merci aussi de m'avoir poussé quand je ne pensais pas pouvoir aller plus loin. Enfin, merci pour tous ces temps conviviaux que tu nous a fait partager.

Abstract

Solid-state NMR with magic angle spinning (MAS) has emerged as a powerful technique for investigating structure and dynamics of insoluble or poorly soluble biomolecules. A number of approaches has been designed for reconstructing molecular structures from the accurate measurement of internuclear proximities, and for probing motions at atomic resolution over timescales spanning several orders of magnitude.

Despite this impressive progress, however, MAS NMR studies are still far from routine. Complete determinations, which are often demonstrated on model microcrystalline preparations, are still rare when it comes to more complex systems such as non-crystalline amyloid fibrils or transmembrane proteins in lipid bilayers.

My work aimed at extending the possibilities of MAS NMR for applications on complex biomolecular systems in different aggregation states. For this, I exploited the unique possibilities provided by high magnetic fields (700, 800 and 1000 MHz 1 H Larmor frequency) in combination with the newest MAS probes capable of spinning rates exceeding 60 kHz. These experimental conditions allow to boost the sensitivity of MAS NMR through 1 H detection at high resolution and to enrich the palette of probes for protein dynamics.

The first part of the thesis reports on my contribution to the development of new strategies for backbone resonance assignment, for structure elucidation, and for investigation of backbone and side-chain dynamics. These methodologies significantly reduce the requirements in terms of experimental time, sample quantities and isotopic labeling, and enlarge the molecular size of systems amenable to NMR analysis.

The second part describes the application of 1 H detected MAS NMR to evaluate the role of protein dynamics in problems such as amyloid fibril formation and membrane protein function.

I first addressed the amyloid fibril formation propensity of human beta-2 microglobulin, the light chain of the major histocompatibility complex I. I performed comparative studies of backbone dynamics of the wild type protein as well as a D76N mutant in crystals, and determined some of the structural features of the fibrillar form. This allowed to identify the presence of pathological folding intermediates and to formulate hypotheses on the mechanism of fibrils formation.

Finally, I studied the local and global dynamics of membrane proteins in lipid bilayers. In particular, I investigated the mechanism of action of the alkane transporter AlkL from P. putida in lipid bilayers. The measurement of parameters for fast (ps-ns) and slow (µs-ms) backbone dynamics of the protein in presence or in absence of a substrate highlights possible routes for molecular uptake and lays the basis for a more detailed mechanistic understanding of the process.

Résumé

La Résonance Magnétique Nucléaire (RMN) à l'état solide avec rotation à l'angle magique (MAS) est une technique de choix pour l'étude de la structure et de la dynamique de molécules biologiques peu ou non solubles. Un grand nombre d'approches ont été développées pour la reconstitution de structures tridimensionelles à partir de mesures précises de proximités internucléaires, ainsi que pour la détection de mouvements moléculaires avec une résolution atomique sur des échelles de temps couvrant plusieurs ordres de grandeur.

Malgré d'impressionnants progrès, les études par RMN MAS sont cependant loin d'être réalisées en routine. Les déterminations structurelles et de dynamique sont souvent démontrées sur des préparations microcristallines modèles, mais sont encore rares pour des systèmes plus complexes tels que les fibrilles amyloïdes non cristallines ou les protéines trans-membranaires insérées dans des bicouches lipidiques.

Mon travail a pour objectif d'étendre les possibilités de la RMN MAS pour l'étude de systèmes biomoléculaires complexes dans différents états d'agrégation. Pour cela, j'ai exploité les possibilités uniques offertes par les hauts champs magnétiques (fréquence de Larmor du 1 H 700, 800 et 1000 MHz) combinés avec les sondes MAS de dernières générations capables d?atteindre des vitesses de rotations supérieures à 60 kHz. Ces conditions expérimentales permettent d'augmenter la sensibilité de la RMN MAS à l'aide de la détection 1 H à haute résolution et d'enrichir la palette de paramètres RMN rapporteurs de la dynamique des protéines.

La première partie de cette thèse décrit le développement de nouvelles stratégies pour l'attribution des résonances du squelette de protéines, pour l'élucidation de structures, et pour l'étude de la dynamique du squelette peptidique et des chaînes latérales. Les méthodes présentées réduisent significativement les besoins en termes de temps expérimental, de quantités d'échantillon et de marquage isotopique, et permettent d'analyser par RMN des systèmes de plus hauts poids moléculaire.

La seconde partie décrit l'application de la RMN MAS avec détection en 1 H pour l'évaluation du rôle de la dynamique des protéines dans des processus tels que la formation de fibrilles amyloïdes et le fonctionnement de protéines membranaires.

Une première application est l'étude de la tendance de la β-2 microglobuline humaine à former des fibrilles amyloïdes. Une comparaison de la dynamique du squelette peptidique de la protéine sauvage et du mutant D76N dans leur forme cristalline, ainsi que la détermination de propriétés structurales de la forme fibrillaire m'ont permis d'identifier la présence de repliements pathologiques et de formuler des hypothèses sur le mécanisme de formation des fibrilles.

Finalement, la dynamique locale et globale de protéines membranaires dans des bicouches lipidiques a été étudiée. En particulier, le mécanisme d'action d'un transporteur d'alkanes, AlkL, de P. putida a été examiné dans un environnement lipidique. La détermination de paramètres pour la dynamique rapide (ps-ns) et lente (µs-ms) du squelette peptidique de la protéine en présence ou en absence de substrat met en évidence des acheminements possibles pour le transfert de molécules vers la membrane et jette les bases pour une meilleure compréhension du processus.

Introduction

Atomic characterization of biomolecules is a major goal for modern biology. For decades, structural biologist have mostly focussed on static molecular structures. However, biomolecules and in particular proteins -which are the subject of this thesis -have a high degree of conformational freedom, or, according to a famous definition, are "kicking and screaming stochastic molecules" (G. Weber, [1]). What is often missed is that these fluctuations of the molecular structure are actually finely tuned through evolution to achieve molecular function. There is therefore a growing interest in understanding how molecules move and how these motions coordinate specific functions.

Experimental determination of protein dynamics at the atomic level is however challenging. It requires the characterization of proteins in the 4D space defined by the three space coordinates and the time dimension. Most biophysical techniques are able either to obtain high-resolution structural ensembles (e.g. Xray diffraction and electron microscopy) or to provide information about their time evolution (e.g. infrared, fluorescence and electron paramagnetic resonance spectroscopies), but not both at the same time. In addition, any biophysical study has to isolate the object of interest from its native environment, which may strongly affect protein behavior. When performing in vitro studies, the experimentalist has to find a balance between the requirements inherent to the technique and the most native conditions for the protein.

Nuclear Magnetic Resonance (NMR) has a particular place among biophysical methods because of its power of revealing protein dynamics with both spatial and temporal resolution. A large arsenal of NMR methods is routinely used to characterize motions of proteins in solution at different timescales. Applications in structural biology include determination of entropic changes upon ligand binding or mutation, characterization of allosteric effects on enzyme function or the indirect observation of folding intermediates. For a large number of systems which are either not soluble or have prohibitively large molecular weight for solution studies (such as amyloid fibrils, membrane proteins, large complexes or viral capsids), solid-state NMR is a technique of choice. In solid preparations however, anisotropic interactions between spins broaden the NMR lines and add unwanted contributions to the observables which report on dynamics. Magic-Angle Spinning (MAS) of the sample helps improving spectral quality and recovering information about dynamics by averaging these interactions.

Progresses in MAS technology have enabled the design of approaches for structural determination and the investigation of dynamics over a broad range of timescales. Despite the rapid evolution of MAS NMR and impressive proofof-concept studies on microcrystalline proteins, applications to more challenging systems such as amyloid fibrils or transmembrane proteins in lipid bilayers are still rare. The objective of my PhD has been to extend the potential of MAS NMR for the investigation of complex biological systems in different aggregation states. In order to reach this goal, I exploited the unique possibilities provided by high magnetic fields (700, 800 and 1000 MHz 1 H Larmor frequency) in combination with the newest MAS probes capable of spinning rates exceeding 60 kHz. These experimental conditions allow to boost the sensitivity of MAS NMR through 1 H detection at high resolution and to enrich the palette of probes for the study of protein dynamics.

In Chapter 1, I review the recent history of MAS NMR for the investigation of protein motions. In order to introduce several key concepts, I first present some of the techniques used for solution NMR studies of dynamics with timescales ranging from ps to s. I then discuss the advantages as well as the challenges of such studies in the solid state and give a general overview of how the limits of MAS NMR have been moved forward in the last ten years, with the presentation of methodological developments as well as latest applications.

In a first experimental part, I present my contributions to the methodological developments that have been carried out in our group during the last four years. In Chapter 2, I first discuss the advantages in terms of sensitivity and resolution of proton detection with fast MAS. At 60 kHz MAS, high-resolution 1 H spectra can be acquired on deuterated samples with high proton content at the amide sites. By taking benefit of this spinning regime, sensitive correlation experiments for the assignment of backbone resonances and structural investigations have been designed. The consecutive advent of 110 kHz MAS probes marked a significant evolution in the field of MAS NMR by enabling side-chain resonance assignment and structure determination on sub-milligram amount of fully protonated samples. In Chapter 3, new opportunities arising from this technological development for the elucidation of both backbone and side-chain dynamics of proteins are evaluated.

A second part of this thesis is dedicated to the application of recent MAS NMR methods to the study of the amyloidogenic properties of the D76N variant of beta-2 microglobulin (β2m). β2m is the light chain of the major histocompatibility complex of type I and is recognized as a model for the study of amyloid fibril formation. A recent case of familial amyloidosis implies the D76N variant of β2m and the propensity of this variant to aggregate in vitro has been shown to be spectacularly higher than the wild type (wt) protein. In order to rationalize this behavior, the dynamics of crystalline wt and D76N β2m is compared. This allows to identify the presence of pathological folding intermediates. Subsequently, the fibrillar form of D76N β2m is investigated and structural features are determined, bridging the observations on the native form to the mechanism of fibrils formation.

Finally, I study the dynamics of two large membrane proteins in lipid bilayers. In Chapter 5, I present the results of the investigation of local and global motions of the outer membrane protein OmpA of Klebsiella pneumoniae. Then, I address the mechanism of action of the alkane transporter AlkL from Pseudomonas putida in lipid bilayers (Chapter 6). The measurement of parameters reporting on fast (psns) and slow (µs-ms) backbone dynamics of the protein in presence or in absence of a substrate highlights possible routes for molecular uptake and lays the basis for a more detailed mechanistic understanding of the process.

Chapter 1

Protein dynamics by Magic-Angle Spinning NMR

Structure and dynamics determine function

Most cellular functions are linked to the action of specific proteins. It is therefore of tremendous interest to rationalize their functions and reactivity, whether for basic understanding of biology, for drug discovery or for development of bioinspired materials. The so-called structure activity relationship states "structure determines function", meaning that the description of the entire arrangement of atoms of a protein triggers its properties.

Three-dimensional structure of proteins and other biological molecules can be determined by X-ray diffraction, nuclear magnetic resonance (NMR) or electron microscopy (EM). The development of these techniques have proven to be necessary to the understanding of biological functions and have been rewarded by many Nobel Prizes: among them, M. Perutz and J. Kendrew in 1962 for the structure of globular proteins by X-ray diffraction; J. Crick, F. Watson and M. Wilkins the same year in medecine for the molecular structure of DNA; D. Hodgkin in 1964 for the structure of biochemical substances; A. Klug in 1982 for the development of electron microscopy; R. Ernst in 1991 for the development of multidimensional NMR spectroscopy; K. Wüthrich in 2002 for the development of biomolecular NMR; P. Agre and R. MacKinnon in 2003 for their discoveries on water/ion channels on cell membranes; R. Kornberg in 2006 for his work on the molecular basis of eukaryotic transcription; V. Ramakrishnan, T. Steitz and A. Yonath in 2009 for the structure and function of the ribosome; R. Lefkowitz and B. Kobilka for their studies on the G-protein coupled receptors; J. Dubochet, J. Frank and R. Henderson in 2017 for high-resolution structure determination of biomolecules by cryo-electron microscopy.

Yet, static pictures of 3D structures do not capture alone the intrinsic dynamical nature of life. Indeed, in a cellular environment, the interactions with neighboring molecules modulate the morphology of macromolecules. Furthermore, the structure of a biomolecule results from averaging all molecular conformations accessible at a given temperature.

A more comprehensive description of proteins requires in fact the extension of the 3D-structures to the time dimension. The time window on which molecular processes happen in a protein stretches from picoseconds to more than seconds. A handy way of classifying the conformational fluctuations is to introduce a hierarchy between local flexibility, which involves only little groups of atoms, and collective motions of larger domains [START_REF] Frauenfelder | The energy landscapes and motions of proteins[END_REF][START_REF] Henzler-Wildman | Dynamic personalities of proteins[END_REF][START_REF] Huang | Understanding enzymes: function, design, engineering, and analysis[END_REF]. The first type of motions occurs in a ps-ns timescale and generally engages numerous conformations separated by small energy barrier which are constantly excited via thermal agitation. These states are treated statistically and participate to the entropy of the system. Large domains motions or local protein unfolding can, in principle, be described as a discrete set of conformations exchanging typically within µs-ms or slower. While the averaged conformation is the one usually reported by structural investigations, less-populated states may be more relevant for function. For example, populating alternative states can, for a given time, change the surface properties of the protein, unveiling the catalytic center of an enzyme [START_REF] Huang | Understanding enzymes: function, design, engineering, and analysis[END_REF][START_REF] Eisenmesser | Intrinsic dynamics of an enzyme underlies catalysis[END_REF] or opening a gate in membrane-bound protein [START_REF] Patel | Coupling of retinal isomerization to the activation of rhodopsin[END_REF][START_REF] Cafiso | Identifying and quantitating conformational exchange in membrane proteins using site-directed spin labeling[END_REF][START_REF] Wylie | Transmembrane allosteric coupling of the gates in a potassium channel[END_REF]. Conformational exchange is finely tuned by natural selection to optimize protein function [START_REF] Huang | Understanding enzymes: function, design, engineering, and analysis[END_REF] and may be even a driving force of protein evolution [START_REF] Tokuriki | Protein dynamism and evolvability[END_REF][START_REF] Pabis | Cooperativity and flexibility in enzyme evolution[END_REF].

Half a century of advances in structural biology has enabled to determine the precise location of atoms in space. However, the present technology does not permit to directly follow the evolution of all the atomic coordinates during the time course. Instead, biophysical techniques generally rely whether on the deduction of motions from a bundle of static images or on the observation of related physical parameters, which are interpreted in terms of molecular motions. An illustration of this situation is given by the single-molecule fluorescence resonance energy transfer (FRET) [START_REF] Diez | Proton-powered subunit rotation in single membrane-bound F 0 F 1 -ATP synthase[END_REF] and electron-microscopy [START_REF] Zhao | Electron cryomicroscopy observation of rotational states in a eukaryotic V-ATPase[END_REF] studies of the subunit rotations of, respectively, F-and V-ATPases These protein assemblies couple the transport of ions through membranes within the cell to the synthesis and/or the hydrolysis of molecules of adenosine triphosphate (ATP).

In the first mentioned study, by monitoring the florescence intensity of fluorophores grafted on two parts of the F-ATPase complex, the authors were able to follow step by step the rotations of a set of subunits with respect to the molecular framework during ATP hydrolysis [START_REF] Diez | Proton-powered subunit rotation in single membrane-bound F 0 F 1 -ATP synthase[END_REF] (Figure 1.1a and b). Interestingly, they observed that the direction of rotation is reversed when ATP is synthesized. Such study proves the importance of following protein conformational changes over time. A fundamental limitation of FRET experiments however is that, except the distance between the fluorescent probes, no structural information can be obtained. The figures are reproduced with permissions from [START_REF] Diez | Proton-powered subunit rotation in single membrane-bound F 0 F 1 -ATP synthase[END_REF] and [START_REF] Zhao | Electron cryomicroscopy observation of rotational states in a eukaryotic V-ATPase[END_REF].

In contrast, cryo-electron microscopy allows to determine high-resolution structures of large proteins [START_REF] Liao | Structure of the TRPV1 ion channel determined by electron cryo-microscopy[END_REF][START_REF] Amunts | Structure of the yeast mitochondrial large ribosomal subunit[END_REF]. By illuminating the sample with a beam of electrons, a series of images of the protein are obtained, which represents an ensemble of projections of the actual structure. The reconstruction of the 3D structure becomes complicated if several conformational substates are present in the sample. Thanks to sophisticated softwares, the authors of the above mentioned study of the V-ATPase [START_REF] Zhao | Electron cryomicroscopy observation of rotational states in a eukaryotic V-ATPase[END_REF] classified the different conformations of the protein assembly and determined their structure with a resolution of ∼ 7 Å (Figure 1.1c). These snapshots revealed how each subunit adapts to the global conformational changes.

These two examples are cited to give a taste of how it is possible to describe the trajectory of protein assemblies with two orthogonal state-of-the-art techniques. They also illustrate the fact that proteins are not comprehensively described by their average structure. Instead, we have to take into consideration an array of different conformations perpetually interconverting between each other, each conformation being itself agitated by random motions. Even if the precise role of dynamics in protein function is not always completely clear, we should complement the postulate given at the beginning of this text: "structure and dynamics determine function".

Exploring the dynamics of proteins by solution NMR

Among biophysical methods used to investigate molecular dynamics, NMR has a unique place, due to its capacity to describe molecular motions at high spatial resolution with timescale information. A variety of experiments has been designed to study motions on a broad time range. These experiments determine NMR observables related to specific motion types. Once combined, these data allow to obtain a complete picture of the dynamical behavior of the system.

We will first focus on spin relaxation which is a phenomenon used in NMR to probe fast fluctuations of a protein conformation. A second part is dedicated to techniques for the study of exchange of conformations.

Spin relaxation

The basics

In NMR, interactions that each individual spin experiences can be described with an Hamiltonian composed by isotropic (Zeeman interaction, isotropic chemical shift, scalar coupling) and anisotropic (dipolar and quadrupolar coupling, chemical shift anisotropy) contributions. During an experiment, the system is carried out of its equilibrium by radio-frequency (RF) pulses. The magnetization vector trajectory then results from the coordinated precession around the magnet During an NMR experiment, the spins are synchronously precessing around the external magnetic field. Because of local fluctuations of the experienced magnetic field, each spin will have a stochastically modulated precession rate, leading to a global dephasing and a subsequent loss of transverse magnetization M x/y . Longitudinal relaxation: In a case where the system is out of equilibrium with randomly oriented spins, there is no net macroscopic magnetization. Reorientations of the local magnetic field induces transitions between the Zeeman energy states. After a certain amount of time, spins are polarized in the (energetically favored) sense of the external magnetic field, resulting in a macroscopic magnetization M z along the magnet axis. The polarization depends on the temperature and the energy gap between the Zeeman levels. For illustration purposes only, the bottom right panel represents a fully polarized system.

axis of the ensemble of microscopic nuclear spins at a frequency corresponding to the eigenvalue of their Hamiltonian. This "motion" would last for ever if each nuclei is surrounded by the same environment. However, in solution molecules experience local structural fluctuations and global reorientations (molecular tumbling) which stochastically modulate the anisotropic terms of the NMR Hamiltonian. Local fields may therefore vary in strength and orientation with two consequences: (i) desynchronization of the precessions of each individual spin and therefore the annihilation of the components of the total magnetization in the transverse plane (M x/y ) (ii) return of the system to thermal equilibrium with the (re)-apparition of longitudinal magnetization (M z ) (Figure 1.2). These processes are called spin relaxation (transverse and longitudinal relaxation, respectively) and follow the phenomenological laws introduced by Bloch in 1946 [START_REF] Bloch | Nuclear induction[END_REF] (here expressed in the rotating frame):

dM x/y dt = M x/y T 2 dM z dt = M z -M z,eq T 1
The corresponding relaxation times (T 2 and T 1 ) directly report on the dynamics of the molecule. In the following, we may also refer to the relaxation rates R 2 and R 1 simply defined as 1/T 2 and 1/T 1 .

Spin relaxation is usually treated in the framework of the Redfield theory [START_REF] Redfield | On the theory of relaxation processes[END_REF]. Here, without going into details, we will give an overview of the key concepts used in relaxation studies and present some applications (for reviews, see [START_REF] Palmer | NMR characterization of the dynamics of biomacromolecules[END_REF][START_REF] Jarymowycz | Fast time scale dynamics of protein backbones: NMR relaxation methods, applications, and functional consequences[END_REF][START_REF] Kleckner | An introduction to NMR-based approaches for measuring protein dynamics[END_REF]). In particular, we will show broadly how to relate the parameters describing molecular motions to the expression of the relaxation rates.

Spin relaxation, as stated above, arrises from the random (local) fluctuations of anisotropic interactions. In Redfield theory, the spatial parts of the Hamiltonian representing these interactions are treated separately from the spin parts and in a classical way. This Hamiltonian, also called the stochastic Hamiltonian, can therefore be written as:

H 1 (t) = q F q (t)A q (1.1)
in which F q (t) are the randomly modulated spatial components of the Hamiltonian and A q the spin operators. Figure 1.3a and c show the random fluctuations of such function F q (t) with two different fluctuation timescales. From a visual inspection, it is obvious that the signal in Figure 1.3a is modulated at a lower frequency than the signal of Figure 1.3d, but the difference can not be easily interpreted in a quantitative manner. A more convenient way to treat the information content is to introduce the autocorrelation function C q (τ ) defined as the overlap of the function F q at a time t with the same function at a time t + τ :

C q (τ ) = F q (t)F * q (t + τ ) (1.2)
As indicated by the brackets, the relevant macroscopic property is obtained by taking an average over the spins ensemble. As the motions probed by spin relaxation are supposed to be a stationary process, the autocorrelation therefore does not depend on the origin of time. We can therefore re-write equation 1.2 as:

C q (τ ) = F q (0)F * q (τ ) (1.
3)

The correlation function can be seen as the memory of the spatial fluctuations of the spin-spin interactions. When τ = 0, the autocorrelation function represents the magnitude of the spatial part of the interaction (normalized to 1). Then, as τ increases, the overlap between F q (0) and F q (τ ) decreases (and so does C q (τ )) since the time-evolution of F q (t) is stochastic: at longer times, the function contains less and less information about its original value. If the motion is unrestricted (e.g. for rotational diffusion of a molecule in solution), the "memory" of the function is completely lost for infinite times and C q (∞) = 0 (Figure 1.3b and e).

0 1 0 1 τ τ t t F q (t) F q (t) C q (τ) C q (τ) (a) (d) (b) (e) 0 0 J q (ω) J q (ω) ω ω (c) (f)
Figure 1.3: Autocorrelation function for random fluctuations of a spatial component (F q ) of an anisotropic Hamiltonian resulting from unrestrained motions. The evolution over time of F q (t) is shown for fluctuations with long (a) and short (c) correlation times. The respective autocorrelation functions (C q ) are shown panels b and d, and the spectral density functions (J q ) in panels c and f.

A key function for treating spin relaxation in NMR is the spectral density J q (ω) which is the Fourier transform of the autocorrelation function. J q (ω) represents the probability of finding a motion at a given angular frequency ω, and as such is directly involved in the mathematical expression of the measurable relaxation rates. For fluctuations of the spin-spin interaction with long correlation times, the autocorrelation function decays slowly and the spectral density function is sharp (Figure 1.3c). For shorter correlation times, C q (τ ) decreases faster, and J q (ω) is broader (Figure 1.3f).

Relaxation rates are determined by the different terms of the stochastic Hamiltonian, which for a spin 1 /2 nucleus includes terms such as the chemical shift anisotropy (CSA) and internuclear dipolar couplings. They are expressed as a Relaxation rate Contributing interaction Formula

R 1 IS dipolar coupling R D 1 =(d 2 /4)[J 0 (ω I -ω S ) + 3J 1 (ω S ) + 6J 2 (ω I + ω S )] CSA R CSA 1 = c 2 J 1 (ω S ) R 2 IS dipolar coupling R D 2 =(d 2 /8)[4J 0 (0) + J 0 (ω I -ω S ) + 3J 1 (ω S ) + 6J 1 (ω I ) + 6J 2 (ω I + ω S )] CSA R CSA 2 = (c 2 /6)[4J 0 (0) + 3J 1 (ω S )] η IS IS dipolar coupling η IS = d 2 4R 1 γ I γ S × [6J 2 (ω I + ω S ) -J 0 (ω I -ω S )]
Table 1.1: Mathematical expressions of relaxation parameters commonly used in solution NMR. Here, we only consider the contributions from the fluctuations of heteronuclear dipolar coupling between a spin I and a spin S and the CSA of spin S. d = (µ 0 γ S γ I )/(4πr 3 IS ) and c = (ω S δσ)/ √ 3 where µ 0 is the magnetic permeability of the free space, is the reduced Planck constant, γ I , γ S are the gyromagnetic ratio of the I and S spins, r IS the internuclei distance and δσ the chemical shift anisotropy of S. These two last parameters are generally fixed to a rigid value determined experimentally or theoretically. For 15 N relaxation, commonly used values are r NH = 1.02 -1.04 Å [START_REF] Ottiger | Determination of Relative N-HN, N-C , Cα-C , and Cα-Hα Effective Bond Lengths in a Protein by NMR in a Dilute Liquid Crystalline Phase[END_REF][START_REF] Case | Calculations of NMR dipolar coupling strengths in model peptides[END_REF] and δσ = -160 to -170 ppm [START_REF] Yao | Site-specific backbone amide 15 N chemical shift anisotropy tensors in a small protein from liquid crystal and cross-correlated relaxation measurements[END_REF]. linear combination of the spectral density function sampled at the eigenfrequencies of the spin operators present in the Hamiltonian (Table 1.1) [START_REF] Abragam | Principles of nuclear magnetism[END_REF].

Let us first consider the CSA contribution to the relaxation of a spin S. The operators describing the spin part of the CSA are I z and I ± . The effect of the operator I ± is to transfer populations between the Zeeman energy states α and β. The corresponding term of the stochastic Hamiltonian thus contributes to the return to equilibrium of the spin-state populations. Consequently, R CSA 1 samples the spectral density function at the characteristic frequency of I ± , ω S , the Larmor frequency of the spin S. On the contrary, I z is not associated to any spin-state transition and corresponds to the zero frequency. Therefore, J(0) does not appear in the expression of R CSA

1

. In contrast, both J 0 (0) and J 1 (ω S ) terms contribute to R CSA

2

. Indeed, longitudinal fields, represented here by the spin operator I z , induce a phase to the oscillation of the individual spins in the transverse plane. Their local and stochastic modulation by molecular motions dephase the coherences.

Let us now consider the case where heteronuclear dipolar coupling between two spins 1 /2, I (e.g. 1 H) and S (e.g. 15 N) drives the relaxation of spin S. Figure 1.4 shows the energy levels of a coupled IS system and the transition frequencies. The spin operators present in the dipolar coupling Hamiltonian are I z S z , I ± S ∓ , I z S ± , I ± S z , and I ± S ± . As for I z , I z S z is associated to the zero frequency. The rest of the operators allow the transitions numbered respectively, ( 4), (1,[START_REF] Patel | Coupling of retinal isomerization to the activation of rhodopsin[END_REF], [START_REF] Frauenfelder | The energy landscapes and motions of proteins[END_REF][START_REF] Eisenmesser | Intrinsic dynamics of an enzyme underlies catalysis[END_REF], and (3) in Figure 1.4. Only the transitions involving a change of the state of spin S, namely (1,[START_REF] Henzler-Wildman | Dynamic personalities of proteins[END_REF][START_REF] Huang | Understanding enzymes: function, design, engineering, and analysis[END_REF][START_REF] Patel | Coupling of retinal isomerization to the activation of rhodopsin[END_REF] affect the longitudinal relaxation. The expression of the longitudinal relaxation rate due to IS dipolar coupling, R D 1 , as given in Table 1.1, indeed brings up the spectral density at the frequencies ω S , ω I + ω S and ω I -ω S . At first sight, it may be intriguing when considering R D 2 that the spectral density at the Larmor frequency of spin I (transitions 2 and 5 in Figure 1.4) also contributes to the transverse relaxation rate of spin S (Table 1.1). This can be interpreted by taking into account the fact that the different spin states of I create different local fields for S. Therefore, transitions between the spin states of I also induce coherences dephasing.

Finally, let us briefly introduce another relaxation parameter, often measured in complement to R 1 and R 2 (vide infra), the steady-state nuclear Overhauser effects (NOE) enhancement, η IS . During steady-state NOE measurements, the transitions ( 2) and ( 5) are saturated (the populations of the two levels are equalized by applying a continuous RF field). The populations of the states are then governed by longitudinal relaxation of S (transitions 1 and 6) and by the so-called cross relaxation (transitions 3 and 4). When the steady state is reached, NOE may lead to the enhancement of the magnetization of S. The enhancement factor, η IS , therefore depends on the density function at ω I + ω S and ω I -ω S (Table 1.1).

Overall, the expressions of the relaxation rates given in Table 1.1 are informative on the characteristic frequency of motions at which each relaxation process is sensitive, independently of the precise form of the spectral density function. In order to infer more quantitative information about the motion, the form of the correlation function and subsequently of the spectral density function has to be defined. Different approaches have thus been developed to this end [START_REF] Lipari | A Model-Free Approach to the Interpretation of NMR Relaxation in Macromolecules[END_REF][START_REF] Lipari | Padé approximants to correlation functions for restricted rotational diffusion[END_REF][START_REF] Bremi | Locally anisotropic internal polypeptide backbone dynamics by NMR relaxation[END_REF]. The most popular approach was developed by Lipari and Szabo in 1981 [24]. In their treatment, the autocorrelation function accounts for both internal motions and the overall tumbling of the molecule. In the case that these two types of motion are independent, the correlation function can be expressed as the product of the correlation functions of the rotational diffusion (C O ) and of the internal motion (C I ):

C(t) = C O (t)C I (t) (1.4)
An implicit assumption is that for a molecule in solution, the correlation function does not depend on the index q, i.e. C q = C for all q.

For isotropic rotational diffusion, the correlation function is mono-exponential and has the following expression:

C O (t) = 1 5 e -t/τ M (1.5)
where τ M is the correlation time of the rotational diffusion (τ M = 1/6D M , D M being the corresponding diffusion constant).

For restricted motions, the correlation function does not decay to 0 for infinite τ but instead reaches a plateau, identified as the order parameter S 2 which describes the degree of restriction of the motion. The absence of motion would give an order parameter of 1 (by definition), and the autocorrelation function tends to 0 for unrestricted motions. Assuming the autocorrelation function for internal motions also decays exponentially, the autocorrelation function can be written:

C I (t) = S 2 + (1 -S 2 )e -t/τe (1.6)
where τ e is the effective correlation time. τ e depends not only on the timescale of the internal motion but also on the amplitude. A model of the motion is therefore needed for its interpretation.

From equations 1.4, 1.5 and 1.6, we deduce the total spectral density function:

J M F (ω) = 2 5 S 2 τ M 1 + (ωτ M ) 2 + (1 -S 2 )τ 1 + (ωτ ) 2 τ -1 = τ -1 M + τ -1 e (1.7)
The total spectral density function depends on 2 × N + 1 parameters (N the number of protein sites observed): N × τ e , N × S 2 and τ M . A common implementation consists in measuring R 1 and R 2 relaxation rates together with NOE enhancements for each site, which is sufficient to determine a solution to the problem. If the internal correlation time is very fast (typically τ e < 100 ps), equation 1.7 simplifies to:

J M F (ω) = 2 5 S 2 τ M 1 + (ωτ M ) 2 (1.8)
The number of parameters reduces to N+1 and the information about correlation times disappear. Equally, if the internal motions are slower than the molecular tumbling, τ -1 ∼ τ -1 M and therefore no information about dynamics timescale can be extracted. Finally, the internal motions timescale has to be separated enough from the overall tumbling for accurate determination of the motional parameters.

In the general case, where the motions are more complex, the model-free analysis can be extended to take into account multiple timescales. The correlation function is therefore a sum of n exponentials (where n is the number of timescales):

C I (t) = S 2 + n i=1 A i e -t/τ e,i (1.9)
with

                         τ 1 < τ 2 < ... < τ n A 1 = (1 -S 2 1 ) A i = i-1 j=1 S 2 j (1 -S 2 i ), ∀i > 1 S 2 = n i=1 S 2 i n i=1 A i = 1 -S 2
The spectral density becomes:

J EM F (ω) = 2 5 S 2 τ M 1 + (ωτ M ) 2 + 2 5 n i=1 A i τ i 1 + (ωτ i ) 2
(1.10)

The number of fit parameters here is 2 × N × n + 1. Even if some assumptions made in a case-by-case basis can reduce the number of variables, increasing the complexity of the model also drastically expands the size of the required experimental dataset.

Practical implementation

15 N relaxation is the most generally used probe of backbone dynamics because it is mainly triggered by two interactions that are well studied: the dipolar coupling with the bonded 1 H and its own CSA. The experimental implementation of the above-described approach, introduced by Kay and coworkers in 1989 N correlation spectra acquired with variable z-storage delay (50, 210, 370, 530, 690, and 850 ms, respectively). Adapted with permission from reference [START_REF] Kay | Backbone dynamics of proteins as studied by nitrogen-15 inverse detected heteronuclear NMR spectroscopy: application to staphylococcal nuclease[END_REF]. [START_REF] Kay | Backbone dynamics of proteins as studied by nitrogen-15 inverse detected heteronuclear NMR spectroscopy: application to staphylococcal nuclease[END_REF], exploits the sensitivity of proton detection combined with an indirect 15 N dimension to reach site resolution. Since then, RF irradiation schemes have been improved but have always kept the same principles. R 1 parameters are generally measured with a z-storage mixing period inserted after the Insensitive Nuclei Enhancement by Polarization Transfer (INEPT) block of the Heteronuclear Single Quantum Coherence (HSQC) irradiation scheme. During the mixing time, the polarization of 15 N relaxes toward thermal equilibrium. Similarly for R 2 relaxation rates measurement, during the mixing period, a Carr-Purcell-Meiboom-Gill (CPMG) pulse train is applied. It is composed of a series of inversion pulses separated by a period of 2τ CPMG (see Figure 1.8f) which refocusses chemical shifts, heteronuclear J-coupling and inhomogeneous contributions (dephasing due to spatial inhomogeneities of the magnetic field, not stochastically modulated during the experiment and therefore non related to motion). Multiple decay points are sampled so to add an additional pseudo dimension (Figure 1.5). The signal intensity decay curve is fitted to a mono-exponential function. In general, this dataset is complemented by measuring heteronuclear steady-state NOE enhancement, η IS . In order to measure 1 H-15 N η IS , two spectra are acquired: one with saturation of 1 H before the direct excitation of 15 N and one without. The saturation delay is generally set to 3-4 seconds to achieve significant saturation transfer efficiency. η IS values are then related to the ratio of the intensity of the peak with (I sat ) and without (I eq ) saturation:

η IS = I sat I eq -1 (1.11)
Altogether, these measurements allow to extract parameters describing the molecular motions by assuming a certain spectral density function. The order parameters obtained from the model-free analysis of 15 N relaxation rates are plotted versus the sequence for the apo (white) and bond (black) forms. High increase in order in site II where the ion first binds was observed compared to more modest changes in site I. The position of the α-helix as well as the binding sites are shown above the panel. Adapted with permission from reference [START_REF] Koerdel | Backbone dynamics of calcium-loaded calbindin D9k studied by twodimensional proton-detected nitrogen-15 NMR spectroscopy[END_REF] This approach has been widely used to understand entropic effects of ligand binding or mutations [START_REF] Jarymowycz | Fast time scale dynamics of protein backbones: NMR relaxation methods, applications, and functional consequences[END_REF]. One of the first attempts to link entropy to spin relaxation was made by the group of Palmer. In a series of papers, they investigated the entropic cost of ion binding to the calcium binding protein Calbindin D 9k [START_REF] Koerdel | Backbone dynamics of calcium-loaded calbindin D9k studied by twodimensional proton-detected nitrogen-15 NMR spectroscopy[END_REF][START_REF] Akke | Effects of ion binding on the backbone dynamics of calbindin D9k determined by nitrogen-15 NMR relaxation[END_REF][START_REF] Akke | NMR order parameters and free energy: an analytical approach and its application to cooperative calcium (2+) binding by calbindin D9k[END_REF]. Calbindin D 9k possesses two Ca 2+ binding site (I and II). The authors measured 15 N relaxation on the apo and (Ca 2+ ) 2 -bond forms of Calbindin D 9k . The analysis of dynamics revealed that presence of Ca 2+ greatly increases the order in site II where the ion first binds (Figure 1.6). Additional data on the half-saturated protein (Cd 2+ -Calbindin D 9k ) allowed the investigation of cooperative effects. The authors estimated in this way that binding site II has a high entropic cost and rigidifies the structure, facilitating the second binding event. The contribution from the observed bond reorientations to the Gibbs free-energy was quantified with different motional models by assuming that each bond moves independently from the others. The authors evaluated this contribution to ∼ -13 kJ.mol -1 . Even if translations or rotations around the bond vector were not taken into account as they do not lead to spin relaxation, the methodology developed in this study gives strong insight into the mechanism of ion binding and was extended to many other biological problems.

Alternatively, an approach named "spectral density function mapping" has been developed to avoid assumptions about the exact form of the spectral density function [START_REF] Peng | Mapping of the spectral densities of nitrogenhydrogen bond motions in Eglin c using heteronuclear relaxation experiments[END_REF][START_REF] Peng | Mapping of spectral density functions using heteronuclear NMR relaxation measurements[END_REF][START_REF] Farrow | Spectral density function mapping using 15 N relaxation data exclusively[END_REF]. In the approach described in the previous paragraphs, three relaxation parameters are generally measured, which are dependent on the spectral density function at five frequencies: 0, ω H , ω N , ω H + ω N and ω H -ω N (Figure 1.7). In order to determine the values of the spectral density at these frequencies without any a priori on its shape, Peng et al. proposed to measure five linearly independent relaxation parameters [START_REF] Peng | Mapping of the spectral densities of nitrogenhydrogen bond motions in Eglin c using heteronuclear relaxation experiments[END_REF][START_REF] Peng | Mapping of spectral density functions using heteronuclear NMR relaxation measurements[END_REF]. More conveniently, the conservative assumption that the spectral density function varies slowly at high frequencies, between ω H + ω N and ω H -ω N , allows to simplify the problem [START_REF] Farrow | Spectral density function mapping using 15 N relaxation data exclusively[END_REF]. With this condition, ω H + ω N , ω N and ω H -ω N are proportional to a same term and the spectral density function is thus determined at the five different frequencies by measuring only 15 N R 1 , R 2 and 1 H-15 N steady-state NOE values.

As evidenced by Figure 1.7, spin relaxation measured at a single external magnetic field leads to a sparse sampling of the spectral density function. Longitudinal relaxation at low fields provides supplementary information about the density function, in particular at intermediate frequencies. Unfortunately, measuring relaxation at low fields is prohibited by the costs in resolution and sensitivity. A method, dubbed high-resolution relaxometry, have been developed to combine the advantages of relaxing the spins at variable low fields and acquiring at high magnetic field [START_REF] Redfield | Shuttling device for high-resolution measurements of relaxation and related phenomena in solution at low field, using a shared commercial 500 MHz NMR instrument[END_REF]. The general setup combines conventional detection of the NMR signal in a high field spectrometer with the shuttling of the sample in the zone of the magnet with lower magnetic field during the mixing time at which relaxation happens. An application to the measurement of site-specific measurement of 15 N R 1 at three different "high fields" and seven "low fields" (from 5 to 0.5 T) of the protein ubiquitin was reported by Ferrage and colleagues [START_REF] Charlier | Nanosecond time scale motions in proteins revealed by high-resolution NMR relaxometry[END_REF]. The shuttle system allowed a transition between different part of the bore of the

0 J(ω) ω N ω H + ω N ω H -ω N ω H 15 N R 1 15 N R 2 1 H-15 N σ
(σ = γ S γ I R 1 η IS )
magnet within tens of milliseconds. Interestingly, their results highlighted (low frequency) ns-motions of ubiquitin which are underestimated when using only high-field relaxation.

Conformational exchange

In the previous sections, I reported some concepts developed in the framework of Redfield theory, and historical as well as more recent applications. I will discuss in the following additional contributions to transverse relaxation rates, not comprised in the latter theory, which result from exchange of protein conformations. This possibly leads to abnormal line broadening and may bias the analysis by overestimating J(0) contribution. This contribution can be quantified and reveals a slower regime of dynamics with a high impact for proteins function.

Effects on the NMR spectrum

Relaxation in solution reports on sub-ns fluctuations of protein sites. Above few nanoseconds, the overall tumbling of the molecule hides the contribution of internal motions to relaxation. Yet, slow interconversion between different conformations crucially form the bedrock of most protein functions [START_REF] Henzler-Wildman | Dynamic personalities of proteins[END_REF]. These changes may lead to a consequent modulation of the chemical shifts. Their effect on the NMR signal depends on the ratio between the rate of exchange, k ex , and the chemical shift difference between the different states, ∆ω [START_REF] Palmer | NMR characterization of the dynamics of biomacromolecules[END_REF][START_REF] Kleckner | An introduction to NMR-based approaches for measuring protein dynamics[END_REF][START_REF] Palmer | Chemical exchange in biomacromolecules: past, present, and future[END_REF]. For clarity purposes, let us consider two states of a same protein (prosaically numbered 1 and 2) in exchange. This situation is very often encountered, as a result of an exchange between either two conformations (active and non-active, folded and partially unfolded, open and closed, ...) or between apo-and ligand-bound states, or monomer and dimer, or other possibilities. State 1 converts into state 2 with a rate k 12 and state 2 to state 1 with k 21 . The rate of exchange is then defined as k ex = k 12 + k 21 . We attribute to each state a relative population p 1 or p 2 and a chemical shift ω 1 or ω 2 (with ∆ω = |ω 1 -ω 2 |). More complex situations are (although more sparsely) found in literature but can be treated in a similar way.

In the slow exchange regime (k ex ∆ω), the two resonances are well defined and the volume of each peak is proportional to the population of the corresponding conformer (Figure 1.8a-b). As long as the two states are sufficiently populated, information about structure and thermodynamics are readily available. It is additionally possible to obtain the rate of interconversion by exchange spectroscopy where the chemical shifts of the two species are encoded in a direct and an indirect dimension. A variable mixing time is introduced between the two detection periods during which magnetization is stored along the z-axis.

When the exchange is faster than longitudinal relaxation, a cross peak appears in the spectrum. The exchange between the two states is then monitored using the variation of cross-peak intensity with mixing time.

R 2 = R • 2 + R ex (1.12)
where R • 2 is the transverse relaxation rate in the absence of conformational exchange (as described by Redfield theory).

Relaxation dispersion

A standard way to determine R ex is to vary the pulse train frequency of the CPMG experiments ν CPMG (ν CPMG = 1/τ CPMG ) during R 2 measurement (Figure 1.8f). Indeed, magnetization is incompletely refocused by the spin echoes if random changes of chemical shift occur during the inter-pulses delay. However, if the pulsing frequency is high enough, the magnetization gets properly refocussed and R ex tends to zero. The dependency of R ex upon ν CPMG is calculated using the McConnell equations [START_REF] Mcconnell | Reaction rates by nuclear magnetic resonance[END_REF]. Different analytical formula have been proposed with specific assumptions. For a two-sites fast exchange process with highly skewed populations (p 1 p 2 ), the exchange contribution R ex can be expressed as [START_REF] Ishima | Estimating the time scale of chemical exchange of proteins from measurements of transverse relaxation rates in solution[END_REF]: [START_REF] Liao | Structure of the TRPV1 ion channel determined by electron cryo-microscopy[END_REF] where ω e is the effective magnetic field defined as: (c), 500 s -1 (d) and 5000 s -1 (e). Basic pulse sequences for the acquisition of CPMG (f), R 1ρ (g) and CEST (h) experiments. Starting magnetization is on z. (i) and (j), respectively CPMG and R 1ρ dispersion curves calculated using equation 1.13 with p 1 = 0.9, p 2 = 0.1 and, respectively k ex = 1000 s -1 and 10000 s -1 . (k) CEST profile simulated with equation 1.17 with p 1 = 0.95, p 2 = 0.05 and k ex = 50 s -1 . The spin lock is set at a frequency of 10 Hz during 0.5 s. In all simulations, R • 2 is set to 10 Hz and ∆Ω/2π to 100 Hz. Green and orange dotted lines in pannels (a-e) and (k) correspond, respectively, to the main and second resonances. Narrow and broad black rectangles indicateπ/2 and π pulses, respectively, and white rectangles continuous irradiations.

R ex = p 1 p 2 ∆ω 2 k ex k 2 ex k 2 ex + ω 2 e (1.
ω 2 e = 144 16τ 4 CPMG + p 2 1 ∆ω 4 . ( 1
This defines a so-called "relaxation dispersion" curve. A typical curve calculated with k ex = 1000 s -1 and p 1 = 0.9 is shown in Figure 1.8i. The timescale probed by this type of experiments depends on the range of available effective fields. For CPMG experiments, accessible ν CPMG frequencies are typically on the order of 10-1000 Hz. They are therefore mainly used to probe ms timescale motions. It is possible to probe faster µs motions with R 1ρ relaxation dispersion techniques.

R 1ρ is the rate at which a transverse magnetization decays when it is locked in the transverse plane by a continuous RF field at a frequency ω SL (also called spin-lock field). R 1ρ can be expressed as a combination of R 1 and R 2 [START_REF] Palmer | Characterization of the dynamics of biomacromolecules using rotating-frame spin relaxation NMR spectroscopy[END_REF]:

R 1ρ = R 2 sin 2 θ + R 1 cos 2 θ (1.15)
where θ is the tilt angle in the rotating frame defined as tan(θ) = ω SL / Ω. Here and in the following we consider the offsets relative to the carrier frequency ω rf :

Ω = ω -ω rf , Ω 1 = ω 1 -ω rf and Ω 2 = ω 2 -ω rf .
The contribution of two-site fast exchange process is then the same than given by equation 1.13 with:

ω 2 e = ω 2 1,eff ω 2 2,eff /ω 2 eff (1.16)
ω 1,eff , ω 2,eff and ω eff are the effective field for the first and second states and the population average.

ω 1,eff = Ω 2 1 + ω 2 SL ω 2,eff = Ω 2 2 + ω 2 SL ω eff = Ω2 + ω 2

SL

It is noteworthy that for a fast exchange process, the populations and differences in chemical shifts cannot be separated (see equation 1.13). They are generally grouped under a single parameter Φ ex = p 1 p 2 ∆Ω 2 . Additionally, Φ ex and k ex are also highly correlated in equation 1.13. Acquiring dispersion curves at least two different external magnetic field is therefore necessary (but not always sufficient) to separate the two parameters.

A basic pulse sequence to measure R 1ρ consists in transferring the magnetization to the transverse plane prior to applying a spin-lock field with typical powers comprised between 1-6 kHz for solution probes (Figure 1.8g). The effect of exchange on the observed rate can be understood similarly than for CPMG experiments, except that a continuous refocussing of the magnetization at a frequency ω SL needs to be considered instead of separated quasi-instantaneous inversions.

The main differences with CPMG dispersion experiments is the range of ω e values accessible and the fact that the spin-lock can be applied on-or off-resonance.

When the spin-lock is off-resonance, the effective field is reduced, so that information about slower timescale exchange may be recovered. When intermediate or slow exchange is present, varying the carrier offset permits to disentangle the populations and ∆ω parameters and to obtain the sign of the difference of the resonance offsets between the two populations [START_REF] Palmer | Characterization of the dynamics of biomacromolecules using rotating-frame spin relaxation NMR spectroscopy[END_REF].

Chemical Exchange Saturation Transfer

Chemical exchange saturation transfer (CEST) experiments, first developed in the field of magnetic resonance imaging, have been implemented for the study of conformational exchange in biomolecules [START_REF] Vallurupalli | Studying "invisible" excited protein states in slow exchange with a major state conformation[END_REF]. They are best suited for cases where the population of the alternative conformational state is too low or the exchange with the main conformer is too slow to produce an effect on relaxationdispersion type experiments. The experiments are conducted by saturating the transition at a certain frequency (Figure 1.8h). When the irradiation offset is on the chemical shift of a minor conformer, the intensity of the peak corresponding to the main conformer decreases with the time T of the irradiation following the approximate formula [START_REF] Palmer | Chemical exchange in biomacromolecules: past, present, and future[END_REF]:

I(T ) I(0) = cos 2 (θ)e -R 1ρ T (1.17)
where R 1ρ is as in equations 1.12, 1.13 and 1.16, and the factor cos 2 (θ) arrises from the projection of the magnetization from the z-axis onto the tilted frame. An example of CEST profile is given in Figure 1.8k for a rate of exchange of 50 s -1 and relative populations of the minor state of 0.05. Two deeps are observed when the RF field offset is on resonance with the major or the minor state, respectively. When the difference of chemical shifts is little or when the exchange is faster, the profile contains a single deep whose shape still is informative on the dynamical process. CEST may also complement dispersion relaxation for unraveling the population from the chemical shift difference as fitting parameters.

In summary, we reviewed a representative portion of the thick portfolio of NMR experiments developed for determining protein dynamics in solution. On one hand, parameters describing fast ps-ns internal motions can be extracted from the measurement of relaxation rates, with upper limit imposed by the molecular tumbling. On the other hand, relaxation dispersion techniques report on exchange broadening, a phenomenon which is extremely informative on slower µs-ms motions.

High-resolution NMR investigation of protein

dynamics in the solid state: a recent history

On the advantages of using MAS NMR

We had a quick glimpse on the large repertoire of sophisticated solution NMR techniques to study biomolecule dynamics built over the last few decades. An impressive number of applications covers a large part of stuctural biology and has strengthened the idea that NMR is of paramount importance in the field. Nevertheless, we see three important limitations to solution NMR of proteins:

• The protein of interest needs to be soluble. This condition already excludes a large number of biological systems from large insoluble assemblies such as amyloid fibrils to membrane proteins. In addition, many proteins are soluble only at low concentrations, hindering NMR investigation.

• As the molecular weight increases, the overall tumbling gets slower. The consequences are unfortunately shorter T 2 which sacrifices resolution. Even if the use of isotope labelling schemes and the development of Transverse Relaxation-Optimized SpectroscopY (TROSY)-based pulse sequences for high-field NMR have pushed the size barrier [START_REF] Pervushin | Attenuated T 2 relaxation by mutual cancellation of dipole-dipole coupling and chemical shift anisotropy indicates an avenue to NMR structures of very large biological macromolecules in solution[END_REF][START_REF] Fernández | TROSY in NMR studies of the structure and function of large biological macromolecules[END_REF], the investigation of large protein assemblies in solution remains challenging.

• The range of timescales available is limited by the molecular tumbling.

There is therefore a "blind spot" hiding motions ranging from ns to µs such as loop dynamics or side-chain rotations which are potentially relevant for function.

None of these points represents an intrinsic limitations for solid-state NMR. However, while in solution NMR, the molecular tumbling averages out the anisotropic interactions, in the solid state, those same interactions contribute to the NMR signal in addition to the isotropic components, compromising spectral resolution and sensitivity. Indeed, in a solid-state NMR sample, the molecule of interest is generally present with different orientations. The presence of anisotropic interactions broadens the lines since each orientation has a different resonance frequency. Spinning the sample at the so-called "magic angle" (arccos 1 √ 3 ) is a way to recover resolution in the NMR spectra [START_REF] Andrew | Nuclear magnetic resonance spectra from a crystal rotated at high speed[END_REF][START_REF] Lowe | Free induction decays of rotating solids[END_REF]. The versatility of MAS NMR opens the possibility to study at atomic resolution complex biological systems such as membrane proteins, viral capsids, amyloid fibrils or biomolecular complexes. In the following sections, we will discuss the specificities of spin relaxation under MAS conditions before giving an overview of the MAS NMR techniques for the study of protein dynamics.

Spin relaxation under MAS conditions

In Section 1.2.1, we saw that the correlation functions and their Fourier transform, the density functions, are key concepts to interpret spin relaxation in terms of molecular motions. When analyzing spin relaxation under MAS conditions, one has in principle to handle them by taking into account the specificities of MAS experiments. A detailed treatment is provided in reference [START_REF] Giraud | Quantitative analysis of backbone dynamics in a crystalline protein from nitrogen-15 spin-lattice relaxation[END_REF]. We will present some of the key points.

The spatial part of the stochastic Hamiltonian is usually decomposed in a basis of spherical tensors as it is more convenient to treat rotations than a cartesian basis. The components are, in the case of dipolar coupling and CSA interaction, rank-2 spherical harmonics Y 2l . The corresponding correlation functions in the molecular frame are therefore:

G l (t) = Y 2l (Ω(0))Y * 2l (Ω(t)) (1.18)
where Ω(t) is the orientation of the NH vector in an arbitrary but fixed molecular frame.

In MAS experiments, the hypothesis that the correlation function is independent of l made in solution is in principle not valid. In order to calculate the correlation functions in the laboratory frame, multiple operations have to be applied to equation 1.18, as illustrated in Figure 1.9. First, assuming that we do not work with monocrystals or aligned samples, each possible orientation of the molecule has to be taken into account. Then, one has to consider the rotation at the Magic Angle of the sample in the laboratory frame. This series of coordinates changes are mathematically obtained thanks to the Wigner rotation matrix D (2) . The correlation function in the laboratory frame therefore reads:

C a (t) = 2 b,b ,l=-2 D (2) * lb (α, β, γ)D (2) * ba (-ω R t, θ m , 0)D (2) lb (α, β, γ) D (2) b a (-ω R t, θ m , 0) × G l (t) (1.19) in which D (2)
lb are Wigner rotation matrix elements, α, β and γ the Euler angles of each crystallite, ω R the rotor spinning frequency and θ m the Magic Angle. In solids, because of the absence of overall tumbling, the correlation function C a (t) does not decay to zero but instead admits an asymptote for long times, corresponding to the order parameter S • with a diffusion time of 66 ns. Reprinted with permission from reference [START_REF] Giraud | Quantitative analysis of backbone dynamics in a crystalline protein from nitrogen-15 spin-lattice relaxation[END_REF] The correlation functions C a (t) also depends on the orientation of the molecule in the rotor frame. As a consequence, each crystallites (molecule orientation if the sample is not crystalline) relaxes at a different rate. The anisotropy of the longitudinal relaxation rates R 1 is shown in Figure 1.10. A crystallite with a given orientation with respect to the rotor axis (β) experiences different relaxation rates over a sample rotation (as γ is varied). As the intensity decay is generally much slower than the MAS frequency, it can be safely assumed that R 1 is averaged over γ angles for each β. Therefore, each "carousel" (ensemble of crystallites defined by a same β orientation) relaxes with a different rate R MAS 1 (β) defined as:

R MAS 1 (β) = (d 2 /4)[J MAS 0 (ω I -ω S ) + 3J MAS 1 (ω S ) + 6J MAS 2 (ω I + ω S )] +c 2 J MAS 1 (ω S )
(1.20) 1.22) in which d and c are defined in Table 1.1 and d ab (β) are the reduced Wigner matrix elements.

J MAS i (ω) = 2 j=2 C MAS ij (β)J PAS j (ω) (1.21) C MAS ij = 2 b=-2 [d jb (β)] 2 [d bi (-θ m )] 2 (
The observed relaxation process is multi-exponential unlike in solution. The signal intensity in function of the mixing time τ k is obtained by performing powder averaging:

I(τ k ) = I 0 π 0 exp(-R MAS 1 (β)t)sin(β)dβ π 0 sin(β)dβ (1.23)
In practice, the intensity decay so obtained significantly diverges from a monoexponential behavior only for long times, where the signal has decayed to about 1% of its initial value [START_REF] Giraud | Quantitative analysis of backbone dynamics in a crystalline protein from nitrogen-15 spin-lattice relaxation[END_REF][START_REF] Schanda | Studying Dynamics by Magic-Angle Spinning Solid-State NMR Spectroscopy: Principles and Applications to Biomolecules[END_REF]. As the difference is generally below the level of noise in MAS NMR experiments, the decay curves are usually fitted to a monoexponential function. Many studies have preferred a MAS-adapted version of the model free analysis to the more complicated treatment presented here. Schanda et al. evaluated the systematic error induced by the use of the model free approach [START_REF] Schanda | Studying Dynamics by Magic-Angle Spinning Solid-State NMR Spectroscopy: Principles and Applications to Biomolecules[END_REF]. They found the error to be dependent on the way motions are modeled but generally less than 10% and therefore concluded that the model free approach can be reasonably used for spin relaxation analysis in MAS NMR. In a study of the dynamics of Cu + , Zn 2+ -superoxide dismutase (SOD), the authors analyzed backbone 15 N R 1 and R 1ρ relaxation rates using either a model taking into account the anisotropy of the relaxation rates (diffusion in a cone or gaussian axial fluctuations) or the model free approach. Interestingly, the order parameters obtained with the different approaches were the same within experimental error. The model free approach has the benefits to be handled more easily, to demand less computational power and to enable a straightforward extension to multiple timescales of motions.

Quantitative analysis of spin relaxation in MAS NMR

Drawbacks (and the way to overcome them)

Several obstacles have until now prevented the wide application of NMR to the detailed investigation of the dynamics of complex bio-solids.

(i) Methods in MAS NMR using detection on heteronuclei have difficulty to provide sufficient sensitivity for measuring widespread multiple site-specific relaxation parameters. As a result, most site-specific relaxation studies in the solid state have been limited to small model proteins (GB1, SH3, ubiquitin). 1 H-detected multidimensional experiments at fast MAS combined with sample deuteration represent a promising way to overcome this problem, providing simultaneously the resolution and sensitivity required for the efficient determination of site-specific relaxation decays in larger proteins (see Chapter 2). This point was recently illustrated by the study of microcrystalline Cu + , Zn 2+ -SOD, a homodimeric enzyme of 153 residues [START_REF] Knight | Structure and backbone dynamics of a microcrystalline metalloprotein by solid-state NMR[END_REF]. In this study, in a 15 N-1 H CP-based HSQC (CP-HSQC) correlation acquired in about 15 min (on 3.5 mg of sample) on a 23.5 T spectrometer, the author were able to identify 136 (of which 103 baselineresolved) resonances. 15 N R 1 decay curves were sampled (8 time points) in 2.5 days. As a comparison, more than 2 weeks on a 700 MHz spectrometer were necessary by 13 C-detected methods to sample 15 N R 1 decays (4 time points) in microcrystalline Crh, a dimeric protein of 85 residues [START_REF] Giraud | Site-specific backbone dynamics from a crystalline protein by solid-state NMR spectroscopy[END_REF]. In that study, despite the smaller protein size and the larger amount of sample necessary, only 30 15 N rates were determined with sufficient accuracy.

(ii) Magnetization decay in MAS NMR results from two distinct processes: relaxation due to the stochastic modulation of the anisotropic interactions by the molecular motions and coherent effects (dipolar dephasing and spin diffusion) that are not related to the dynamics. Very often, the latter is driven by the strong 1 H-1 H dipolar coupling network and dominates the evolution of magnetization, hiding the actual relaxation and leading to systematic errors on the reported rates. Weakening of the coupling between protons via specific labelling schemes and faster MAS spinning are two complementary strategies that have been adopted to reduce the loss of magnetization through coherent processes.

In the following sections, we will discuss how these methods allow to extend the palette of accessible dynamics probes in MAS NMR.

Longitudinal relaxation

Longitudinal relaxation is exquisitely sensitive to fast ps-ns processes. However, when measuring R 1 in solids, one has to take into account the effect of spin diffusion. Spin diffusion causes magnetization transfer through the large 1 H homonuclear coupled network and leads to an increase and an homogenization of the relaxation rates over the different sites along the protein sequence. The effect of 1 H-driven spin diffusion on 15 N R 1 relaxation rates determination at 10 kHz MAS was investigated by Giraud et al. [48]. The authors found spindiffusion rates to be in the same order of magnitude than the 15 N R 1 relaxation rates in a [ 13 C, 15 N]-labelled protein. Deuteration of the sample is a way to weaken the network of 1 H-1 H interactions and therefore to slow down spin diffusion. This is usually achieved by expressing the protein in a fully deuterated medium (perdeuteration) and then by (partially) exchanging the amide deuterons with protons using buffers with controlled H 2 O/D 2 O ratios. The study of Giraud et al., showed that protein perdeuteration indeed decreases the observed 15 N R 1 relaxation rates, proving that spin-diffusion perceptibly contributes to the observed relaxation rates in fully protonated samples. Chevelkov et al. [START_REF] Chevelkov | Measurement of 15 N-T 1 relaxation rates in a perdeuterated protein by magic angle spinning solid-state nuclear magnetic resonance spectroscopy[END_REF] showed that At 20 kHz MAS spin diffusion is still present in protonated chicken α-spectrin SH3 domain and that reducing the proton content with perdeuteration and 10% reprotonation at the exchangeable sites leads to decay curves that are fully due to relaxation. 15 N spin diffusion, already modest at moderate MAS rates, appears to be negligible at fast MAS (> 50 kHz) even for protonated proteins [START_REF] Knight | Structure and backbone dynamics of a microcrystalline metalloprotein by solid-state NMR[END_REF][START_REF] Haller | Amplitudes and time scales of picosecond-to-microsecond motion in proteins studied by solid-state NMR: a critical evaluation of experimental approaches and application to crystalline ubiquitin[END_REF].

The measurement of 13 C R 1 relaxation rates are an interesting complement to 15 N relaxation. However, 13 C spin diffusion is expected to be stronger than 15 N due to its higher gyromagnetic ratio. A study of the effect of spin diffusion on 13 C relaxation at 60 kHz MAS was conducted using an alanine sample and microcrystalline GB1 [START_REF] Lewandowski | Measurement of Site-Specific C-13 Spin-Lattice Relaxation in a Crystalline Protein[END_REF]. 13 C R 1 were found to be similar for alanine crystals uniformly [ 13 C, 15 N]-labelled and 13 C-labelled at a single site, indicating that spindiffusion can be neglected. At fast MAS, 13 C longitudinal relaxation can therefore be used as a supplementary probe of backbone dynamics. The situation gets more complicated for aliphatic 13 C. Indeed, 13 C benefits from the large chemical shift difference with the aliphatic carbons which slows down spin diffusion. Coherent effects indeed lead to an overestimation of aliphatic 13 C relaxation rates even at 60 kHz MAS. The authors recommend the use, if available, of faster spinning rates, higher magnetic fields or more sparse labelling. Reif and coworkers introduced a new isotope labelling scheme where the 1 H and 13 C contents in side-chains are lowered, enabling accurate 13 Cα R 1 relaxation measurement [START_REF] Asami | Access to Cα Backbone Dynamics of Biological Solids by 13 C T 1 Relaxation and Molecular Dynamics Simulation[END_REF].

NOE enhancement factors have also been used in MAS NMR to quantify protein dynamics [START_REF] Giraud | Observation of Heteronuclear Overhauser Effects Confirms the 15 N-1 H Dipolar Relaxation Mechanism in a Crystalline Protein[END_REF][START_REF] Del Amo | Site-specific analysis of heteronuclear Overhauser effects in microcrystalline proteins[END_REF]. Notably, site-specific quantification of heteronuclear NOE enhancements was shown to be possible in highly deuterated samples and has been used to quantify backbone and side-chain dynamics of SH3 [START_REF] Del Amo | Site-specific analysis of heteronuclear Overhauser effects in microcrystalline proteins[END_REF].

Transverse relaxation

Longitudinal relaxation samples the high frequency range of the spectral density. To obtain further information on the slower dynamics components, it is therefore crucial to measure transverse relaxation. Site-specific determination of transverse relaxation has been avoided until very recently because of coherent dephasing. This phenomenon arrises from the insufficient averaging of anisotropic interactions, in particular dipolar coupling to protons, even at the fastest MAS rates nowadays available [START_REF] Bockmann | Spinning proteins, the faster, the better?[END_REF]. This static contribution strongly impacts 15 N T 2 relaxation even at very low proton contents, with fast MAS and under state-ofthe-art decoupling schemes [START_REF] Lewandowski | Site-Specific Measurement of Slow Motions in Proteins[END_REF]. A way to overcome this problem is to measure related parameters such as 15 N dipole CSA cross-correlated relaxation [START_REF] Chevelkov | Differential line broadening in MAS solid-state NMR due to dynamic interference[END_REF][START_REF] Chevelkov | Quantitative analysis of backbone motion in proteins using MAS solid-state NMR spectroscopy[END_REF][START_REF] Schanda | Quantitative analysis of protein backbone dynamics in microcrystalline ubiquitin by solid-state NMR spectroscopy[END_REF]. This method is based on the fact that dipolar coupling and CSA can interfere constructively or destructively in function of the spin state of the proton. It requires to resolve the J NH multiplets by avoiding proton decoupling during 15 N evolution. Cross-correlated relaxation rates Γ D/CSA are then obtained by subtracting the decay rate of each subpeak. Γ D/CSA directly reports on dynamics if we consider the two lines to be subjected to the same coherent contribution. Because the error on the relaxation rates of the two components are additive, these measurements require a very high experimental precision. In addition, the absence of decoupling reduces the resolution of the spectra. An alternative consists in measuring 15 N R 1ρ relaxation rates as reporters of ns-µs dynamics [START_REF] Lewandowski | Site-Specific Measurement of Slow Motions in Proteins[END_REF][START_REF] Krushelnitsky | Microsecond Time Scale Mobility in a Solid Protein As Studied by the 15 N R 1ρ Site-Specific NMR Relaxation Rates[END_REF]. In fully protonated proteins spin-lock field above 10 kHz are sufficient to measure accurate 15 N R 1ρ by using >45 kHz MAS [START_REF] Lewandowski | Site-Specific Measurement of Slow Motions in Proteins[END_REF].

R 1ρ experiments under MAS involve different processes with potentially similar time dependence : sample rotation (10 4 Hz), spin-lock irradiation (10 3 -10 4 Hz) and µs-ms dynamical processes. The possible interferences between these phenomena are expressed in the analytical expressions derived for R 1ρ in rotating solids [START_REF] Kurbanov | The nuclear magnetic resonance relaxation data analysis in solids: General R 1 /R 1ρ equations and the model-free approach[END_REF]:

R D 1ρ = d 2 8 1 3 2J(ω 1 -2ω R ) + 4J(ω 1 -ω R ) + 4J(ω 1 + ω R ) + 2J(ω 1 + 2ω R ) + J(ω I -ω S ) + 6J(ω I ) + 3J(ω S ) + 6J(ω I + ω S ) R CSA 1ρ = c 2 18 2J(ω 1 -2ω R ) + 4J(ω 1 -ω R ) + 4J(ω 1 + ω R ) + 2J(ω 1 + 2ω R ) + 3J(ω I ) (1.24)
In this equation, the density function is sampled at ω 1 ± ω R and ω 1 ± 2ω R . This dependency of R 1ρ on the spinning rate has a non-negligible influence for motions slower than µs when approaching rotary resonance conditions (i.e. ω 1 = ω R and ω 1 = 2ω R ). Measuring R 1ρ relaxation rates at different MAS frequencies can therefore be used to detect processes with long correlation times even when no fluctuations of the isotropic chemical shifts are involved [START_REF] Lamley | Intermolecular Interactions and Protein Dynamics by Solid-State NMR Spectroscopy[END_REF]. This situation, not encountered in solution, raises the question of the validity of Redfield theory in this time range. Indeed, one of the fundamental assumption of this theory is that correlation times are an order of magnitude faster than the evolution time of the density operator which may not be fulfilled in these conditions. Comparisons between numerical simulations and the analytical treatment from Redfield theory showed that the latter only slightly (if not at all) overestimate R 1ρ relaxation rates for motions between 10 -6 and 10 -3 s [START_REF] Schanda | Studying Dynamics by Magic-Angle Spinning Solid-State NMR Spectroscopy: Principles and Applications to Biomolecules[END_REF][START_REF] Smith | Characterization of fibril dynamics on three timescales by solid-state NMR[END_REF].

Implementation

The combined determination of 15 N R 1ρ and R 1 relaxation rates to detect backbone mobility has recently become a popular strategy and has been used for the quantification of the amplitude and timescale of motions in different systems (Table 1.2).

The inherent long acquisition times of relaxation data generally obstruct the detailed analysis of protein dynamics by MAS NMR. Recent implementations of relaxation studies took advantage of the boost in sensitivity provided by proton detection (see Chapters 2 and 3) together with perdeuteration and fast MAS to investigate site-specific dynamics. An example of this strategy is a study of microcrystalline Cu + , Zn 2+ -SOD [START_REF] Knight | Structure and backbone dynamics of a microcrystalline metalloprotein by solid-state NMR[END_REF]. 15 N R 1 and R 1ρ relaxation rates were used to quantify backbone dynamics (Figure 1.11). A 1D Gaussian Axial Fluctuation (1D-GAF) model was used for the calculations (Appendix D). The order parameters and timescale so obtained followed a gradient of flexibility in the loop regions in this approach, the anisotropy of the relaxation rates is taken into account [START_REF] Giraud | Quantitative analysis of backbone dynamics in a crystalline protein from nitrogen-15 spin-lattice relaxation[END_REF].

and are quite uniform in the β strands, as expected. Noteworthy, the internal motions described in this study were characterized by ns "model free" correlation times and would therefore belong to the "blind spot" of solution NMR. Readapted with permission from [START_REF] Knight | Structure and backbone dynamics of a microcrystalline metalloprotein by solid-state NMR[END_REF].

More complex models of motions can be elaborated similarly to what has been proposed in solution to take into account the multiplicity of timescales. Notably, measuring 15 N-1 H dipolar coupling was shown to be important for constraining the spectral density [START_REF] Chevelkov | Quantitative analysis of backbone motion in proteins using MAS solid-state NMR spectroscopy[END_REF][START_REF] Schanda | Quantitative analysis of protein backbone dynamics in microcrystalline ubiquitin by solid-state NMR spectroscopy[END_REF]. Following this line, another study proposed to measure relaxation of both 15 N and 13 C to precise backbone dynamics [START_REF] Lamley | Unraveling the complexity of protein backbone dynamics with combined 13 C and 15 N solid-state NMR relaxation measurements[END_REF]. In particular, the authors showed that 13 C R 1ρ can be used without undesired coherent effects for protonated proteins at 60 kHz MAS by applying strong (>15 kHz) spin-lock frequencies.

The impressive number of degrees of freedom and interactions present in a protein greatly complicates the analysis. Even the most complex models used for interpreting relaxation data use no more than 2-3 timescales of motions. These assumptions simplify the nature of the dynamics, in particular because they assume a discrete time distribution. However, this approach permits to elaborate a picture of the molecular processes and the resulting order parameters can be compared to computational simulations [START_REF] Mollica | Atomic-Resolution Structural Dynamics in Crystalline Proteins from NMR and Molecular Simulation[END_REF]. Noteworthy, Smith et al. proposed another strategy to catch protein dynamics based on the production of "optimized detectors" by linear combinations of NMR observables which minimize their overlap in the time domain and on the observation of the "response" of the system to these detectors [START_REF] Smith | detectors" for dynamics analysis in solid-state NMR[END_REF]. No specific model is therefore required as the analysis only reports on how much motion there is in different time windows.

In fine, the multiplication of relaxation-based probes of dynamics helps to better define motional processes in the ps-µs range whether one uses a model or not. We reviewed some of the advances on MAS NMR toward the description of protein dynamics by the measurement of relaxation. They are enabled by the use of specific labelling schemes and/or fast MAS rates since both factors limit coherent contributions to the decay of magnetization. Because MAS NMR is not restricted by the molecular tumbling, relaxation rates can also report on slow dynamics. Transverse relaxation rates may additionally have a contribution from fluctuations of the isotropic chemical shifts as in solution NMR.

Conformational exchange

As stated in Section 1.3.2, relaxation dispersion experiments are exquisitely sensitive to processes involving slow chemical shift fluctuations. A first application of CPMG dispersion in the solid-state was performed on a highly deuterated protein (perdeuterated with 20% reprotonation at the exchangeable sites) at 50 kHz MAS [START_REF] Tollinger | Site-Resolved Measurement of Microsecond-to-Millisecond Conformational--Exchange Processes in Proteins by Solid-State NMR Spectroscopy[END_REF]. Important coherent contributions to the absolute T 2 values may be expected in these conditions. Nevertheless, the authors showed that this contribution does not depend significantly on the CPMG frequency. Since relaxation dispersion only takes into account differences between the rates, coherent effects may simply cancel during the analysis. Simulations indicated that, in addition, anisotropic interactions impact differently transverse relaxation rates according to ν CPMG even if this systematic error may generally be absorbed in the experimental error.

Alternatively, the possibility to use 15 N R 1ρ relaxation dispersion to determine protein dynamics in MAS NMR was investigated by Emsley and colleagues on protonated GB1 [START_REF] Lewandowski | Site-Specific Measurement of Slow Motions in Proteins[END_REF]. Interestingly, MAS NMR probes are usually capable of much higher RF powers during spin-lock periods than solution NMR probes. This permits to sample further the relaxation dispersion curves and therefore to access a broader range of timescales. In the above-mentioned work, the experimental conditions enabled the measurement of 15 N R 1ρ relaxation rates without coherent effect only with high RF fields (ν SL >10 kHz). Later on, 15 N R 1ρ relaxation dispersion curves were acquired on a perdeuterated (with subsequent 50% proton back exchange) at 40 kHz MAS, showing that, in these conditions, 15 N R 1ρ with spin-lock powers as low as 2 kHz are free from coherent effects [START_REF] Ma | Probing Transient Conformational States of Proteins by Solid-State R11 Relaxation-Dispersion NMR Spectroscopy[END_REF]. The dispersion curves were analyzed with the Bloch-McConnell formalism (equation 1.13) for the lowest spin-lock frequencies (Figure 1.12a) and with numerical simulations for spin-lock frequencies approaching rotary resonances (Figure 1.12b). Interestingly, taking into account both parts of the relaxation dispersion curve improved the fitting procedure and gave additional model-dependent structural information on the process such as the jump angle of the NH bond vector between the two conformations.

Very recently, Linser and coworkers proposed to complement on-resonance 15 N R 1ρ relaxation dispersions with off-resonance 15 N R 1ρ relaxation rates and CEST experiment [START_REF] Rovó | Microsecond Timescale Protein Dynamics: a Combined Solid-State NMR Approach[END_REF]. By doing so, they quantified SH3 exchange parameters by using a single external magnetic field. The obtained CEST profiles did not exhibit a second deep corresponding to a minor conformer but instead were broadened in the presence of exchange with rates in the kHz range. The introduction of CEST experiments for conformational exchange determination in high-resolution MAS NMR is an important step forward for studying systems possessing "invisible", slowly-exchanging conformers.

Applications

A notable application of the recent developments of dynamics by MAS NMR is the study of a protein-antibody complex performed by Lewandowski and coworkers in 2015 [START_REF] Lamley | Intermolecular Interactions and Protein Dynamics by Solid-State NMR Spectroscopy[END_REF]. In this work, the authors compared the dynamical behavior of [100% 1 H N , 2 H, 13 C, 15 N]-labelled GB1 in crystals and in a protein-antibody 15 N R 1ρ relaxation is enhanced at the rotary resonance condition: 15 N R 1ρ rates (red) obtained at 20 kHz MAS with simulated curves for an exchange rate of 2900 s -1 and a minor state relative population of 0.093. Reprinted with permission from reference [START_REF] Ma | Probing Transient Conformational States of Proteins by Solid-State R11 Relaxation-Dispersion NMR Spectroscopy[END_REF] complex of more than 300 kDa. By measuring 15 N R 1 , 15 N R 1ρ at multiple MAS rates and 15 N R 1ρ relaxation dispersion, they were able to obtain insights into flexibility, slow conformational dynamics and rigid-body motions of GB1 in the two forms. While the local flexibility is unaffected by the intermolecular interactions, some residues experience conformational exchange only when bound to antibodies. Interestingly, the 15 N R 1ρ were found to be about six times higher in the complex than in the crystals. The authors attributed this difference to overall tumbling motion in the complex which was quantified by fitting 15 N R 1ρ rates with variable MAS frequency. Overall, this paper showed the capacities of MAS NMR to disentangle different types of dynamics of a protein diluted in a large supramolecular complex. Importantly, solid-state NMR evolves today toward more and more complex systems, potentially lacking symmetry, where interactions with a protein and its environment (protein-protein, protein-lipids, proteinnucleic acids, ...) and dynamics are important for the function.

Anisotropic interactions as additional probes of dynamics in MAS NMR

Measurement of anisotropic interactions is an important complement to spin relaxation. Let us consider an interaction for which the anisotropic component is noted δ µ . In presence of molecular motions with characteristic timescale τ so that τ δ µ 1, the interaction tensor is partially averaged and the effective value of the anisotropy, δ µ,eff , is reduced compared to the rigid limit. The order parameter can then be directly calculated as S 2 = (δ µ,eff /δ µ,rigid ) 2 . The determination of δ µ,eff is therefore a direct reporter of the amplitude of motions faster than the coupling constant.

Anisotropic interaction

Notation used here Typical value † CSA scales with the external magnetic field and is therefore expressed in ppm.

In solution, anisotropic interactions are averaged by a stochastic process (rotational diffusion). The value of the anisotropy of interaction can be retrieved by the measurement of cross-correlated relaxation [START_REF] Tjandra | Protein backbone dynamics and 15 N chemical shift anisotropy from quantitative measurement of relaxation interference effects[END_REF][START_REF] Fushman | Direct measurement of 15 N chemical shift anisotropy in solution[END_REF][START_REF] Yang | A study of protein sidechain dynamics from new 2 H auto-correlation and 13 C cross-correlation NMR experiments: application to the N-terminal SH3 domain from drk1[END_REF][START_REF] Chiarparin | Relaxation of two-spin coherence due to cross-correlated fluctuations of dipole-dipole couplings and anisotropic shifts in NMR of 15 N, 13 C-labeled biomolecules[END_REF] or by using an aligning medium [START_REF] Yao | Site-specific backbone amide 15 N chemical shift anisotropy tensors in a small protein from liquid crystal and cross-correlated relaxation measurements[END_REF][START_REF] Tjandra | Direct measurement of distances and angles in biomolecules by NMR in a dilute liquid crystalline medium[END_REF]. In this latter case, special care has to be provided to avoid effects from solvent change when combining these data to relaxation measurements. In MAS NMR, measurement of anisotropic interactions is more straightforward. The averaging of these interactions by MAS is indeed controlled by the user so that it is possible to specifically recouple anisotropic interactions with RF pulses interfering with the rotation at the magic angle. In this way, 1 H-X (X= 15 N or 13 C) dipolar coupling, 2 H quadrupolar interaction, and 15 N CSA (Table 1.3) can be reintroduced and measured to determine motions of protein backbone and side-chains.

Dipolar coupling

The first experiment introduced to recouple heteronuclear dipolar interaction under MAS conditions is cross-polarization under MAS (CP-MAS) [START_REF] Schaefer | Carbon-13 nuclear magnetic resonance of polymers spinning at the magic angle[END_REF]. CP was first designed to enhance the magnetization of low-γ nuclei (S) by the transfer of polarization from higher-γ nuclei (I) in static ssNMR [START_REF] Pines | Proton-enhanced nuclear induction spectroscopy. A method for high resolution NMR of dilute spins in solids[END_REF]. Magnetization transfer is allowed by using a continuous wave irradiation on both channels under the socalled Hartmann-Hahn conditions, stipulating that the frequency of the RF field on both channels should be equal (ν 1,I = ν 1,S ). When the rotor is spun at the magicangle, these conditions translate to: [START_REF] Lipari | Padé approximants to correlation functions for restricted rotational diffusion[END_REF] where ν R = 1/τ R , ν 1,I and ν 1,S the intensity of the RF pulses being applied respectively on I and S, and n ∈ N. The parameter defines the zero-quantum (ZQ, = -1) and double-quanta (DQ, = +1) CP conditions. Variations on the theme of CP was used for motion-averaged dipolar coupling determination on biomolecules. For example, Lee-Goldburg CP uses a tilted spin-lock on the 1 H channel to suppress 1 H-1 H homonuclear dipolar coupling [START_REF] Lee | Nuclear-magnetic-resonance line narrowing by a rotating rf field[END_REF][START_REF] Hong | Investigation of molecular motions by Lee-Goldburg cross-polarization NMR spectroscopy[END_REF][START_REF] Lorieau | Conformational dynamics of an intact virus: order parameters for the coat protein of Pf1 bacteriophage[END_REF]. The effective field on 1 H during CP is tilted from the z-axis by the Magic Angle so that interactions described by rank-2 tensors vanish. This scheme is therefore extremely sensitive to offset and pulse miscalibration. Alternatively, Cross-Polarization with Polarization Inversion (CPPI) [START_REF] Wu | Cross polarization with high-speed magic-angle spinning[END_REF] was implemented to study heavily deuterated proteins [START_REF] Chevelkov | Accurate determination of order parameters from 1 H, 15 N dipolar couplings in MAS solid-state NMR experiments[END_REF]. During CP, the Hartmann-Hahn conditions alternate between n = +1 and -1 to reduce the effect of RF inhomogeneity.

|ν 1,I + ν 1,S | = nν R (1.
Since its introduction in 1989 [START_REF] Gullion | Rotational-echo double-resonance NMR[END_REF], Rotational-Echo, DOuble-Resonance (RE-DOR) scheme has been widely used in MAS NMR of proteins. Its popularity comes from its robustness to RF missettings and resonance offsets over a large range of MAS rates as well as its straightforward setup and analysis [START_REF] Schanda | Accurate measurement of one-bond H-X heteronuclear dipolar couplings in MAS solid-state NMR[END_REF]. One of the possible pulse sequences consists in a train of rotor synchronized inversion pulses on the I channel which re-introduces the IS dipolar coupling and a central pulse in S channel to refocus the isotropic chemical shifts (Figure 1.13a). When no pulse is applied on I channel, the magnetization is subject to a simple spin echo. REDOR experiments are generally executed such that for each mixing time the signal with (S) and without (S 0 ) rotor-synchronized pulses are successively acquired so to have a control experiment and to remove unwanted contribution from transverse relaxation. It is then possible to directly extract the magnitude of the dipolar coupling, δ D , from the modulation of ∆S/S 0 (∆S = S 0 -S) (Figure 1.13b): 1.26) in which J is a Bessel function of the first kind and N c is the number of rotor periods. This equation is valid only for an isolated spin pair and for infinitely short pulses (i.e. τ R τ π,I ). In reality, this second condition is generally not met, in particular when using fast MAS. However, it is possible to correct for the deviation from the ideal case by scaling up the effective dipolar coupling δ D,eff by a constant 1/κ with [START_REF] Jaroniec | Recoupling of heteronuclear dipolar interactions with rotational-echo doubleresonance at high magic-angle spinning frequencies[END_REF]:

∆S S 0 = 1 - √ 2π 4 J 1/4 ( √ 2λ)J 1/4 (- √ 2λ) λ = N c τ R δ D 2π (
κ = δ D,eff δ D = cos((π/2)(2τ π,I /τ R )) 1 -(2τ π,I /τ R ) 2 (1.27)
More importantly, 1 H-15 N/ 13 C REDOR is sensitive to the presence of remote 1 H spins and is therefore generally used in proteins where a high number of hydrogens are substituted by deuterons at relatively fast MAS (>30 kHz). It is noteworthy that dipolar coupling tensor may possess an asymmetric component in the cases where the fluctuations of the dipolar vector are not symmetric. This is the case for example for methyl rotameric jumps where the populations of the different isomers are not equilibrated [START_REF] Schanda | Solid-State NMR Measurements of Asymmetric Dipolar Couplings Provide Insight into Protein Side-Chain Motion[END_REF]. In such cases, simulation of REDOR curves gives access to the properties of side-chain rotameric states populations by fitting a single experimental dataset. (d,e) Time evolution of the CP-VC signal calculated for a 1 H-15 N spin pair with the equations given in [START_REF] Park | Resolution and measurement of heteronuclear dipolar couplings of a noncrystalline protein immobilized in a biological supramolecular assembly by proton-detected MAS solid-state NMR spectroscopy[END_REF] (d) and its Fourier transform (e). For sake of clarity, we omit here the 13 C channel which is used for decoupling during indirect acquisition.

The last years have witnessed the development of MAS probes reaching a new regime of fast rotation (ν R > 100 kHz) [START_REF] Lamley | Intermolecular Interactions and Protein Dynamics by Solid-State NMR Spectroscopy[END_REF][START_REF] Agarwal | De Novo 3D Structure Determination from Sub-milligram Protein Samples by Solid-State 100 kHz MAS NMR Spectroscopy[END_REF][START_REF] Nishiyama | A. 3D 15 N/ 15 N/ 1 H chemical shift correlation experiment utilizing an RFDR-based 1 H/ 1 H mixing period at 100 kHz MAS[END_REF][START_REF] Andreas | Structure of fully protonated proteins by proton-detected magic-angle spinning NMR[END_REF]. These advancements open the possibility of high-resolution 1 H detection with fully protonated samples, as it will be reviewed in Chapter 3. It would therefore be appealing to perform structural and dynamical characterization on the same sample and avoiding at the same time costly deuteration schemes. A recoupling strategy has recently been proposed for ultra-fast MAS applicable for cases where neighboring protons are present [START_REF] Park | Resolution and measurement of heteronuclear dipolar couplings of a noncrystalline protein immobilized in a biological supramolecular assembly by proton-detected MAS solid-state NMR spectroscopy[END_REF][START_REF] Paluch | Simple and accurate determination of X-H distances under ultra-fast MAS NMR[END_REF][START_REF] Paluch | Theoretical study of CP-VC: A simple, robust and accurate MAS NMR method for analysis of dipolar C-H interactions under rotation speeds faster than ca. 60 kHz[END_REF]. The scheme is a simple use of CP between I and S spins with a varying contact-time (Figure 1.13c) and is therefore dubbed Cross-Polarization with Variable Contact-time (CP-VC). The time dependance of the signal (Figure 1.13d) can be Fourier transformed to obtain a dipolar lineshape as presented in Figure 1.13e. The distance between the two horns of the profile, ∆, is directly related to the magnitude of the dipolar coupling:

∆ = δ D,eff 2π √ 2 (1.28)
The robustness of the sequence to the presence of CSA and close-by protons is related to the use of fast MAS rates.

Chemical shift anisotropy

The measurement of the anisotropy of chemical shifts has attracted growing interest in the last decades since they contain information about local structure (backbone geometry, hydrogen bonding, rotameric states of the sidechains) and dynamics. In MAS NMR, CSA parameters can be measured either by fitting the CSA spinning side-bands at slow MAS rates [START_REF] Herzfeld | Sideband intensities in NMR spectra of samples spinning at the magic angle[END_REF] or by specific recoupling at faster MAS rates [START_REF] Chan | Recoupling of chemical shift anisotropies in solidstate NMR under high-speed magic-angle spinning and in uniformly 13 C-labeled systems[END_REF]. Two studies from Rienstra's group attracted the attention on the advantages of determining the tensor elements with site resolution in MAS NMR by combining two isotropic chemical shifts dimension with a CSA dimension [START_REF] Wylie | Determinations of 15 N chemical shift anisotropy magnitudes in a uniformly 15 N, 13 C-labeled microcrystalline protein by three-dimensional magic-angle spinning nuclear magnetic resonance spectroscopy[END_REF]100].

CSA tensor elements have a complex dependence on the local molecular structure and on dynamics which complicates their use to analyze protein motions. In a recent paper, a sophisticated computational analysis was presented, which allow to unravel those contributions and interpret the experimental data [101]. 15 N CSA were measured with site resolution on HIV-capsids (Figure 1.14a and b). A comparison with density functional theory (DFT) computing of CSA based on the structure of the monomeric unit showed divergences between experimental and calculated parameters. In contrast, incorporating molecular dynamics (MD) trajectories in the simulations improved the agreement, demonstrating the importance of taking into account motion averaging in the analysis. This work therefore showed how using computational simulation allows to integrate CSA measurements to the analysis of protein dynamics.

Quadrupolar coupling

As 2 H is a quadrupolar spin-1 nucleus, it is challenging to study by NMR. At the same time, the large anisotropy of its quadrupolar coupling (Table 1.3) is a wealth of information about dynamics. Numerous solid-state NMR investigations have used 2 H lineshape analysis to determine biomolecules motions [103][104][105][106]. These studies generally relied on the incorporation of 2 H at a single site. In 2006, Hologne et al. proposed a 2 H-13 C-13 C correlation scheme to measure site-specifically 2 H signals in uniformly labelled α-spectrin SH3 domain [107]. This approach is particularly important as samples with high 2 H content in the sidechains are commonly prepared to recover resolution in MAS NMR. This was further illustrated by a recent study of Rienstra's group where the authors present a novel RF irradiation scheme to obtain quadrupolar parameters for 140 sites of GB1 [102]. Figure 1.14 C shows a 2D 2 H-13 C plane extracted from the 3D spectrum acquired on [ 2 H, 13 C, 15 N]-GB1. Fitting the resolved line-shapes gives the averaged quadrupolar constant C Q and the asymmetry of the quadrupolar coupling η. By doing so, the authors drew a complete picture of backbone and side-chain dynamics of GB1, with implications of such motions for its biological function.

Conclusive remarks

In conclusion, the potential of MAS NMR for studying protein dynamics was highlighted. Spin relaxation in solids is not limited by the overall tumbling of the protein but instead is sensitive to all timescales ranging from picoseconds to milliseconds. In addition, much higher RF powers can be employed during 15 N spin-lock periods in MAS NMR compared to its solution counterpart. This results in the definition of relaxation dispersion curves with much broader ranges, which in turn translates into the possibility of monitoring slow exchange phenomena on a larger window. Finally, motion averaged anisotropic interactions can be measured with tailored RF irradiation schemes, complementing spin relaxation experiments.

The analysis of protein dynamics in MAS NMR is challenged by the presence of coherent contributions to the observed relaxation rates. The development of isotope labelling schemes and NMR probes with fast MAS capacities have increased the amount of accessible information on molecular dynamics, to the point that parameters probing the whole range of timescale of motions above 10 -12 s can be now measured with site resolution (Figure 1.15).

While further developments are still needed, in particular for the determination of side-chains dynamics, we expect that this impressive arsenal of MAS NMR techniques will contribute to the detailed understanding of the role of dynamics in the stability and/or function of complex biological systems such as viral capsids, amyloid fibrils or even asymmetric biomolecular complexes. Resonance assignment and structural investigation of proteins by 1 H-detection and MAS at 60 kHz

Introduction

Since its introduction in 1958 [START_REF] Andrew | Nuclear magnetic resonance spectra from a crystal rotated at high speed[END_REF][START_REF] Lowe | Free induction decays of rotating solids[END_REF], MAS has been successfully applied for determination of structure and dynamics of proteins [108][109][110][111][112][113][114]. Most of the MAS NMR studies to date have combined moderate MAS rates (∼ 10 kHz) with multidimensional experiments detected on heteronuclei. The advantage of detecting either 13 C or 15 N coherences for obtaining high resolution spectra is twofold. On one hand, the low gyromagnetic ratio of both 13 C and 15 N compared to 1 H reduce the strength of the dipolar couplings they experience, so that low spinning frequencies are sufficient for efficient averaging. In the other hand, 13 C and 15 N have higher dispersion of chemical shifts. However, for the same reason 13 C/ 15 N detected experiments are rather insensitive and therefore demanding for long acquisition times and large amount of samples.

An obvious way to enhance sensitivity in MAS NMR is to detect proton signals [115,116]. This approach has met a growing success in the last decades thanks to progress in MAS capacities of the modern NMR probes. A first section (Section 2.2) is dedicated to the discussion of the benefits in terms of sensitivity and resolution of 1 H-detection at fast MAS [117].

In the following sections, I show how these benefits can be exploited for backbone assignment and structural investigation of perdeuterated and selectively reprotonated samples at 60 kHz MAS. Section 2.3 describes an assignment procedure based on 13 C resonances matching [118] which I applied on microcrystalline and fibrillar samples. Additionally, I contributed to the design and implementation of an alternative set of experiments for robust sequential assignment via 15 N-15 N matching in a single spectrum [119] (Section 2.4). Finally, Section 2.5 presents new experiments for the detection of H-bonds in solids. Such experiments are used to provide insights into the registry of amyloid fibrils.

Sensitivity and resolution of 1 H-detected NMR spectra at ultra-fast MAS

The sensitivity advantage of proton detection was recognized early by the solution NMR community [121]. Linewidths and other factors being equal, the sensitivity of detection scales with a factor γ 3/2 . Therefore 1 H detection is approximately 8 or 31-fold more sensitive compared with 13 C or 15 N, assuming that initial polarization is on 1 H in both cases.

In solid proteins, however, the network of dipole-coupled protons results in severe homogeneous broadening that is difficult to overcome using MAS alone, due to in part the narrow chemical shift range in the 1 H dimension, which is not well separated even at the highest magnetic fields available [122]. Nevertheless, the last two decades have witnessed an increase in the MAS frequency from 10 to 110 kHz and an increase in the magnetic field to up to 1 GHz, both of which have driven a significant improvement in spectral resolution required for investigation of global protein structure. The effect of MAS and magnetic field are illustrated in Figure 2.1 on a 15 N-1 H dipolar-based heteronuclear single-quantum correlation (CP-HSQC) for the Single-Stranded-DNA Binding (SSB) protein from E. coli [120]. An increase in MAS rates causes a linear improvement in 1 H coherence lifetimes, resulting in narrower lines (Figure 2.1a-c and h), while by increasing the magnetic field, nearby protons are separated in frequency, which results in less efficient spin flip-flops, and further narrowing the homogeneous line width (Figure 2.1d-f and h).

High resolution can be reached by combining MAS with a decrease of the proton content in the sample [130][131][132]. Resolution enhancement following protein deuteration was first demonstrated at a MAS rate of 10-20 kHz on samples with full protonation of the amide sites [133], and Reif's group showed that a further improvement in resolution was achieved by lowering the levels of amide protonation (typically 10-40%, see Table 2.1) [124,125]. With faster MAS, however, the network of 1 H dipolar interactions is more efficiently averaged and less 1 H dilution is necessary to achieve the same 1 H line widths (Table 2.1) [128,134]. In fully back-exchanged (100% 1 H at exchangeable sites) and otherwise [ 1 H, 13 C, 15 N]-labeled microcrystalline proteins, high-quality CP-HSQCs (with typical 1 H linewidths of 50 Hz) can be obtained in a very short time (typically few minutes) on very small amounts of samples, typically 2-3 mg at 60 kHz [128], or even less than 1 mg at 100 kHz [127]. Figure 2.2 shows the fingerprint correlation spectra of two small microcrystalline domains, a sedimented nucleocapsid, and two α-helical and β-barrel membrane proteins. The high resolution of these correlations requires a high degree of sample homogeneity, which is often achieved by preparing crystals (or 2D crystals in the case of membrane proteins), although these preparations do not need to possess a macroscopic three-dimensional order. For example, sedimented proteins and fibrils can exhibit high-resolution spectra due to a high degree of local order.

δ( 15 N) / ppm ν R =22.5 kHz ν R =40 kHz ν R =60
In order to achieve faster MAS rates, the rotor size and, correspondingly, the sample volume, must be reduced. At first glance, this may appear as a limitation, given the loss in sensitivity that occurs with a reduction in the amount of sample. However, although the sample volume decreases by a factor of about 12 between a 1.3 mm rotor and a 3.2 mm rotor, the resulting loss in sensitivity is significantly lower due to more effective inductive coupling of the sample to the smaller coil. Without considering changes in the number of turns in the coil, or the spacing, the sensitivity of a typical-sized NMR coil scales as the inverse of the diameter [135]. Thus the 12-fold reduction in sample volume is partially compensated by a 2.5-fold improvement in sensitivity per mg of sample, amounting to a 5-fold reduction in sensitivity overall. Without considering changes in line width, the sensitivity of proton detection in a 1.3 mm coil is still nearly twice as sensitive than carbon detection in a 3.2 mm coil.

Of great importance for biological samples, this ten-fold reduction in sample volume allows the investigation of only 1-2 mg of sample, extending the application scope of MAS NMR to the study of proteins that are difficult to produce in large amounts. At full proton back-exchange in deuterated samples, maximal sensitivity is found near 40 kHz MAS (without a drastic reduction in resolution). This implies that 1.9 mm rotors are ideally suited for low dimensionality spectra involving few magnetization transfers. When additional coherence transfers are considered, higher sensitivity is observed with faster rotation in 1.3 mm rotors [136]. Additionally, above 40 kHz MAS, heteronuclear decoupling are efficiently performed with low-power irradiation [137,138]. This produces a further increase in sensitivity by enabling a reduction in the interscan delay, in order to obtain highly resolved spectra without rf-induced heating and subsequent sample deterioration. Overall these gains in sensitivity offset the loss in sample volume when using small diameter rotors for high frequency MAS.

In summary, proton detection provides a boost of sensitivity for ssNMR spectra provided that high resolution is recovered by a combination of fast MAS and/or perdeuteration with back protonation at the exchangeable sites or alternative labelling schemes. The increase in MAS rate capacities of modern NMR probes correlates with less need for proton dilution and high-power decoupling schemes. In addition, the volume of the sample is reduced with the size of the rotor. The loss of sensitivity due to the reduction of material amount is compensated by better inductive couplings of the sample with the coil, the higher acceptable level of protonation and higher efficiency of transfers in multidimensional experiments. The advantages of proton detection at ultra-fast MAS (> 60 kHz) are demonstrated in the following with examples of strategies developed in our group for resonance assignment, structure investigation and dynamics studies.

Backbone resonance assignment via 13 C linking 2.3.1 Introduction

Assignment of resonances is a prerequisite for the structure and dynamics investigations at atomic resolution. This step is done by applying magnetization transfer schemes that are defined in function of the atom arrangement through the primary sequence. In MAS NMR, this generally translates into the acquisition of 2D or 3D experiments correlating amide 15 N to backbone and side-chain 13 C spins [139,140]. However, this approach suffers from high level of ambiguity and/or long experimental times.

The availability of resolved 1 H resonances offers the possibility to increase the resolution at amide sites by adding a third dimension and therefore reduces ambiguity of the spectra. Backbone assignment strategies based on the acquisition of multidimensional correlations involving 1 H, 13 C and 15 N spins, in a similar spirit to routine procedures used in solution, were first developed for highly deuterated proteins [141]. Assignment of perdeuterated and 100% proton back-exchanged proteins was subsequently performed with 1 H detection at >40 kHz MAS rates [118,120,128,142,143]. In particular, a set of six 3D experiments has been shown to be highly efficient for different sample preparations and aggregation states, yielding data of high quality and redundancy that is amenable to unsupervised data analysis [118]. In the following, I will describe this assignment procedure in more detail and present its application to the D76N variant of the protein β2m in microcrystalline and fibrillar states.

Radiofrequency irradiation schemes

As mentioned above, 1 H-15 N CP-HSQC spectra obtained at 60 kHz MAS and a high magnetic field (optionally combined with sample perdeuteration) generally display high resolution (Figure 2.2) and can serve for "fingerprinting" the protein. Additional 13 C dimension can be added to the spectra by transferring magnetization from 1 H to 13 C and then from 13 C to 15 N (Figure 2.3a and c). For heteronuclear transfer steps, CP-based experiments are more efficient than IN-EPT experiments. By changing the offset of the CP RF pulses on 13 C, it is possible to detect either 13 Cα or 13 C resonances. The two pulse sequences designed in this way are named (H)CANH (Figure 2.3a) and (H)CONH (Figure 2.3c). They correlate the 1 H and 15 N resonances of the i th residue to the 13 Cα resonance of the same residue or to the 13 C resonance of the (i -1) th residue.

In order to link pairs of amide 1 H and 15 N resonances, it is necessary to use additional experiments with 13 C-13 C transfer steps. These experiments, namely (H)(CO)CA(CO)NH and (H)CO(CA)NH, connect 1 H i and 15 N i resonances to 13 C α,(i-1) and 13 C i ', respectively (Figure 2.3b and d). The increase in coherence lifetimes at faster MAS allows to exploit J-couplings to trigger coherence transfers. Due to the relatively strong one-bond 13 Cα -13 C scalar coupling (55 Hz), INEPT is a competitive alternative to dipolar-based methods for 13 C-13 C transfers [144]. INEPT is typically performed by using band-selective inversion pulses with the offset at the center of either 13 Cα or 13 C frequency range.

These four experiments have been shown to be sufficient for backbone resonance assignment of the 153-residue metalloprotein SOD [128]. However, 13 Cβ chemical shifts are more sensitive to the chemical environment in side-chains, thus they are powerful reporters of residue type. The experimental scheme used to obtain intra-residue 1 H, 15 N and 13 Cβ correlations is conceptually similar to the (H)CANH experiment with the addition of an "out-and-back" 13 Cα-13 Cβ- 13 Cα transfer (H)(CA)CB(CA)NH, Figure 2.3e). The addition of a 13 Cα to 13 C transfer step allows to obtain inter-residue correlations ( 1 H i , 15 N i and 13 C β,(i-1) , (H)(CA)CB(CA)(CO)NH see Figure 2.3f).

Long 13 C coherence lifetimes observed for perdeuterated samples at > 60 kHz MAS increase the efficiency of 13 C-13 C and therefore the performance of

τsat t2 t1 z-filter 13 CO 1 H 15 N 13 C β 13 C α τsat t2 t1 z-filter 1 H 15 N 13 C β 13 C α τsat t2 t1 1 H 15 N 13 CO 13 C α 1 H 15 N 13 CO τ sat t2 t1 13 C α τsat t2 t1 1 H 15 N 13 C α 13 CO τsat t2 t1 1 H 15 N 13 C α 13 CO O N H C Cα N H Cβ Cβ Cα O C i-1 i O N H C Cα N H Cβ Cβ Cα O C i-1 i O N H C Cα N H Cβ Cβ Cα O C i-1 i O N H C Cα N H Cβ Cβ Cα O C i-1 i O N H C Cα N H Cβ Cβ Cα O C i-1 i O N H C Cα N H Cβ Cβ Cα O C i-1 i (H)CONH (
CO i-1 N i H i ), (B) intraresidue (H)CO(CA)NH experiment (CO i N i H i ), (C) intraresidue (H)CANH experiment (CA i N i H i ), (D) interresidue (H)(CO)CA(CO)NH experiment (CA i-1 N i H i ), (E) intraresidue (H)(CA)CB(CA)NH experiment (CB i N i H i ),

Application on microcrystals and amyloid fibrils

We applied this assignment strategy to [100% 1 H N , 2 H, 13 C, 15 N]-labelled microcrystals and fibrils of D76N β2m. The experimental details are listed in Table 2.2. Because of shorter coherence lifetimes, the sensitivity of the experiments is lower in the case of the amyloid fibrils sample, increasing the required acquisition time. Figure 2.4 illustrates the assignment procedure for a stretch of five residues with strip-plots extracted from each of the six 3D experiments. The procedure is performed in the following steps: (i) Spin systems comprising 1 H, 15 N, 13 Cα, 13 Cβ and 13 C chemical shifts are identified. (ii) 1 H-15 N spin pairs are linked by simultaneously matching the shifts of three different 13 C nuclei. (iii) Stretches of usually three or more spin systems are aligned with the sequence. This last step is performed by comparing the chemical shifts with statistical data. Here, for example, the characteristic 13 Cβ shift of the Serine-33 residue lowers the ambiguity of the alignment of the spin systems to the sequence. 70 backbone amide cross-peaks were assigned for microcrystalline β2m (Table A.2 in Appendix A), corresponding to 100% of the observed backbone resonances in the 1 H-15 N correlation spectrum, in which proline residues and flexible/disordered parts are absent. In the case of the fibrillar sample, broader lines are observed (most likely due to inhomogeneity of the sample) and increased ambiguity impedes the complete assignment of all observed resonances. In total, 47 backbone amide resonances were assigned (Table A.3 in Appendix A), which we estimate to be more than 80% of the residues present in the rigid core of the fibrils (see Chapter 4).

Conclusions

The set of experiments presented here allows backbone resonance assignment of proteins with high amide proton content at >60 kHz MAS within a reasonable experimental time (1-2 weeks) and with low ambiguity. The application to microcrystals of β2m was a part of the evaluation of the protocol in the original work [118], and and the beginning of my PhD studies I applied it also to D76N β2m fibrils. With only a few days of acquisition time, a high degree of assignment completeness was obtained, which served then as the basis for further investigation of structure and dynamics of D76N mutant of β2m. The extended discussion of the assignment and other aspects of the study of D76N β2m are presented in Chapter 4.

Sequential resonance assignment in a single spectrum 2.4.1 Introduction

As an alternative to the approach described above, we proposed a strategy based on direct matching of amide 1 H and 15 N resonances. This strategy enables to reduce the number of experiments required for chemical shift assignment and increases the reliability of the result by providing double linking, i.e. in both the forward and backward directions [119].

In solution, the introduction of multiple 15 N frequencies has been shown to greatly reduce the ambiguity in crowded spectra of disordered proteins [145][146][147]. These pulse sequences rely on weak 3 J cc scalar couplings, or require many scalar transfer steps. The efficiency is high for small molecules or disordered proteins, but generally prohibitively low for larger molecules. For proteins in the solid state, through-space magnetization transfers have been applied at both slow [148] and fast spinning conditions [START_REF] Nishiyama | A. 3D 15 N/ 15 N/ 1 H chemical shift correlation experiment utilizing an RFDR-based 1 H/ 1 H mixing period at 100 kHz MAS[END_REF], but the resulting spectra contain both sequential and long-range cross-peaks and are therefore of limited use for resonance assignment. 13 Cα-13 C transfer is highlighted in green. We set the carrier frequency of the 13 C channel to about 100 ppm, and the offset of 13 C on resonance with either 13 C or 13 Cα for CP. Bell shapes represent selective inversion pulses, here 350 ms Q3 pulses [149] with the frequency offset applied at the center of the range for either 13 C or 13 Cα. Excitation and inversion pulses are shown as narrow and wide black rectangles, respectively. Grey rectangles represent heteronuclear decoupling. Water suppression [150] is shown as purple and grey rectangles on the 1 H channel, and was typically applied for 100 ms. Delays δ 1 and δ 2 were set to 3.5 and 3.0 ms, respectively.

Radiofrequency irradiation schemes

N H C Cα N H Cβ (b) (a) 13 CA 1 H 13 CO (H)N(CACO)NH t δ 2 δ 2 δ 1 δ 1 x x y y 13 CA 13 CO (H)N(COCA)NH δ 1 δ 1 δ 2 δ 2 Cα O C N H 15 N 1 t 2 τ sat
In contrast to the case of scalar transfer, the new sequence yields diagonal free spectra, which decreases spectral crowding. This is relevant when neighbouring residues show close 15 N shifts.

Application to ubiquitin

We recorded amide correlation spectra of [100% 1 H N , 2 H, 13 C, 15 N]-labelled microcrystalline ubiquitin, a 76-residue protein commonly used for method development in solution and solid-state NMR [151]. Experimental sensitivities with respect to the (H)NH spectrum are 7% and 6% for (H)N(CA)(CO)NH and (H)N(CO)(CA)NH, respectively, at 60 kHz MAS on a 800 MHz 1 H Larmor frequency spectrometer.

In contrast to the 13 C matching approach, here every signal encodes a pair of nitrogen shifts, and a single spectrum is in principle sufficient for linking amide resonances along the protein backbone. This is illustrated in Figure 2.6a, where sequential correlations are shown for microcrystalline ubiquitin. 41 and 43 peaks were identified in the (H)N(CA)(CO)NH and (H)N(CO)(CA)NH spectra, respectively. This corresponds to all of the 54 amide 15 N resonances that could be assigned using 13 C matching and 1 H-detected spectra. When both 3D datasets are recorded, the simultaneous analysis of signals from the pair builds triplets of 15 N chemical shifts. As illustrated in Figure 2.6b and c the simultaneous matching of these triplets establishes connections spanning three residues. The alignment of stretches relies on two 15 N frequencies that are directly correlated in each spectrum. Comparable reliability of sequential assignment would require two pairs of 13 C-based spectra, or double the set of experiments required here for resonance assignment. Additionally, matching dimension. Of particular relevance is also the low correlation of sequential 15 N chemical shifts as compared with intra-residue correlation of C , Cα, and Cβ.

F45-A46 A46-G47 G47-K48 K48-Q49 Q49-L50 L50-E51 I44-F45 L43-I44 R42-L43 Q41-R42 Q40-Q41 D39-Q40 O N H C Cα N H Cβ O N H C Cα N H Cβ NNH CNH δ 15 N (ppm) δ 15 N (ppm) (a) (b) (c)
Once the signals are identified by peak-picking, the sequence-specific assignment of resonances can be quickly obtained with automated protocols, such as those developed for solution NMR data and implemented in UNIO-MATCH [152]. Inter-residue linking was determined by UNIO-MATCH for 38 15 N-1 H pairs. The resulting assignment shows little deviations with respect to that of non His-6 tagged microcrystals [START_REF] Schanda | Quantitative analysis of protein backbone dynamics in microcrystalline ubiquitin by solid-state NMR spectroscopy[END_REF].

Amide 15 N matching in two directions resulted in the unambiguous identification of sequential stretches of residues. For long stretches and highly complete data, correct assignment is possible. With less complete data, both correct residue placement and sequential linking can be done with the addition of datasets incorporating 13 Cα or 13 Cβ shifts because these shifts are particularly sensitive to residue type and secondary structure. This is particularly relevant when large portions of the protein (often flexible loops or tails) are not observed in the MAS spectra.

Conclusions

In conclusion, we have described a method that simplifies protein resonance assignment by providing spectra that can be rapidly and automatically assigned using UNIO-MATCH. Amide signals offer particularly unambiguous sequential connectivity because of the low overlap of resonances. Where sensitivity is a limiting factor, we expect carbon based matching with Cα and C to remain the primary method of assignment due to a significant sensitivity advantage of about two to fourfold. However, we found similar efficiency for the nitrogen matching experiments and the (H)(CA)CB(CA)(CO)NH sequence used for Cβ matching. We therefore expect that when Cα and C matching are not sufficient for assignment, nitrogen matching will be a useful approach comparable or even better than Cβ matching.

Internuclear proximities over H-bonds 2.5.1 Introduction

Once amide resonances are assigned, high-quality 1 H-15 N spectra can be extended to include dipolar coupling-based mixing periods in order to obtain longrange distance restraints. At very fast MAS rates (above 60 kHz), efficient polarization transfers are usually obtained by applying first-order recoupling schemes such as Radio Frequency-Driven Recoupling (RFDR) [153] and Dipolar Recoupling Enhanced by Amplitude Modulation (DREAM) [154], or alternatively by spin diffusion in the rotating frame [127].

The first report by Zilm and coworkers introduced a 3D (H)NHH experiment, comprising a 1 H-1 H dipolar mixing period to correlate nearby proton spins, and 15 N editing to enhance resolution, as demonstrated on microcrystalline ubiquitin at 25 kHz MAS [133]. Rienstra and coworkers employed 1 H-1 H mixing in combination with two hereronuclear ( 13 C and 15 N) dimensions in a 3D CONHH sequence [142]. The constraints recorded on microcrystalline GB1 at 40 kHz MAS were sufficient to determine the 3D structure of the protein. This approach was then applied to investigate significantly larger targets by exploiting the resolution enhancement offered by high magnetic fields and fast MAS. At 1 GHz magnetic field, the resolution and sensitivity of a 3D (H)NHH enabled identification of 297 1 H-1 H constraints and structure calculation of human SOD (153 residues) by using automatic tools implemented in UNIO [128].

A good definition of the β-sheet structure is generally required to elucidate the proper fold of a protein. This is particularly true for amyloid fibrils in which intermolecular H bonds between backbone atoms is a common feature. In NMR, H bonds between C and amide N are evidenced by the presence of long-range scalar couplings between the involved nuclei ( 3h J NC ). In a similar approach to what has been proposed in solution NMR [155], Schanda et al. showed that such weak couplings (< 1 Hz) can be exploited to probe H bonds in solids thanks to high levels of deuteration of the sample and fast MAS rates (55 kHz) [156]. Alternatively, Agarwal et al. proposed to use the stronger 1 H-13 C dipolar couplings to deduce the presence of H bonds [157]. Their approach, based on 1 H-13 C CP transfers, was successful for detecting side-chain and backbone hydrogen bonds, and quantifying the side-chain bond lengths. However, because of the many cross-peaks present in the 1 H-13 C plane, resolution is a bottleneck for a large application of this methodology. Here, we extend this approach by designing multi-dimensional experiments which allow to probe 1 H-13 C proximities across H bonds at high resolution. The distance between H and C involved in the same H-bond is approximately 2.9 Å. Closer H-C distances are found between H i or H i+1 and C i (approximately 2.7 Å and 2.0 Å, see Figure 2.8a). We ran SIMPSON simulations [158] to determine the appropriate length of the corresponding CP transfer. After about 3 ms, magnetization transfer reaches a plateau (Figure 2.8b). As expected, the most efficient transfer is found between sequential H i+1 and C i while intraresidue and through-H-bond transfers have similar efficiencies. Interestingly, protons distant more than 4 Å transferred magnetization only marginally, highlighting the specificity of these experiments.

Radiofrequency irradiation schemes

Application to GB1

We first performed both H(CO)NH and (H)CO(H)NH experiments on [100% 1 H N , 2 H, 15 N, 13 C]-labelled microcrystalline GB1 at 60 kHz MAS. Perdeuteration of side-chains is important for obtaining specific 1 H-13 C contacts through H-bonds. If this condition is not met, i.e. in fully protonated samples, magnetization leaks to aliphatic 1 H spins during the 13 C -1 H CP, leading to a poor sensitivity for the resonances of interest. The length of the CP is set to balance the build-up and relaxation losses due to 1 H R 1ρ relaxation rates. The experimental parameters for all recorded spectra are given in Table 2.3. The sensitivity of the two experiments with respect to (H)NH are correspondingly 0.2 and 0.08 (Figure 2.7c and f), thus H(CO)NH may therefore be the experiment of first choice for samples of unknown structure. However, the H-bond donor group is determined by a single 1 H resonance frequency which leads to high assignment ambiguity due to relatively larger line-width (50-100 Hz) and narrower spectral window (approx. 4 ppm) in comparison to heteronuclei. Sensitivity permitting, (H)CO(H)NH lifts this ambiguity since the H-bond donor is encoded here in two dimensions ( 1 H and 15 N) and the acceptor in a single 13 C dimension but with high resolution (at least for homogeneous samples, [119]). Representative planes of the experiments performed on GB1 are shown in Figure 2.9a. Despite the distance between 13 C i and 1 H i+2 is often larger than 4 Å, corresponding correlations were frequently observed in the spectra. Interestingly, we were able to identify the contact between backbone atoms of residue G9 and the side-chain of N8 (see Figure 2. 9b andc). Such an observation may be of particular interest in the case of amyloid fibrils where Asparagine ladders are a usual structural feature which consolidates the intermolecular packing.

Figure 2.10 shows the contact map observed for GB1 in H(CO)NH and (H)CO(H)NH experiments. In this figure, the points located on the diagonal correspond to "trivial contacts", defined as intraresidue contacts or contacts between H i+1 or H i+2 and C i . The expected contacts over H-bonds are shown by orange crosses. The contact map shows that the proposed experiments can reproduce the H-bond network across the four beta strands. Additional correlations are observed which mainly correspond to contacts between 13 C of the i th residue and 1 H of up to the (i + 4) rd residue in the helix. These results prove the capacity of [START_REF] Andreas | Structure of fully protonated proteins by proton-detected magic-angle spinning NMR[END_REF] these two experiments to clearly define α-helices and β-strands in a microcrystalline protein. The approach may therefore be of high interest for the determination of the registry of the intermolecular β-sheet structure of amyloid fibrils. In the following, we test this hypothesis on the amyloidogenic protein D76N β2m [159]. We recorded H(CO)NH spectra on [100% 1 H N , 2 H, 15 N, 13 C]-labelled microcrystals and fibrils of D76N β2m (see Table 2.3). For microcrystalline D76N β2m, we found 33 H-bond correlations (selected strips are depicted in Figure 2.11a). Compared to GB1, only few additional peaks (other than "trivial" or through-Hbond) are observed, which we mainly attribute to transfers from 1 H i+2 to 13 C i . As it can be noticed in Figure 2.11a (in the middle strip), long-range contacts can also be observed (here N42-Y78). Strips from H(CO)NH experiment recorded for the fibrils sample are shown in Figure 2.11b. The resolution in 1 H dimension, which determines the ambiguity of observed contacts in H(CO)NH is inferior to the case of microcrystals. The significantly larger 1 H line-widths (>160 Hz) than observed for microcrystals of D76N β2m (∼ 70 Hz), are probably due to heterogeneity of the fibrillar sample. We indeed estimated the homogenous contribution to the line-width to be approximately only 50 Hz (as estimated from the bulk 1 H T 2 ' of 6.4 ms). No contacts were observed between residues separated by more than 2 residues in the sequence. As it will be detailed in Chapter 4, this finding is in agreement with RFDR-based recoupling experiments and indicates a parallel in register arrangement of the intermolecular β-sheet structure of the fibrils.

Discussion

We presented here two 3D experiments to specifically probe proximities through H-bonds. H(CO)NH exploits the same magnetization pathway as (H)CONH experiment reported in references [118,128] with indirect detection on 1 H instead of 13 C , and thus features similar high sensitivity. We showed the power of this experiment to detect H-bond correlations on two microcrystalline and a fibrillar sample. Measuring 1 H-13 C dipolar couplings by varying the corresponding CP contact time may therefore be affordable in terms of acquisition time with a straightforward implementation. Alternatively, REDORtype block can be inserted as proposed by Reif and coworkers [157]. For systems showing lower spectral resolution, we expect (H)CO(H)NH experiment to be a valuable complement as the H-bond donor is encoded in two dimensions. However, the sensitivity of the latter experiment is lower due to the additional 13 C -1 H and 1 H-15 N CP transfers. Application of this strategy on amyloid fibrils provides important structural insights. In order to discriminate between intra-and inter-molecular contacts, mixed labelling strategies should be used. For example, fibrils grown from a 1:1 mixture of [100%-1 H N , 2 H, 13 C]-and [100%-1 H N , 2 H, 15 N]labelled monomers would specifically show intermolecular correlations in the proposed (H)CO(H)NH experiments. The same sample can be used for the measurement of unambiguously intramolecular 1 H N -1 H N contacts.

Conclusions

In summary, 1 H detection in ssNMR has been enabled by the availability of fast MAS probes, proton dilution by deuteration and high magnetic fields. At 60 kHz spinning frequency, the need for proton dilution is less stringent and highly resolved spectra can be obtained with full back-exchange at labile sites of otherwise perdeuterated samples in H 2 O. Under these conditions, sensitive experimental schemes involving multiple efficient magnetization transfer steps can be designed for resonance assignment and structure investigation. The advantages of this approach has been demonstrated on a variety of proteins and sample preparations.

Chapter 3

Methods for the determination of protein dynamics at >100 kHz MAS

Introduction

The previous chapters illustrate the advantages provided by fast MAS either for enhanced spectral resolution or for increasing the number of available probes of dynamics. The use of fast MAS together with sample deuteration enables the detection of protons at high resolution by disrupting the otherwise strong 1 H-1 H dipolar network. For the same reason, deuteration has also been used for suppressing coherent contributions to spin relaxation in the solid state.

A consequence of increasing MAS rates is the reduction of the deuteration requirements. With the advent of MAS NMR probes with >100 kHz spinning rates, therefore emerges the question of the necessity of deuteration for MAS NMR experiments, opening new possibilities for the investigation of structure and dynamics.

A first section of this chapter is dedicated to the discussion of the relative advantages of deuteration and protonation at fast MAS by reviewing some recent articles on this topic. In the following section, 1 H-detected 3D CP with Variable Contact time (CP-VC) experiments is used for the rapid determination of sidechain 1 H-13 C order parameters MAS rates above 100 kHz. We then investigate if >100 kHz MAS rates are sufficient to remove coherent contributions to aliphatic 13 C R 1 relaxation rates in a fully protonated sample (Section 3.4). Finally, we test the potential of 1 H R 1ρ relaxation as a probe for long-range dynamic, in conditions where the proton coupling network is highly reduced, at 100 kHz MAS with perdeuteration of the sample (Section 3.5).

Relative advantages of protonation and deuteration at >100 kHz MAS

The experiments for resonance assignment and structure investigation presented in Chapter 2 perform optimally on perdeuterated proteins fully backexchanged with H 2 O at 60 kHz MAS. However, in perdeuterated samples, most of the side-chain signals are not observed, precluding high-resolution structure determination and limiting investigation of side-chain dynamics. Different strategies have been developed to overpass this drawback by incorporating a limited amount of protons in the side-chains. One approach is to express proteins with either [ 2 H, 13 C]-glucose and controlled low concentration of H 2 O (5-15%) in a D 2 O buffer (Reduced Adjoining Protonation (RAP) [126,160]) or with [ 1 H, 13 C]-glucose in D 2 O (Fractional Deuteration, FD [161]). Both approaches result in high proton resolution at the expenses of lower sensitivity. This is particularly true for experiments commonly employed for the collection of protonproton distance restraints where the signal to noise ratio of the cross-peaks scales down with the square of the proton content (as the probability to simultaneously find two given sites occupied by protons). Alternatively, it is possible to selectively introduce protons at the methyl sites (CH 3 or CHD 2 ) of Isoleucine, Leucine and Valine residues with 100% efficiency by adding specific metabolic precursors to the deuterated growth medium (ILV labeling) [162].

All the methods described above rely on protein expression in a costly deuterated medium, which generally slows down bacterial growth, reduces the protein yields and is not compatible with certain expression systems. In addition, the incorporation of 1 Hs at amide sites may not be possible for solvent-protected regions of a folded protein. A way to remedy this problem is to inverse the FD scheme by adding [ 2 H, 13 C]-glucose in a protonated medium for protein expression (inverse fractional deuteration, iFD labelling). A rigorous comparison of perdeuteration with 100% proton exchange at the amide sites, iFD and full protonation strategies is presented in reference [163] at 60 kHz and 110 kHz MAS for two different samples (GB1 and Acinetobacter phage 205 coat protein, AP205CP). In this work, the authors showed that 1 H, 15 N and 13 C linewidths of the proteins prepared either with perdeuteration, iFD or full protonation benefit from increasing spinning frequency from 60 kHz to 110 kHz (Table 3.1). When comparing spectral features obtained for iFD-labelled and protonated samples, it appears that, whatever the MAS rate, deuteration is beneficial for 1 H linewidths and coherence lifetimes, although the differences between the two labelling schemes are attenuated at 111 kHz compared to 60 kHz. Noteworthy, the gain in resolution obtained with fractional deuteration is offset by the loss in resolution of the aliphatic 13 C resonances due to isotopomer distribution. Overall, the work of Cala-De Paepe, Stanek et al. brings elements for deciding the most suited labelling scheme. Deuteration is highly beneficial at 60 kHz and below. At above 100 kHz however, this requirement is less stringent, opening the possibility to perform high-resolution ssNMR experiments on fully protonated samples.

Sample

Labelling MAS (kHz) Production of fully protonated samples is the simplest procedure and can be applied to a wide range of systems. Moreover, the information content of such samples is maximized as much as it is not limited by resolution. Of considerable importance for structural studies, a recent publication reports a strategy for assignment of proton resonances of backbone and side-chains using 110 kHz MAS [164]. The high resolution of the 1 Hα- 13 Cα correlations under these conditions makes them an ideal detection scheme for sequential assignment experiments in 3D. Stanek et al. therefore tailored five RF irradiation schemes for 1 Hα, 15 N, 13 Cα, 13 Cβ and 13 C' resonance assignment, complementing the set of amide 1 Hdetected experiments presented in reference [118]. The 1 Hα- 13 Cα detection block was used for the assignment of side-chain 1 H and 13 C resonances by the addition of a J-based 13 C-13 C isotropic mixing to the 1 H-13 C CP-HSQC. This procedure allowed the assignment of 87% of side-chain 1 H and 13 C of AP205CP and 99.1% for GB1.

The advantage of side-chain resonance assignment is illustrated by a recent study which demonstrates the possibility to perform structure determination with 0.5 mg of protonated samples at >100 kHz MAS in less than two weeks of experimental time [START_REF] Andreas | Structure of fully protonated proteins by proton-detected magic-angle spinning NMR[END_REF]. 3D correlation experiments performed on GB1 and AP205CP using RFDR recoupling scheme allowed the direct detection of a large array of long-range 1 H-1 H contacts (236 and 410 contacts, respectively). These restraints were used for the determination of the structure of the two proteins (Figure 3.1). All-heavy atoms RMSD of the NMR bundles are, respectively, 1.04 Å and 1.84 Å. The number of contacts observed in this study severely decreases when considering separately amide-amide (Figure 3.1b and f) and ILV methylmethyl contacts confirming the importance of the presence of protons in the sidechains (Figure 3.1c and g). These studies inaugurate a substantial evolution of the field of MAS NMR where deuteration of the sample is no more a pre-requisite for high spectral resolution. The advantages of the new MAS regime (110 kHz) for structural investigations have been clearly demonstrated by the work of Andreas and coworkers and we anticipate many more applications in the next future. Studies of protein dynamics have also benefited from 110 kHz MAS. In particular, the MAS rate dependence of 15 N R 1ρ has been used to detect and quantify µs motions of ubiquitin backbone [165]. To date however, the possibilities of using 110 kHz MAS for investigation of side-chain dynamics are still under-explored.

Side-chain flexibility and fast MAS

The possibility to assign side-chain 1 H and 13 C resonances gives rise to new opportunities to investigate protein dynamics. In 2013, Paluch et al. proposed to use the oscillatory behavior of the 1 H-13 C/ 15 N CP signal to determine accurate related dipolar parameters [START_REF] Paluch | Simple and accurate determination of X-H distances under ultra-fast MAS NMR[END_REF]. The approach named CP with variable contact time (CP-VC) does not suffer from the presence of additional 1 Hs if the MAS rate is sufficiently high (> 30 kHz). The performance of the experimental setup was demonstrated on a tripeptide using 13 C detection. Later on, the same group proposed a detailed theoretical study of CP-VC [166]. In this work, they established the robustness of the sequence to RF inhomogeneity, 13 C CSA and offset.

The CP-VC approach was successfully applied on protein samples [START_REF] Park | Resolution and measurement of heteronuclear dipolar couplings of a noncrystalline protein immobilized in a biological supramolecular assembly by proton-detected MAS solid-state NMR spectroscopy[END_REF]167]. In a relatively early work, Park et al. showed the applicability of the approach on the 46-residues coat protein of Pf1 bacteriophage [START_REF] Park | Resolution and measurement of heteronuclear dipolar couplings of a noncrystalline protein immobilized in a biological supramolecular assembly by proton-detected MAS solid-state NMR spectroscopy[END_REF]. Perdeuteration of the protein at 50 kHz MAS allowed to resolve 6 glycine residues in 1 H-detected 1 H-15 N correlation spectra. Based on this observation, a 3D 1 H-detected version of CP-VC was implemented, which allowed the determination of 1 H-15 N dipolar coupling constants. Paluch et al. proposed alternative 3D CP-VC experiments based on 13 C detection to obtain information about the side-chain dynamics of a tripeptide and fully-protonated GB1 [167]. In this paper, they reported a detailed analysis of the aromatic sites motional modes which was further supported by DFT calculations.

The practical interest of using CP-VC with 1 H detection was highlighted in different publications [START_REF] Park | Resolution and measurement of heteronuclear dipolar couplings of a noncrystalline protein immobilized in a biological supramolecular assembly by proton-detected MAS solid-state NMR spectroscopy[END_REF]168,169]. Additionally, the robustness of the sequence was shown to increase with the MAS rate [166]. Nishiyama et al. used the conjugated benefits of the two factors to determine order parameters in amino acids and peptides at natural abundance [169]. In the following, we test the performance of this experiment for the study of side-chains dynamics in a fully protonated protein sample with 1 H detection at 110 kHz MAS.

Experimental setup

CP-VC experiments exploit the spin dynamics happening during a standard CP transfer step to determine dipolar coupling constants. CP is performed by applying RF irradiations on two channels with RF frequencies matching the Hartmann-Hahn conditions (see Chapter 1, Section 1.3.4) and allows the magnetization of one spin (here, 13 C) to be transferred to the other (here, 1 H). During the contact time, the energy is alternatively exchanged between the two coupled spins before equilibrating, leading to an oscillatory behavior of the 13 C magnetization [170]. The frequency of the oscillations is the dipolar coupling constant scaled by a factor depending on the number of coupled 1 Hs [171].

The 3D-version of the CP-VC experiment used in this work relies on chemical shift encoding in 1 H and 13 C dimensions (with direct detection performed on 1 H) and evolution of the dipolar coupling during 13 C-1 H CP transfer (Figure 3.2a). The contact time of the second CP step (τ CP ) is varied by increments of 15 µs (corresponding spectral width of 33 kHz) so to ensure that the complete lineshape is sampled. RF powers during CP are set so that (i) they match ZQ-Hartmann-Hahn (for which CP-VC is the most robust) (ii) 13 C RF power is maximal (to limit the effect of offsets due to the large side-chain 13 C frequency range) [167,169]. The high efficiency of inductive couplings between the coil and the sample in fast MAS probes allows RF field strengths up to about 200 kHz for 1 H and 100 kHz for 13 C (for Bruker 0.7 mm probe). Figure 3.2b displays the experimental 1 H-13 C CP matching conditions at 110 kHz MAS. The plot is obtained by monitoring the intensity of the 1 H signal for different RF field strengths. The different Hartmann-Hahn conditions have opposite phases so that the signal appears either positive (here, ZQ) or negative (here, DQ). For CP-VC, ZQ conditions are used with 13 C power of 80 kHz. 13 C spectral width in our experimental setting (external magnetic field of 23.5 T) is 16 kHz (excluding aromatic resonances). The maximum offset (8 kHz) is therefore one order of magnitude less than the 13 C RF frequency used during CP-VC, avoiding undesirable offset effects.

t 1 1 H 13 C (a) (b) 15 N τ CP ZQ SO DQ DQ ν 1,C (kHz) ν 1,H (kHz) (ν 1,H -ν 1,C = ν R ) (ν 1,H + ν 1,C = 2ν R ) (ν 1,H = ν 1,C ) (ν 1,H + ν 1,C = ν R ) 90

Data analysis

A typical experimental CP-VC curve is shown Figure 3.3a. As expected, the signal first builds up quickly (70-80 µs) and oscillates. We can also observe a decay of the signal during the mixing period due to 1 H R 1ρ relaxation. The Fourier transform of the raw signal leads to the dipolar pattern, together with a strong peak at zero frequency (Figure 3.3b), which can be reduced by subtracting the signal to the mean value of the last few points (Figure 3.3c and d). To display the data, we also perform zero filling (Figure 3.3d and e). The fitting procedure was performed in the frequency domain by excluding the central part of the spectrum (evaluated on a case-by-case basis) in order to remove all unwanted contributions which generally appear at the zero-frequency [166].

The CP-VC curves of the 13 C 1 H spin systems are simulated using equations presented in [START_REF] Wu | Cross polarization with high-speed magic-angle spinning[END_REF]167]. For a single crystallite, the signal is given by:

S(τ ) = 1 -cos ω CP τ (3.1)
The frequency of the oscillations ω CP can be expressed as a function of the dipole coupling constant δ D and the asymmetry of the dipolar coupling η D as follows:

ω CP = 1 6 2πδ D 2 cos 2 (β-(3 + η D cos (2α)) 2 + 2η 2 D sin 2 (2α) (3.2)
where (α, β) are the Euler angles corresponding to the frame transformation from the Principal Axes System (PAS) linked to the 13 C-1 H bond to the rotor frame and δ D is the anisotropy of the dipolar coupling. In all simulations, we performed powder averaging with the planar grid defined in reference [172] with 50 × 50 α and β angles. Exponential line-broadening is subsequently applied in order to take into account relaxation.

Results and discussion

3D-CP-VC experiments were acquired on microcrystalline GB1 in one day and five hours at 110 kHz MAS on a 1 GHz 1 H Larmor frequency spectrometer, providing about 120 dipolar lineshapes. This number is close to the number of 2 H quadrupolar couplings signatures observed in 13 C-detected spectra [102]. Here, we only considered isolated resonances in the 2D 1 H-13 C projection of the 3D experiments (Figure 3.4a). As CP-VC evolution adds a new frequency dimension to the spectra, it may in some cases be possible to deconvolute the dipolar lineshapes of two correlations which partially overlap in the 2D projection.

Typical dipolar lineshapes are depicted in Figure 3.4b-d. CH groups are the most straightforward to analyze since the curves fit well with the analytical formula given by equations 3.1 and 3.2, allowing to directly extract the dipolar coupling constant as well as the asymmetry parameter (Figure 3.4b). For CH 2 and CH 3 groups, the magnetization is transferred faster as multiple 13 C-1 H dipolar couplings are present [171] and the presence of 1 H-1 H couplings splits the dipolar pattern into multiple horns [173]. Even if the coherence lifetimes are rather long at 110 kHz MAS, the resolution of the dipolar pattern does not allow to clearly observe the horn splitting for CH 2 groups (Figure 3.4c). Instead, the presence of such a splitting results in broader signals. For methyl groups, the pair of horns in the low field region is separated from the two other pairs (Figure 3.4d). The value of the dipolar coupling constant was extracted by measuring the distance between horns. We first investigated 1 Hα-13 Cα dipolar couplings. The results of the CP-VC experiments are depicted in Figure 3.5. The rigid limit of the 1 Hα-13 Cα dipolar coupling is set to 22.8 kHz, which corresponds to a bond length of 1.09 Å. The order parameters are very uniform along the primary sequence of GB1 and very close to 1, in perfect agreement with previous work [174]. Contrarily to what has been observed in reference [174], few residues contravene the trend (namely, K10, T11, D40 and G41), located in more flexible loops. Comparing 13 Cα and amide 15 N mobility is not strictly speaking rigorous because of free rotation around the 13 Cα-15 N axis. However, our results do correlate very well with amide S 2 obtained by the analysis of 15 N R 1 and R 1ρ relaxation (Figure 3 Side-chains possess a more complex dynamical behavior which does not only depend on the position in the sequence but also on the residue type (Figure 3.5c). It is interesting to compare these results with a recently published description of GB1 side-chain dynamics obtained by 2 H NMR [174]. The correlation between the dipolar coupling constants measured here and the quadrupolar couplings measured by Rienstra and coworkers is very high considering the different techniques used, experimental temperatures and sample preparation procedures (Figure 3.6a). Interestingly, a single residue, N35, deviates significantly from the observed trend. In the work of Franks et al., N35 is singled out from the other asparagine because of its characteristic mobility. As a validation of this observation, the authors note that N35 indeed points toward the solvent in the ssNMR structure [176] (Figure 3.6b). In contrast, our data suggest higher rigidity for this residue with a dipolar coupling constant close to the rigid limit value. The crystals we used here are prepared with the same protocol than in [176]. However, subtle differences in sample preparation and in particular during crystallization step may influence structure and dynamics properties of the protein. Interestingly, in the MAS NMR structure solved by our group [117], side-chain of N35 is folded in the direction of D36 (Figure 3.6c). This structural change may therefore be correlated to the different observed dynamical behaviors. A standard way to recover resolved signals from side-chains in both solution and solid-state NMR is to specifically label methyl sites in an otherwise deuterated matrix (ILV labeling) [162]. In order to estimate the capacity of CP-VC experiment to probe methyl dynamics of protonated proteins, we compared the results obtained with this approach to the analysis of REDOR curves obtained on an ILVlabelled sample spinning at 60 kHz. The methyl region of the 1 H-13 C correlation spectra is depicted in Figure 3.7a. 7 sites are present on the spectrum acquired on the ILV-labelled sample. All of them have uniformly low order parameters due to the three fold rotation of the methyl group. Interestingly, residue I6 has a particularly low S 2 in both samples (Figure 3.7d), a feature which was not detected in 2 H experiments [174]. The low value of the effective dipolar coupling constant is beneficial for fitting REDOR curves since it allows to finely sample the slowly evolving dephasing curve (Figure 3.7b). In contrast, because CP-VC analysis is performed in the frequency domain, the determination of the dipolar coupling constant becomes less precise when its absolute value is lower (Figure 3.7c). For residue I6, the dipolar coupling constant is so low that the first horn appears too close to the zero frequency domain to be detected. The value of the dipolar coupling constant was therefore extrapolated from the second pair of horns.

Are aliphatic 13 C longitudinal relaxation rates a robust probe of dynamics at 110 kHz MAS?

Importantly, the order parameters do not give any information by themselves about the timescale of motion. For this reason, the measurement of relaxation rates is necessary to provide more information on the mobility of the side chain. Yet, in protonates proteins, 13 C relaxation rates are obscured by coherent contributions even for MAS rates as high as 60 kHz [START_REF] Lewandowski | Measurement of Site-Specific C-13 Spin-Lattice Relaxation in a Crystalline Protein[END_REF]. In the following, we aim at determining to which extent 13 C R 1 relaxation rates report on the proper dynamics of the aliphatic sites when spinning at the highest MAS rates today available (110 kHz).

MAS rate (kHz) mean [START_REF] Liao | Structure of the TRPV1 ion channel determined by electron cryo-microscopy[END_REF] Cα R 1 (s -1 ) mean [START_REF] Liao | Structure of the TRPV1 ion channel determined by electron cryo-microscopy[END_REF] Cβ R 1 (s - 1) To access side-chain 13 C resonances with maximal sensitivity at high resolution, we used again the 1 H-13 C CP-HSQC as a detection block. The longitudinal magnetization was allowed to relax thanks to a z-storage block inserted before the indirect evolution delay. During this time, 1 H-driven spin diffusion may lead to unwanted decay of longitudinal 13 C magnetization and by consequence to enhanced effective relaxation rates and a homogenization of the relaxation rates over different sites (see Chapter 1). MAS reduces the extent of this phenomenon by weakening the 1 H-1 H dipolar network. We therefore investigated the evolution of 13 C R 1 relaxation rates with the spinning speed to evaluate the reduction of this coherent contribution. Site-specific R 1 relaxation rates for Cα and Cβ at 4 MAS rates comprised between 80 and 110 kHz are shown Figure 3.8a and b.

mean 13 CX R 1 (s -1 ) MAS rate (kHz) MAS rate (kHz) (c) (d) (e)
While for some residues, no significant changes in relaxation rates are observed, for the majority of them we observe a clear reduction of R 1 between the slowest and the fastest MAS rates. The variations follow the same order for all residues. For a better visualization, the mean value of the longitudinal relaxation rates for Cα, Cβ and the other aliphatic carbons are depicted in Figure 3.8c-e. It appears that going from 80 kHz to 90 kHz MAS has a significant impact on the relaxation rates while the latter are more stable in the 90-110 kHz MAS range. Inconsistently with a decrease of coherent contribution at faster MAS, 13 C R 1 values are lower at 100 kHz than at 110 kHz MAS. We did not find any rationalization for this fact except that in spite of the attention we paid to the conservation of the experimental conditions, the temperature inside the rotor may have varied when moving from one MAS condition to the other.

Variation of sample temperature of few degrees does not drastically change the motional modes of the protein. As a consequence, we should expect that the uncontrolled changes in experimental conditions we report should impact the timescale but not the amplitude of the dynamical processes. We therefore pushed further our investigation by comparing 13 C R 1 relaxation rates obtained at different MAS rates to the order parameters S 2 obtained from CP-VC experiments. Assuming a distribution of amplitude of motions (1 -S 2 )θ(τ ), R 1 relaxation rates can be written in the form [START_REF] Smith | detectors" for dynamics analysis in solid-state NMR[END_REF]:

R θ,S 1 = (1 -S 2 ) θ(τ )R • 1 (τ )dτ (3.3)
where R • 1 (τ ) is the longitudinal relaxation rate as reported in Table 1.1 in Chapter 1 for a given correlation time τ and S 2 = 0. Assuming an even distribution of motions for 13 Cα-1 Hα bonds along the primary sequence of GB1 (which according to the results reported in reference [START_REF] Lamley | Unraveling the complexity of protein backbone dynamics with combined 13 C and 15 N solid-state NMR relaxation measurements[END_REF] is close to reality), we expect a significant correlation between 13 Cα R 1 relaxation rates and (1 -S 2 ) obtained from 13 Cα-1 Hα dipolar coupling measurements. Figure 3.9 shows the correlations obtained at variable MAS rates. While at 80 kHz MAS relaxation rates and order parameters correlate very poorly (R 2 = 0.54), the situation improves monotonically at higher MAS speeds. Considering the low dispersion of flexibility among GB1 residues and the fact that the distribution of timescales is not perfectly homogeneous over the primary sequence, the relaxation rates and the order parameters can be considered to highly correlate at 110 kHz MAS (R 2 = 0.83).

These exciting preliminary results call for a deeper investigation to check their reproducibility, in particular because of the out-of-trend behavior of the experiments performed at 100 kHz or 110 kHz MAS. However, it stands out from this study that while even at 80 kHz MAS, coherent contributions clearly blur the analysis of side-chain motions, 13 C R 1 relaxation at 110 kHz MAS in a protonated sample appear to reflect mostly dynamics and can be used to complement the information obtained from the motion-averaged 1 H-13 C dipolar couplings.

1 H R 1ρ relaxation dispersion at 110 kHz MAS

In very recent papers, the groups of Schanda and Linser drew the attention on the interest of using 1 H R 1ρ relaxation rates as reporters of dynamics [START_REF] Rovó | Microsecond Timescale Protein Dynamics: a Combined Solid-State NMR Approach[END_REF]177]. In contrast to heteronuclei, 1 H relaxation does not depend only on interactions with the bonded nuclei and on its own CSA, but it is also influenced by longer range interactions with other 1 Hs. One interest of measuring long-range dynamics is the possibility to detect internal translational motions which do not influence either 15 N or 13 C relaxation. In the following, we estimate the applicability of this approach by measuring amide 1 H R 1ρ relaxation dispersion on a perdeuterated sample of the model protein GB1 spinning at 100 kHz.

Rovo et al. derived the expression of the R 1ρ relaxation rates due to homonuclear dipolar coupling between two I spins, being:

R D,II 1ρ = 3d 2 16 J(2ω 1 -2ω R ) + 2J(2ω 1 -ω R ) + 2J(2ω 1 + ω R ) + J(2ω 1 + 2ω R ) + 10J(ω I ) + 4J(2ω I ) (3.4)
It is interesting to note that the homonuclear dipolar coupling contribution to 1 H R 1ρ rates has two resonance conditions: the rotary resonance (ω e = ω R ) and the homonuclear rotary resonance (HORROR) condition (ω e = 1 2 ω R ). The presence of 10 -5 -10 -3 s dynamics is accompanied by an increase of the effective relaxation rates when approaching these conditions. A typical experimental dispersion is shown in Figure 3.10a. All measured dispersions display similar profiles with enhanced relaxation rates for spin-lock fields approaching 50 kHz and 100 kHz. Without considering coherent contributions, this finding would suggest the presence of slow motions for all residues of GB1, in obvious contradiction with the 15 N R 1ρ relaxation dispersion experiments conducted by Lamley and coworkers [START_REF] Lamley | Intermolecular Interactions and Protein Dynamics by Solid-State NMR Spectroscopy[END_REF]. On the contrary, as discussed in reference [177], coherent effects can explain the observed dispersions. These effects are indeed described by the same hamiltonians than 1 H relaxation and are therefore also enhanced at the resonance conditions.

We then investigate if qualitative information can be extracted from 1 H R 1ρ relaxation rates. When the HORROR and rotary resonance conditions are avoided, 1 H R 1ρ relaxation is mainly affected by the dipolar coupling with the bound 15 N spin. Albeit additional information is present in 1 H relaxation compared to its 15 N counterpart, one expects these two phenomena to correlate to some extent.

In Figure 3.10, both 15 N and 1 H R 1ρ values are plotted as a function of the sequence. It is evident from the comparison of these two plots that the two types of measurements do not correlate. In particular, the loop containing residues 10-14, known to possess particular flexibility, does not have particularly enhanced 1 H R 1ρ relaxation rates. In contrast, particularly high apparent 1 H R 1ρ relaxation rates at 80 kHz spin-lock field are found for residues T44 and Y45, while they display only moderate 15 N relaxation rates. The inspection of the difference ∆ 1 HR 1ρ between 1 H R 1ρ rates measured with conditions close to the half-rotary resonance (ω 1 = 55 kHz) and further away (ω 1 = 80 kHz) gives a similar picture. Interestingly, ∆ 1 HR 1ρ values are particularly high at the N-terminal part of the α-helix and, again, for residues 44-45, reflecting the fact that 15 N R 1ρ relaxation dispersions were observed in crystalline GB1 for residues 17-27 and 44-46 [62].

The data presented here were gathered on deuterated samples at the fastest MAS rates today available. They clearly show that coherent effects still dominate observed 1 HR 1ρ relaxation rates under these conditions. Even if one cannot completely disregard the contribution of dynamics to the observed relaxation rates, no clear qualitative picture of protein motions emerges from such measurements.

Conclusions

In summary, we first discussed the relative advantages of protonated or deuterated samples for MAS rates above 100 kHz. While the network of dipolar coupling is not fully disrupted in this MAS regime, 1 H line-widths observed for protonated samples are sufficiently narrow for a wide application of 1 H-detected strategies. This has been brightly demonstrated by the rapid structure elucidation of the microcrystalline protein GB1 and the phage coat protein AP205CP.

We then evaluated the potentialities of MAS above 100 kHz for the investigation of protein dynamics using GB1 as a model system. The results reported in this chapter confirm that CP-VC experiments permit accurate measurement of motion-averaged dipolar couplings. This methodology applied with 1 H detection at 110 kHz MAS allows to determine the magnitude of motional processes of backbone and side-chains with short experimental times. 13 C relaxation rates can complement this information since they also report on the timescale at which motions occur. The evolution of effective 13 C R 1 relaxation rates with MAS speeds suggest that using 110 kHz MAS the coherent contributions to the decay rates are sufficiently suppressed to obtain reliable dynamics parameters.

Finally, we addressed the question of the possible implementation of 1 HR 1ρ relaxation measurements at 100 kHz MAS. From our observations, it appears that this experimental setup does not allow to draw a clear picture of the dynamical events happening in GB1, even when working on a deuterated sample.

Part II

Beta-2 microglobulin

Chapter 4

Molecular basis for D76N beta-2 microglobulin amyloidosis 4.1 Introduction

In the previous chapters, we detailed methodological developments of MAS NMR for the study of proteins relying on the use of high magnetic field, MAS rates above 60 kHz and (potentially) deuteration of the sample. In particular, we highlighted the on-going advances of MAS NMR for the determination of molecular dynamics. In the following, I present a direct application of recent MAS NMR technological developments to investigate the amyloid fibril formation propensity of the D76N variant of beta-2 microglobulin (β2m) [START_REF] Marchand | Conformational dynamics in crystals reveal the molecular bases for D76N beta-2 microglobulin aggregation propensity[END_REF].

Numerous pathologies are related to the transformation of functional wellsoluble proteins into amyloid fibrils. There is therefore a large interest of understanding the underlying mechanism of aggregation and the fibrils architecture as assessed by the abundant literature on this topic. While many of amyloidosis involve disordered peptides or proteins (e.g. Aβ, α-synuclein), more surprisingly, globular proteins (e.g beta-2 microglobulin, superoxide dismutase, lysozyme) can also be associated to amyloidosis diseases [178,179]. Yet, evolution has highly protected their native state from aggregation, and for these proteins special conditions such as low pH or high metal ion concentration are required for their aggregation in vitro. In contrast, specific mutations are able to dramatically increase the potential of these proteins to form fibrils.

During amyloid fibril formation, many intermediate forms come into play (Figure 4.1). Each specie may play a role in the pathology according to their biophysical characteristics, such as stability, rigidity, lifetime or toxicity. Rationalizing the amyloidogenicity of proteins is therefore a challenging task, requiring in principle atomic-scale information about the different stages of aggregation. In this chapter, we present the results we obtained with ssNMR on a prototypic example, the D76N mutant of β2m which allowed us to establish the bases for a molecular understanding of its amyloid fibril formation propensity. At room temperature, some of these partially folded states, generally structurally close to the native fold, are populated. These conformations may have higher amyloid formation propensity than the protected native state and be able to form oligomers (f) which will evolve into protofibrils (g) and then to mature fibrils (h).

β2m is a 99-residues protein involved in the Major Histocompatibility Complex class I (MHCI) and present in most biological fluids. Its monomeric structure has been first solved in 1985 [180], revealing a typical immunoglobulin-like structure comprising two anti-parallel β-sheets linked together by a disulfide bridge between cysteines 25 and 80 (Figure 4.2a). After dissociation from the MHCI, β2m is catabolized in the kidney. In patients suffering from kidney failure, the concentration of β2m strongly increases, leading to the formation of amyloid fibrils and deposition in osteoarticular tissues in a pathology named Dialysis Related Amyloidosis (DRA) [181].

A recent case of systemic amyloidosis involves the highly unstable D76N mutation of β2m [159]. Patients carrying this mutation exhibit large amyloid deposits in internal organs, leading to progressive gastrointestinal symptoms and autonomic neuropathy. In vitro, the native fold of the D76N mutant displays a strikingly low thermodynamic stability compared to the wild type (wt) protein [ 159,182]. The D76N mutant is highly amyloidogenic and, in contrast to the wt protein, aggregates efficiently and abundantly without seeding and under native conditions [183].

Common effects of point mutations are the modification of the backbone, the impairment of the hydrophobic core or the introduction of hydrophobic patches on the protein surface, with potentially very drastic outcomes involving a marked change in size or in the chemical properties. None of these effects can nevertheless explain the enhanced reactivity of D76N β2m. Indeed, the position 76 is located in a solvent exposed loop between β-strands E and F (Figure 4.2).The D76N mutation is not introducing a hydrophobic residue, which could lower protein solubility. From the structural point of view the D76N mutation does not trigger major effects neither in solution nor in crystalline form [159,183]. In particular, the crystal structures of wt β2m and of D76N mutant match very closely (Cα RMSD 0.59 Å) and the residue 76 can be well superposed in the two structures [159].

In this chapter, we report the results of our investigation of the process of amyloid fibril formation of D76N by the two ends of the associated pathway. A first section is dedicated to the analysis of the molecular determinants of the self-assembly of the monomers into amyloid fibrils. For this, we compare the intrinsic dynamics of the native form of both wt and D76N β2m. We performed the analysis on the crystalline form, exploiting the shielding role of the crystal lattice to preserve an overall native conformation. We are therefore able to exclusively probe intrinsic dynamics of individual proteins and decouple them from (pathological) intermolecular interactions. In a second section, I present a ssNMR structural characterization of the amyloid fibril form of D76N. Taken together, these results shed light on the process of amyloid fibril formation of D76N β2m.

Conformational dynamics in crystals reveal the molecular bases for D76N beta-2 microglobulin aggregation propensity

The following section is dedicated to the comparative study of the dynamics of the native fold of wt and D76N β2m. We will discover in it that alternative protein conformations related to the formation of amyloid fibrils can be detected in the crystalline environment, uncovering the early events of fibrillogenesis. Our experimental findings are compared and complemented by computational simulations thanks to collaborations with Dr. Nicola Salvi and Dr. Martin Blackledge at the institute of structural biology (IBS) in Grenoble, and with Dr. Carlo Camillioni (Department of Biosciences of the University of Milan).

Sample preparation and assignment of the resonances

Samples of both D76N and wt β2m were prepared as described in Appendix C. The proteins were expressed in D 2 O with [ 2 H, 13 C]-glucose and 15 N-labelled ammonium chloride. The purification was performed in denaturing conditions in H 2 O so that all amide sites are protonated. Crystals were subsequently grown at the same pH conditions for both forms (pH 5.5). Figure 4.3a and b shows the "fingerprint" 15 N-1 H correlation spectra of the crystalline preparations of [100%-1 H N , 2 H, 13 C, 15 N]-labelled wt and D76N β 2 m acquired using 60 kHz MAS on an 800 MHz 1 H Larmor frequency spectrometer.

The assignment of the resonances was performed using 1 H N -detected experiments as already presented in Chapter 2, Section 2.3. The observed signals correspond to around 70% of the amide groups. The absent signals are associated with residues located in loops, which are subject to either large conformational disorder or extreme mobility, preventing their detection in CP-based experiments. It is noteworthy that two set of peaks are present in the spectra for both proteins. They correspond to residues present in the core of the protein and in spatial proximity to the disulfide bridge. We attribute this peak doubling to the presence of different static conformations of the protein in the different crystallites since (i) only one molecule is found per crystal unit in the X-ray diffraction experiments on monocrystals (ii) no peak doubling was observed in the NMR spectra of previous samples studied in the lab [118] (iii) exchange experiments performed on these samples failed to reveal cross-peaks between the two set of peaks. In the following, we only discuss the results obtained for the set of resonances matching closely the assignment of WT β2m reported in reference [118], the experimental results obtained for the other set being identical.

Fast dynamics probed by 15 N relaxation

Site-specific 15 N spin-lattice relaxation rates in the laboratory frame (R 1 ) and in the rotating frame (R 1ρ ) were measured for the two molecules. The corresponding pulse sequences and typical relaxation curves are shown in Figure 4.4. The obtained rates (Figure 4.5a and b) are mainly determined by the fluctuation of the 15 N chemical shift anisotropy and of the 15 N-1 H dipole coupling. These data were interpreted by using the simple model-free (SMF) approach, determining order parameters (S 2 ) and correlation times (τ ) for all visible amide groups, which respectively report on the amplitude and the timescale of local motions (Figure 4.5c and d). For most of the residues, the order parameters obtained are close to 0.9, and extremely similar for the two proteins. These results indicate that the two proteins are overall very rigid, as expected for a beta-sandwich fold and that the rigidity of the beta-sheet core of the protein is not affected by the mutation. The residues in the EF-loop (containing residue 76) display lower order parameters. While this is in principle not surprising, as loops may feature significant mobility also in the presence of crystal packing, it is intriguing to observe that the fitted order parameters are consistently lower in the mutant with respect to the wild type, reflecting an increase in flexibility following the mutation. Small differences in the order parameters were also observed for residues 11 and 12 at the end of the A-strand.

Molecular dynamics (MD) simulation offers a mechanistic description of fast protein dynamics [START_REF] Mollica | Atomic-Resolution Structural Dynamics in Crystalline Proteins from NMR and Molecular Simulation[END_REF]. We compare the experimental order parameters with these calculated from two 50 ns MD runs (Figure 4.5c) of both wt and D76N β2m in a water box 1 . Overall, the simulation reproduces the experimental sequence dependence with good accuracy. In the simulation, the AB loop and the C-terminus are predicted to be highly flexible, more so in the D76N mutant than in the wt, 1 MD simulations were run by Dr N. Salvi, IBS, Grenoble in agreement with the fact that these regions are not visible in CP-based experiments. The high quality reproduction of dynamics observed throughout the two proteins experimentally and via simulation suggests that the observed motions are intrinsic to the protein, and are essentially independent of immediate environment (solution and crystalline forms). Reduced flexibility is found in the EF loop in both D76N and wt, possibly due to additional slower motions that are not captured by the 50 ns simulations.

Conformational exchange within the crystals

The contribution of conformational exchange on µs-ms timescales to transverse spin relaxation can be characterized by measuring 15 N R 1ρ relaxationdispersion experiments. R 1ρ was measured for all residues of wt and D76N β2m, using 15 N spin-lock RF amplitudes ranging from 3 kHz to 20 kHz. For several residues, mainly located within the β-sheet core, the relaxation rates are independent of the applied RF field (Appendix E). Interestingly, however, the rates of a non-negligible fraction of the signals feature ample dispersion profiles as the RF field is varied. Figure 4.6a shows selected dispersion profiles for residues located at the end of the A-strand and on the BC-and EF-loops. These data are indicative of conformational exchange in the µs-ms range [START_REF] Palmer | Characterization of the dynamics of biomacromolecules using rotating-frame spin relaxation NMR spectroscopy[END_REF]. Despite the quality of the experimental data obtained at different magnetic fields (800 MHz and 1 GHz 1 H Larmor frequencies), our attempts to fit the dispersion curves to simple exchange models failed due to under-determination of the exchange parameters. Figure 4.6b reports the amplitude of the dispersion of the R 1ρ rates (∆R ex ). For the wt, the residues showing the highest dispersion are located in the BC-loop, comprising Proline-32, as well as residues located in the nearby N-terminus of the protein. These residues display a very similar dispersion also in the D76N mutant, as it could be expected given the large distance with respect to the mutation site. However, in D76N, major changes are clearly observed in the EF-loop, as well as for the residues at the end of the A-strand. In the case of residues 75-76 in D76N, 15 N R 1ρ rates were so large for low spin-lock fields that the signals disappeared within the shortest relaxation delay, indicating extensive exchange contribution for these two residues.

These results reveal the presence of significant conformational exchange, consistent with two global processes. A first mode, common to wt and to D76N variant, involves the residues in spatial proximity to Proline-32. A second mode affects the EF-loop and remote residues at the end of the A-strand, and produces a clear experimental signature only for D76N β2m.

Replica-Averaged Metadynamics simulations

In order to gain further insights into the conformational exchange process and its consequences for the amyloid fibril formation, Replica-Averaged Metadynamics (RAM) simulations were performed 2 [185, 186]. RAM is used to integrate experimental data, in this case ssNMR chemical shifts, in molecular force fields to accurately explore the conformational space over timescales longer than these accessible during conventional MD runs.

On the basis of 1 H N , 15 N, 13 Cα, 13 Cβ and 13 C chemical shifts, converged ensembles of structures were calculated for both the wt and the D76N mutant. Free energy surfaces were generated (Figure 4.7a and b) as a function of the side-chain rotamer distribution (AlphaBeta collective variable, AB) and the anti-parallel βstructure content (AntiBetaRMSD collective variable, anti-β). In both cases, the calculated structures clustered around two bundles, describing a major populated substate close to the crystal structure (less than 2 Å full backbone average RMSD) and a minor populated substate, which features significant differences in secondary structural content (Figure 4.7c and e). The major substate (85% and 95% populated for the wt and D76N, respectively) is characterized by a high βsheet content (∼48% for both ensembles) while the minor substates are characterized by a significantly lower β-sheet content (∼38% and 30% for wt and D76N, respectively). Comparison of the secondary structure content and the solubility profiles in the minor substates shows how the larger loss of beta structure in the A-strand for the D76N variant with respect to the wt corresponds to a decrease of solubility for B-strand region that is left more exposed to the solvent (Figure 4.7c-f).

The RAM results provide a molecular framework for understanding the dramatic increase of the aggregation propensity of the D76N mutant. In order to understand the energetic contribution of the D76N mutation, we calculated the nonbonded interaction energies for each pair of residues, averaged over the ensemble. Not surprisingly, it is evident that the largest differences in non-bonded interaction energies between wt and D76N are related to the interactions of residue 76 with the neighboring residues ( a number of interactions with charged residues. In particular a salt-bridge with K41 is removed and the interaction with the R97 side chain that mediates an interaction between the C-and the N-termini of the protein disappears. Overall this explains the higher loss of β structure at the N-and C-termini in D76N minor substate, thus linking the lower fold stability with the increased aggregation propensity in such minor substate.

Discussion

In 2012 the D76N mutant was reported to be responsible for an aggressive form of systemic amyloidosis with large fibrillar deposits in internal organs. Exceptional D76N aggregation propensity was also verified in vitro: this variant abundantly aggregates under native conditions in the absence of seeds [159]. While in principle it is unexpected that a D to N mutation on an exposed loop triggers such a dramatic effect, D76N mutant also displays a dramatic decrease in thermodynamic stability [183]. In contrast, the mutation of one Aspartate to Asparagine -with the consequent loss of a negative charge -in a different part of the protein does not result in any marked effects in terms of fold stability or aggregation propensity [159,187].

A comparison of fast (sub-microsecond) timescale dynamics measured using MAS NMR relaxation reveals similar motional characteristics in the wt and D76N mutant, with slightly higher levels of flexibility in the mutant. The position and amplitude of the dynamic modes observed in the crystalline lattice are well reproduced by fully solvated MD simulation, suggesting that the observed dynamics are intrinsic to the protein. Remarkably, MAS NMR 15 N R 1ρ relaxation dispersion experiments in crystalline β2m [START_REF] Lewandowski | Site-Specific Measurement of Slow Motions in Proteins[END_REF][START_REF] Krushelnitsky | Microsecond Time Scale Mobility in a Solid Protein As Studied by the 15 N R 1ρ Site-Specific NMR Relaxation Rates[END_REF][START_REF] Tollinger | Site-Resolved Measurement of Microsecond-to-Millisecond Conformational--Exchange Processes in Proteins by Solid-State NMR Spectroscopy[END_REF]188] reveal the existence of excited states that exchange with the ground state on the micro to millisecond timescale for both the wt and the D76N variant. The D76N excited state displays more extensive exchange phenomena than wt β2m, with 7 sites exhibiting markedly high relaxation dispersion effects. These are almost all located in the protein region neighboring the mutation: in the EF-loop, in the E-strand and at the end of the A-strand.

In keeping with these observations, ensembles resulting from RAM simulations showed that β2m displays two distinct conformational states in both mutant and wt. As sketched in Figure 4.8, while globally wt β2m excited state resembles very closely the ground state, D76N populates a conformational ensemble, which is less structured. In the excited state the D and the N-terminal A strands lose their beta structure and the C-terminus is partially detached from protein core. The result is that aggregation prone core strands (B, E and F) lose the protection of the aggregation resistant edge strands (A, C, D, G). Remarkably all the data indicate an increased level of dynamics in D76N N-terminus that is reminiscent of the ∆N6 β2m mutant, which is known for a strong aggregation propensity [189,190].

Furthermore, the conformational ensembles suggest a link between the differences in the excited states and the differences in the thermal stabilities where the mutation in position 76 breaks a large network of electrostatic interactions distributed over a large number of residues including the C-and the N-termini. Enhanced aggregation propensity in D76N appears to be correlated with the existence of a high-energy state characterized by a loss of structure in the protective edge β-strands.

In summary, by using a combination of X-ray crystallography, MAS NMR and molecular dynamics simulations, we have been able to precisely determine the impact of the highly pathogenic D76N mutation on the native state of β2m.

Structural insights into D76N beta-2 microglobulin amyloid fibrils

Moving perspective, we investigated the structural features of the fibrillar form of D76N β2m. Amyloid fibrils are resistant to most of biophysical techniques. Solution NMR techniques are intrinsically limited by the non-solubility of the fibrils and X-ray diffraction to the heterogeneity of the samples. It is nevertheless noteworthy that the the generic cross-beta structure and specific features of amyloid fibrils has been determined by X-ray crystallography [191] and that flexible parts of the fibrils can be characterized by High-Resolution Magic Angle Spinning (HRMAS) NMR [192]. Together with cryo-electron microscopy [193], solid-state NMR is the most suited method for high-resolution investigation of amyloid fibrils architecture [110][111][112][194][195][196]. In addition to the possibility to probe the different level of fibrils organization [197], solid-state NMR has the unique possibility to give site-specific information on the dynamics of amyloid fibrils [START_REF] Smith | Characterization of fibril dynamics on three timescales by solid-state NMR[END_REF][198][199][200].

Many studies have addressed the question of the structure of β2m fibrils. In 2006, Iwata et al. elucidated the structure of a 22-residue fragment of β2m amyloid fibrils, consisting of two layers of staggered parallel β-sheets [196]. Alternatively, models of the full length protein have been proposed where the protein preserves a close-to-native fold [201][202][203][204]. In contrast, a 30 Å resolution cryoelectron micrograph of full-length β2m fibrils contradicts the presence of the native monomer in the fibrils [205]. The results presented in this study instead indicate a dimer-of-dimer unit where β2m possesses a possibly globular but nonnative fold. Following this work, the secondary structure of wt and the truncated ∆N6 variant of β2m were determined by MAS NMR showing a drastic structural rearrangement of the protein backbone during aggregation [206].

Interestingly, it has been shown in the case of amyloid β that a single point mutation can dramatically impact the fibrils structure as well as the associated pathology [207]. The amyloidosis related to the deposition of aggregated species of D76N β2m has indeed strikingly different symptoms than DRA. Fibrils of D76N β2m have been found mainly in the spleen and adrenal glands of the patients carrying the mutation while the wt proteins deposit in the joints. These observations call for a detailed structural investigation of the fibrils formed from the D76N mutant. Besides its relevance in the framework of the actual pathology, such a study may be informative on the general mechanism of transformation of soluble globular proteins into fibrils.

Toward this goal, we investigate the backbone conformation and the intermolecular packing of the fibrils and compare our results to the analysis of the native conformations presented in previous section as well as to the literature on wt β2m fibrils. We show that important changes in secondary and tertiary structure happen during the process of fibrillization and that only the buried β-strands are implied in the cross-beta structure, bridging our findings on the first step of amyloidogenesis to the structure of the fibrils.

[100%-1 H N , 2 H, 13 C, 15 N] D76N β2m was grown at neutral pH without seeding at 37 • C under stirring conditions.

First insights from the sequential assignment of D76N β2m amyloid fibrils

2D projections of the 1 H N -detected spectra used for assignment are depicted in Figure 4.10. The assignment procedure is explained in details in Chapter 2, Section 2.3. In total, 47 1 H-15 N correlations have been assigned, all located between residues 17 and 78 in the sequence, indicating significant disorder/flexibility at the terminal parts of the protein. In agreement with this observation, proteolysis experiments conducted by the group of Dr S. Giorgetti in Pavia showed the presence of a protected core encompassing residues 20-80. In order to infer if D76N β2m conserves its native form within the fibrils, we directly compare the NMR signature of native D76N β2m in solution to the amyloid fibrils. Backbone resonance assignment of monomeric D76N β2m was performed in solution (Table A.1 in Appendix A). As shown in Figure 4.11, the 1 H-15 N correlation spectra of native and fibrillar D76N β2m strikingly differ. Amide shifts are very sensitive to the local environments of the protein, it is therefore no surprise that they are perturbed by fibrillization. It is however instructive to compare the chemical shift perturbation due to fibrillization and crystallization (Figure 4.11a). Indeed, upon crystallization, the protein changes its chemical environment but conserves its structure. We observe that, although the spectra of D76N β2m in solution and in crystals do not overlay, the perturbation is less important than between the monomer in solution and the fibrils. The perturbation between cysteine residues 25 and 80 has been shown to be crucial for amyloid fibrils formation of wt β2m [209]. While we could not assign resonances belonging to Cystein-80 (presumably because it is affected by the conformational disorder/dynamics of the C-terminus), 13 Cα and 13 Cβ chemical shifts of Cystein-25 are strongly indicative of an oxidized state (55.6 and 39.3 ppm, respectively) [210]. Another important feature of the fibrils is the isomerization state of Proline-32. Indeed, native β2m (as well as its D76N variant) has an uncommon cis conformation of the Histidine-31-Proline-32 prolyl bond. Cis to trans isomerization has been extensively discussed as a potential pathway for fibril formation of the wt [211-213]. 13 Cβ chemical shifts of proline residues are affected by the isomerization state of the peptide bond. Statistical values of 13 Cβ chemical shift for trans and cis were evaluated to, respectively, 31.75±0. 98 and 34.26±1.15 ppm [214]. We determined 13 Cβ chemical shift of Pro-32 of D76N β2m in both crystalline (34.8 ppm) and fibrillar (32.1 ppm) forms. The values strongly indicate an exchange of the isomerization state between the native conformation (cis) and the amyloid fibrils (trans). Measurement of the 13 Cγ chemical shifts would strengthen this analysis as the difference between 13 Cβ and 13 Cγ chemical shifts is a better reporter of the isomerization state than the 13 Cβ chemical shifts alone. These two features of D76N β2m amyloid fibrils (conservation of the Cys25-Cys80 disulfide bridge and cis-isomerization state of the His31-Pro32 bond) are reminiscent of wt β2m [206].

Secondary and tertiary structures

Petkova et al. classified the amyloid fibril architecture into four structural levels [197]. Primary and secondary structures have similar definition than for monomeric proteins even though the secondary structure elements are in general exclusively β-strands. The tertiary structure corresponds to the parallel or anti-parallel organization of the β-sheets in the cross-beta structure. Finally, the quaternary structure correspond to the relative arrangement of the intermolecular β-sheets. As a first step toward complete structural characterization, we used the H, N, Cα, Cβ and C chemical shifts for the determination of secondary structure elements of D76N β2m by TALOS-N software [215] (Figure 4.12). Five β strands are predicted to be present in the rigid core of the fibrils. They comprise 35 residues, that is to say two third of the assigned residues.

We next aimed at characterizing the tertiary/quaternary structure of the fibrils. We thus performed 1 H-1 H through-space correlations [START_REF] Knight | Structure and backbone dynamics of a microcrystalline metalloprotein by solid-state NMR[END_REF], using RFDR mixing [153]. These experiments were acquired as 3D or 4D datasets, where the contacts between amide protons were encoded with the chemical shifts of the 1 H and/or 15 N atoms of starting and/or landing amide groups. These experiments revealed contacts between residues spaced in sequence by less than 5 residues while no long-range correlation have been unambiguously assigned (Figure 4.13). This pattern is strongly indicative of a parallel, in register β-sheet intermolecular arrangement, where each residue faces its homologous residue in a neighboring molecule. This finding was further supported by the H-bond correlation experiment already presented in Chapter 2. The residues for which Zseq score is greater than 1 are predicted to be aggregation prone. The secondary structure of native wt and D76N β2m is determined from their crystal structure (PDB code: 2d4f and 4fxl, respectively). The sequence is shown on top of the figure with the proteolysis sites of D76N β2m fibrils in red, and residue N76 in blue. The rigid core of the fibrils, protected from proteolysis (residues S20 to C80), is highlighted with a red rectangle.

H31-F30 S61-F62 F62-S61 F62-Y63 Y63-F62 Y63-L64 Y63-L65 L64-Y63 L64-Y66 L65-Y63 L65-Y66 Y66-L65 Y66-Y67 T68-Y67 T68-E69 E69-Y67 E69-T68 E69-F70 F70-E69 F70-T71 T71-E69 T71-F70 T73-E74 E74-T73 E74-K75 K75-T73 K75-N76 E77-N76 E77-Y78 T68-Y66 N76-K75 K48-V49 Y26-C25 N76-E77 130 120 110 130 120 110 L23-F22 Y26-V27 H31-F30 L54-D53 S61-F62 F62-S61 L64-Y63 L64-L65 L65-L64 L65-Y66 Y66-L65 Y67-T68 T68-Y67 E69N-F70N E69-T71 F70-E69 F70-T71 T71-F70 T73-E74 E74-T73 E74-N76 K75-E74 K75-N76 N76-K75 E77-N76 E77-Y78 A79-Y78 δ( 1 H) / ppm δ( 1 H) / ppm δ( 15 N) / ppm δ( 15 N) / ppm (a) (b)

Comparison with fibrils of wild-type β2m

Discussion

The MAS NMR characterization of amyloid fibrils grown in vitro complement the analysis of the native state of D76N β2m presented in Section 4.2. Interestingly, as shown in Figure 4.12, we find a progressive evolution of structural elements between the starting, native conformation derived from the crystal structure, the ensemble representing the excited state derived by RAM simulations, and the final fibrillar state. Notably, the A-and G-edge strands of β2m, which are destabilized by the point mutation, are not part of the cross beta-structure. Additionally, the N-terminal portion of the B-strand loses secondary structure both in the excited state (native*) and in the fibrils. Furthermore, the rearrangement of the secondary structure elements in the core of the fibrils may be facilitated by the loss of structure of the corresponding native β-elements. This is particularly evident for the region encompassing residues 44-56 where the C and D strands are absent in the excited state while they appear to form a unique strand in the fibrillar state. A similar observation can be made for the E-strand and EF-loop region. The loss of interaction between EF-and C D-loops, as evidenced by the network analysis, may therefore help to promote their reorganization in β-structures within the fibrils. Finally, we concluded from the analysis of the NMR data obtained on the monomeric form of D76N β2m and the RAM simulations that the exposure of the aggregation-prone B-and E-strands may be the driving force of amyloid fibril formation. We observe here that these regions of the proteins are indeed part of the β-structure of the fibrils (Figure 4.12).

The secondary and tertiary structures of wt β2m fibrils have previously been characterized by 13 C-detected ssNMR [206, 217-219]. We note common structural and dynamical features in both wt and D76N fibrils, namely the position of the rigid core, a parallel in register beta arrangement, the preservation of the disulfide bridge and flexible termini, but also differences can be observed in the fine details of the secondary structures, namely the positions of the β-strands. This may indicate comparable fibril architecture even in the presence of likely different aggregation pathways.

Conclusions

In conclusions, we report in this chapter an application of state-of-the-art MAS NMR methodologies to rationalize the amyloid fibril formation propensity of the D76N mutant of β2m. On one side, we studied the dynamical behavior of the native fold of β2m. In particular, we detected the presence of alternative conformations of the protein in the crystalline environment. A combined analysis of the MAS NMR data with ensemble modeling revealed a network of interactions weakened by the mutation, leading to the transient exposure of aggregation prone β-strands. On the other hand, we investigated the structure of D76N β2m amyloid fibrils. The structural insights so obtained complemented our observations on the native state of β2m, linking the two extremities of the aggregation pathway. We anticipate that the elucidation of amyloid fibrils structure and dynamics will lead to a more detailed understanding of their mechanism of formation. The work presented here already forms a solid basis for further characterization.

Part III

Membrane protein dynamics

Membrane proteins and MAS NMR

Membrane proteins comprise about a third of the proteome and are the target of half of today available drugs. The importance of membrane proteins for the biology of the cell comes from their location at the interface between the inner and outer cellular environment or between cell compartments. They are responsible of many different functions such as cell-cell recognition, signal transmission or both active or passive transportation of ions and molecules. Even if the membrane environment constrains possible structural features, membrane proteins adopt a richness of topologies and dynamics, adapted to their specific functions [220]. As an example, rhodopsins, a family of proteins which engage signal transduction pathway upon recognition of light are composed of seven α-helixes forming a transmembrane bundle with a ligand-binding pocket in the extracellular part fit for the fixation of a retinal cofactor [221]. When photo-activated, the retinal undergoes cis to trans isomerization, inducing a conformational change of the helical bundle provoking a cascade of signal transduction events.

Despite their high biological and medical interest, membrane proteins are particularly difficult to study by standard biophysical methods. As a marker, they represent today less than 2% of the structures deposited in the Protein Data Bank. Indeed, membrane proteins are difficult to produce in large amount and do not easily form large crystals needed for X-ray crystallography. In addition, the high molecular weight of the preparations used in solution NMR limits the application of this technique. In addition to these issues, membrane proteins are very sensible to their environment. However, all in vitro studies rely on the mimicking of the native environment of the protein by specific preparations to reach technical requirements, possibly inducing non-native conformational changes [222]. For example, micellar preparations usually used in solution NMR to study membrane proteins may affects protein behavior because of different curvature or hydrophobicity compared to cell membranes.

Solid-state NMR is a method of choice to study membrane proteins since it does not require the formation of crystals and samples can be prepared in nativelike lipid bilayers or even native membranes. First ssNMR studies of membrane proteins were performed on aligned samples [223,224]. This strategy permits to recover resolution by selecting one orientation of the sample compared to the magnetic field. Alternatively, MAS NMR has proven to be an efficient and more versatile tool for the study of membrane proteins. MAS NMR was used to elucidate the structure of the ligand-binding site of influenza M2 proton channel [225]. A strategy based on the combination of oriented samples and MAS NMR [226] was successful to solve the structure of a G-protein-coupled receptor (GPCR) [227]. In 2013, Wang et al. determined the structure of the seven-transmembrane protein Anabaena sensory rhodopsin by MAS NMR alone [113]. Subsequently, the sensitivity provided by 1 H-detected MAS NMR was exploited for the structural elucidation of M2 [228] and the Outer Membrane Protein G [229]. The recent developments of 1 H-detected MAS NMR render possible high-resolution structure determination with sub-milligram sample quantities, opening an avenue for membrane protein investigations by NMR [START_REF] Andreas | Structure of fully protonated proteins by proton-detected magic-angle spinning NMR[END_REF].

Beyond structure, NMR has proven to be a powerful tool for investigation of membrane protein motions. The whole palette of dynamics events are relevant for membrane protein functions, comprising hydrogen exchange [230], aromatic side-chains ring flips [231], side-chains rotameric states exchange [START_REF] Patel | Coupling of retinal isomerization to the activation of rhodopsin[END_REF] and larger conformational changes [START_REF] Good | Solidstate NMR provides evidence for small-amplitude slow domain motions in a multispanning transmembrane α-Helical Protein[END_REF]232]. In turn, dynamics influences the quality of ssNMR spectra. Overall spectra sensitivity and resolution are affected by the presence of rotational diffusion [233,234] and the local presence of µs-ms motions may reduce the efficiency of the ssNMR experiments.

In the following chapters, we will present two case studies of membrane protein dynamics. The first one (Chapter 5) results from a collaboration with the team of Prof A. Milon from the Institut de Pharmacologie et de Biologie Structurale (IPBS) in Toulouse 3 . In this work, we address the question of protein dynamics in fluid bilayers on the β-barrel trans-membrane domain of the outer membrane protein A from K. pneumoniae (KpOmpA) [START_REF] Saurel | Local and Global Dynamics in Klebsiella pneumoniae Outer Membrane Protein a in Lipid Bilayers Probed at Atomic Resolution[END_REF]. In particular, we determine that in the highly concentrated preparation used for MAS NMR experiments, the betabarrel of KpOmpA undergoes a rocking motion on the hundreds-of-ns timescale. Even though the loops are invisible in the ssNMR spectra because of important flexibility or disorder, their relative dynamics are extrapolated from the closest visible residues. Enhanced dynamics is assigned to an extracellular loop and confirmed by proteolysis experiments performed in the laboratory of Prof A. Milon, highlighting a potentially important site for the function of KpOmpA. In Chapter 6, we will present a study of the protein AlkL from the alkane degradation system of P. putida. Our team in collaboration with Dr Andreas' group at the Max Planck Institute in Göttingen determined the structure of AlkL in lipid bilayers.

In order to obtain deeper understanding on the molecule conduction mechanism of AlkL, we investigate the site-specific dynamics of the protein with or without substrate. Thanks to experiments encoding the chemical shifts in three dimensions, we highlight a pathway for the molecular transit from the extracellular environment to the lipid bilayer and determine key residues involved in this mechanism.

Chapter 5

Local and Global motions of KpOmpA

Introduction

In this chapter, we investigate the dynamics of the transmembrane domain of the outer membrane protein A from K. pneumoniae (KpOmpA). K. pneumoniae is a Gram-negative bacterium responsible for respiratory and urinary infections. KpOmpA belongs to the OmpA family which is present in many bacteria with diverse cellular functions including ion passive transport and structure stabilization via the interaction of its globular C-terminal domain with peptidoglycan [235]. Its transmembrane domain was characterized by solution NMR (PDB code 2K0L) [236], and is a 8 β-strands barrel with 4 long extracellular loops (L1-4) and 3 short periplasmic turns (T1-3). A flexible hinge region links the barrel with the soluble C-terminal domain. OmpA contributes to pathogenicity via its role in cell-cell adherence, its proinflammatory role, and its implication in innate and adaptive immunity [235]. This is usually facilitated by specific interaction between a cell surface protein on the bacterium and the extracellular matrix (ECM) proteins in the target cell, such as fibronectin. This suggests that the epitope for protein binding and cell adhesion, could be a flexible and well-presented extracellular loop.

Dynamics of KpOmpA by MAS NMR

Fingerprint spectra

The CP-HSQC spectrum of [ 1 H N , 2 H, 13 C, 15 N]-labeled KpOmpA in lipid bilayers is displayed in Figure 5.1a. The chemical shift dispersion in the direct 1 H dimension (between 8 and 10 ppm) is indicative of secondary structure elements, here the beta-barrel. The observed resonances were assigned to 49 residues with .1b, this feature is particularly obvious considering the peak intensity of two isolated glycines located at the top of strand 1 and 2 (e.g., G26 and G53 and shown to be in chemical exchange in a micelle environment [236,237]) compared to G22, G58 and G71 in the middle of the β-strands 1, 2, and 3 respectively. Intriguingly, none of the extracellular loops, neither the residues 138-147 close to the loop 3 are assigned, suggesting transient unfolding of this portion of the β-strand, or the presence of a topology in lipids different from that found in detergent micelles. In order to further investigate residues from the extracellular loops, 1 H-15 N correlations based on INEPT transfer schemes were measured as well. However, they failed to reveal distinct cross peaks arising from loops L1 to L4, showing that none of these loop possess fast, sub-ns motion.

Modelling the rocking motion of KpOmpA within the lipid bilayer

To investigate dynamics in the beta-barrel core, we first measured the amide 15 N-1 H dipolar coupling on every assigned peak (Figure 5.2b,d). The order parameter <S>, defined as the fraction of the rigid limit coupling, is sensitive to any motional process faster than the 15 N-1 H dipolar coupling (< 11 kHz). In this measurement, <S> is fairly homogeneous along the beta-barrel (Figure 5.2b), with all residues displaying a value above 0.8 (average value: 0.88 ± 0.05). This indicates that there is only moderate amplitude motion throughout the assigned portion of N-KpOmpA and that these residues within the barrel display similar motion.

Amides for which there were not observable 15 N-1 H cross peaks indicate more substantial local motion, which occurred for residues 102-111 in turn 2, the periplasmic side of strand 5 and residues 138-146 in the extracellular side of strand 6. The consistently high values of the order parameter along the sequence also demonstrates the absence of fast axial diffusion (Figure 5.2a,b), which under our experimental conditions would impose a modulation of <S> dependent on the angle between the amide bond vector and the axis of rotation illustrated by the bars in Figure 5.2b. As discussed in reference [233], axial diffusion would lead to a drastic reduction of the resolution which is not observed in our experimental conditions, although the measurements were performed above the lipid phase transition temperature. This might be a consequence of the low lipid to protein ratio 0.5 (w/w), which results in a membrane with greater viscosity and a reduced diffusion rate.

An explanation of the experimental values compatible with the low fluidity of the lipid bilayer is provided by rocking motion of the transmembrane domain. We modeled such anisotropic dynamical process of the beta-barrel with the help of a Gaussian axial fluctuation (GAF) formalism [START_REF] Bremi | Locally anisotropic internal polypeptide backbone dynamics by NMR relaxation[END_REF][START_REF] Lewandowski | Anisotropic collective motion contributes to nuclear spin relaxation in crystalline proteins[END_REF]. In this approach, the amide-NH bonds fluctuate around three axes corresponding to the axes of inertia of the protein. The corresponding order parameters depend on the angle between each amide bond vector and the molecular frame, and on the fluctuation amplitudes (σ α , σ β , σ γ ) of the collective motions (Figure 5.2c) (Appendix D).

Figure 5.2d shows that a set of small amplitudes with cylindrical symmetry of the barrel (σ α = σ ∼ 6 • , σ β = σ γ = σ ⊥ ∼ 10 • ) reproduces accurately the experimental order parameter profile. Using this model, a small range of amplitudes (σ and σ ⊥ ) describes equally well the experimental values. This is depicted in Figure 5.2e, which shows the agreement between experimental and back-calculated order parameters with respect to the two individual amplitudes, expressed by the reduced chi square function (χ 2 red ). The best agreement is reached for σ ranging between ∼ 0 • and ∼ 12 • and σ ⊥ between ∼ 9 • and ∼ 13 • . This plot strikingly shows that the two amplitudes of fluctuation are anticorrelated: relatively high values of σ ⊥ (∼ 11 • ) together with smaller values of σ (∼ 8 • ) explain the experimental order parameters similarly well than low values of σ (∼ 0 • ) and higher values of σ ⊥ (∼ 11 • ).

A deeper insight into this motional process is provided by 15 N relaxation rates, which are sensitive both to amplitude and time scales of motions. 15 N R 1 and R 1ρ relaxation rates were measured for all the resolved amide signals, and are reported in Figure 5.2f and g and Table B.3 in Appendix B. 15 N R 1 relaxation rates range from 0.008 to 0.033 s -1 (average value of 0.022 s -1 ) and are relatively constant for the observed peaks, while 15 N R 1ρ relaxation rates range from 2 to 34 s -1 (average value of 12 s -1 ). 15 N R 1ρ relaxation rates are particularly sensitive to ms-µs motions and exchange phenomena. Most 15 N R 1ρ rates are clustered around 7 to 15 s -1 with a few exceptions. Rates higher than 20 s -1 are found for residues G67, A98, T112, and W151, mostly located in turns or close to the strand extremities and indicate the existence of slow internal motions. Within the sensitivity and resolution of these experiments, R 1ρ could only be measured for residues displaying a moderate increase in R 1ρ , while stronger increases lead to peak disappearance, as was observed for the extracellular half of strand 6.

This evidence thus supports the hypothesis that the N-terminal half of βstrand 6 samples a variety of conformations beyond that determined in detergent micelles. It is also interesting to compare the average R 1ρ value with values from the recent literature. Ladizhansky and co-workers studied a seven helices transmembrane protein, sensory rhodopsin, in proteoliposomes made of DMPC and DMPA, at 280 K, i.e., fifteen degrees below the lipid phase transition temperature, the lipids being in gel phase [START_REF] Good | Conformational dynamics of a seven transmembrane helical protein Anabaena Sensory Rhodopsin probed by solid-state NMR[END_REF]. The R 1ρ of residues in the rigid part of the protein ranged from 1 to 5 s -1 , and residues with R 1ρ between 10 and 30 s -1 were identified as subject to slower internal motions, in the 100 ns time scale. Another interesting point of comparison is given by the work of Schanda and coworkers who studied ubiquitin relaxation rates in several crystal forms [238]. In a rigid crystal, most R 1ρ values ranged from 3 to 6 s -1 , while in a more hydrated crystal they ranged from 7 to 17 s -1 (that is very similar to what observed here). These higher relaxation rates were attributed to a rocking motion of ubiquitin within the most hydrated crystal form, with a time scale in the 0.1 to 1 µs range.

A combined analysis of order parameters and relaxation rates allows a precise characterization of the rocking motion of KpOmpA. For sake of clarity, Figure 5.3 illustrates the analysis procedure. Within the GAF model, once the fluctuation amplitudes are defined by the order parameter, 15 N R 1 and R 1ρ relaxation rates depend solely on the time scales (τ and τ ⊥ ) of the motion. These parameters can therefore be extracted by fitting the experimental rates to the ones calculated according to the expressions for dipolar and CSA relaxation. Within the approximation τ GAF = τ = τ ⊥ , this procedure was performed for a pair of amplitudes (red square in Figure 5.2e), and the result of the fit is depicted in Figure 5.2f and g. Comparable results are obtained for different sets of amplitudes (black squares in Figure 5.2e), and are reported in Figure 5.4. The correlation times τ GAF obtained are about 200 ns. These time scales found for KpOmpA are close to the time scale of rocking motion of ubiquitin in hydrated crystals, explaining the similarity of the 15 N relaxation rates between the two studies. As expected for multiaxial fluctuations, both 15 N R 1 and R 1ρ profiles are rather flat. Residues R155, W151, A98, G67, and T112 are thought to possess internal motion and therefore have relaxation rates higher than calculated in the case of pure rocking motion. Overall, a fairly detailed dynamics picture of KpOmpA (Figure 5.1c) results from these data when considered along with previously published observations. The four loops are subject to ms-µs motions, being observable neither via CP based transfer nor via INEPT based transfer under MAS conditions. Dynamics of the β-barrel are precisely described as collective rocking motions thanks to a GAF model. Small amplitude fluctuations (∼ 10 • ) occurring at hundreds of nanoseconds explain the order parameters obtained experimentally, as well as most of the relaxation rates. The β-barrel is homogeneously rigid within strands 1-4 and 7-8, while the periplasmic side of strand 5 and the extracellular side of strand 6 display conformational disorder which leads to peak suppressions in the 15 N-1 H 2D correlations. The observed motion of these strands highlights increased disorder in loop 3 compared to the other loops. These deductions were corroborated by trypsin cleavage experiments which showed that among 16 potential Arg and Lys cleavage sites in the N-terminal domain, proteolysis occurred uniquely at Arg134. Considering that proteolytic cleavage is limited by loop motion and accessibility, this points to particular mobility of loop L3 in its C-terminal half (since Lys 124 at the beginning of loop L3 is not cleaved). It is thus tempting to speculate that the L3 loop is highly flexible and well presented to the outer cell surface, contrary to the other extracellular loops, and thus accessible to trypsin. This makes it an important epitope for protein binding and cell adhesion. )

σ ⊥,GAF =2°σ //,GAF =11°τ GAF =162ns σ ⊥,GAF =2°σ //,GAF =11°τ GAF =162ns σ ⊥,GAF =9°σ //,GAF =9°τ GAF =232ns σ ⊥,GAF =9°σ //,GAF =9°τ GAF =232ns (b) (a) (c) (d)

Conclusions

In conclusion, application of state-of-the-art 1 H detected techniques using ultrafast magic-angle spinning and ultrahigh magnetic fields allow site-specific quantification of the KpOmpA barrel dynamics in lipid membranes. The measurement of 1 H-15 N dipolar couplings by REDOR indicates the presence of relatively uniform motion along the primary sequence. Collective rocking of the β-barrel with small amplitude (∼ 10 • ) is sufficient to explain the data. Measurement of scaled dipolar couplings reports on motions faster than the rigid limit frequency but not the time scales of motions. For this, we measured 15 N R 1 and R 1ρ relaxation rates and found that the rocking model can explain the relaxation rates with a time scale of hundreds of nanoseconds. However, some residues close to the loops showed particularly enhanced R 1ρ , indicating slower local motion related to the dynamics of the loops. In particular, residues at the extremity of strand 6 connected to loop 3 were assumed to transiently unfold as they are not detected with our approach. This finding is supported by trypsin cleavage experiments where a specific feature of the L3 loop of KpOmpA was identified, namely, motion of sufficient amplitude for productive contacts with trypsin.

In the conditions used for solid-state NMR, the dense protein packing constrains the motion of the barrel. The resulting motion is therefore a collective rocking with amplitudes of around ten degrees and a time scale of hundreds of nanoseconds. This may represent a general phenomenon in samples studied by other physical methods such as X-ray crystallography. It is therefore necessary to take this type of motion into account to interpret local dynamics. Here, the residual dynamics of the residues at the edges of the barrel was informative about the motions of the loops. Combination of solid-state NMR with proteolysis experiment highlights the specific dynamics of the L3 loop which by extension is expected to be particularly accessible for interactions with other proteins required for cell-cell adhesion mechanisms.

Chapter 6

The hydrophobic tunnel of AlkL revealed in lipid bilayers

Introduction

AlkL is a component of the alkane degradation pathway of the Gram negative bacteria P. putida and has been shown to facilitate alkane uptake through the outer membrane [239]. As for other Gram negative bacteria, the outer membrane of P. putida possesses a lipopolysaccharide layer which forms a barrier for the permeation of hydrophobic molecules and antibiotics. The mechanism for small molecules transport by AlkL was deduced from the crystal structure of two homolog proteins, E.coli OmpW [240] and P. aeruginosa OprG [241]. Both structures show that the protein is an 8-strands beta-barrel with well-ordered extracellular regions. The extra-cellular part of both proteins are hydrophobic, acting presumably like a funnel for the transit of hydrophobic molecules through the polysaccharide layer. In contrast to ion-conducting channels, the inner part of the barrel is also hydrophobic. Interestingly, polar and charged residues are found in the periplasmic side of the barrel, avoiding the pore to span the whole transmembrane region. The idea that molecules may exit laterally to the lipid bilayer emerged from the observation that conserved proline residues bend the end of the transmembrane β-strands, forming a hole at the surface, connected to the main pore. This hypothesis suggests the molecular transport mechanism of AlkL. However (i) no experimental evidences have been reported of such structural features for AlkL (ii) the static vision arising from structure determination have to be complemented by the determination of local dynamics of the protein in order to have a deeper understanding of the mechanistic details of AlkL function. The extra-cellular part of the protein is an extension of the betabarrel with hydrophobic residues pointing inside, forming a hydrophobic tunnel (Figure 6.1b). The inner volume of the pore is shown in Figure 6.1c. A cluster of residues blocks the tunnel close to the extracellular-membrane interface, strengthening the lateral exit hypothesis. Among them, Arginine-168 and Phenylalanine-72 form the previously described "hydrophobic gate" [240]. The three proline residues surrounding the putative exit hole in OmpW correspond to residues Proline-76 and -100, and Alanine-101 in AlkL. The NMR structure does not contain evidence of the formation of such a pore. Instead, putative exits are found on both sides of the nearby loop connecting strands β 1 EC and β 2 TM (for the notations used here, please refer to Figure 6.3). Interestingly, residues 50 to 53 in the same loop could not be assigned, suggesting the presence of significant dynamics in this portion of the protein. It is tempting to speculate that the exit of AlkL substrate is facilitated by the flexibility of the β 1 EC -β 2 TM loop. In the following, we investigate in a more rigorous way the relation between function and dynamics in AlkL.

MAS NMR structure

Site-specific dynamics of AlkL

Backbone flexibility

We first measured 15 N relaxation on a perdeuterated AlkL sample at 60 kHz MAS. Previously reported studies of dynamics in the solid state (at the exception of a very recent publication from Linser's group [243]), rely on chemical shift encoding in two dimensions and measurement of relaxation in a third "pseudo"dimension, limiting the application of this method to homogeneous samples with highly-resolved 2D spectra. Here however, only about 30 resonances on the 162 assigned residues are resolved in the 2D 1 H-15 N plane of AlkL (Figure 6.2). We therefore chose to increase the dimensionality of the relaxation experiments thanks to an additional 13 C' chemical shift evolution period. The high resolution of the 3D (H)CONH spectrum allows to detect around 80 purely isolated peaks, covering most of structural elements of AlkL. Figure 6.3 shows measured site-specific 15 N R 1 and R 1ρ relaxation rates (Table B.4 in Appendix B. 15 N R 1 and R 1ρ values are close to 0.02 s -1 and 10 s -1 , respectively, very similarly to what we observed for KpOmpA. 15 N R 1 relaxation rates are correlated with the secondary structure, with higher rates in loops (in particular in the loop between strands β 2 EC and β 4 TM ). Interestingly, increased ps-ns dynamics is observed in the extra-cellular part of the protein compared to the beta-barrel even for residues involved in β-sheet structures, as monitored by 15 N R 1 rates (mean 15 N R 1 value is 0.017 s -1 in the beta-barrel, compared to 0.044 s -1 in the extracellular strands). In contrast, 15 N R 1ρ relaxation rates are more uniform along the sequence, even if slightly higher rates are also found in the extracellular part (mean 15 N R 1ρ value is 8.8 s -1 in the beta-barrel, compared to 10.8 s -1 in the extracellular strands). Only six residues have 15 N R 1ρ higher than 20 s -1 , among which Alanine-77 and Arginine-78 are located close to the end of the pore, defining a possible lateral exit-region. Different ways to accommodate the transfer of molecules to the membrane are possible. The protein may possess some inherent flexibility which facilitate the passage of small molecules through the pore. Alternatively, the protein may adopt two defined and relatively rigid conformations of the backbone, one corresponding to an open state and the other to a close state. We expect the first strategy to smoothen the transport of substrates with varying sizes while the second one may imply higher specificity. Both types of dynamics may also be combined, leading to a more complex behaviour. The elevated values of 15 N R 1 relaxation rates in the extracellular part suggests that AlkL may adopt the former strategy. Nevertheless, the relatively high values of 15 N R 1ρ in some parts of the molecule such as for Alanine-77 and Arginine-78 may also be induced by slow dynamics, possibly corresponding to an exchange mode. Additionally, loop β 1 EC -β 2 TM is not detected by the CP-based experiments, denoting high conformational flexibility or disorder. Conformational exchange is investigated by two different techniques, 15 N R 1ρ relaxation dispersion and 15 N CEST experiments, most sensitive for µs-ms and ms-s timescale of exchange, respectively. Both types of experiment can not be run with affordable experimental time with a 3D-detection block because of the high number of increments required for each of them. Fortunately, Alanine-77 and Arginine-78 are both resolved in the 2D 1 H-15 N CP-HSQC. Figure 6.4 shows their respective dispersion profiles. None of them showed significant dispersion, indicating that high 15 N R 1ρ rates may be better explained by either slow ms-s exchange or fast ns-µs fluctuations of the backbone. Interestingly, the 15 N R 1ρ rates at 20 kHz spin-lock fields are consistently higher than the average of the dispersion. This dispersion profile is consistent with overall rocking motion. In this case indeed, the chemical shifts are not significantly modified in the time course of the motion. In contrast, we expect slow molecular rocking to induce a dependency of the R 1ρ rates to the MAS speed [START_REF] Lamley | Intermolecular Interactions and Protein Dynamics by Solid-State NMR Spectroscopy[END_REF]. The enhanced 15 N R 1ρ rates at the highest spin-lock values may therefore reflect the approach of the rotary resonance. Further experiments are needed to check the reproducibility of this result and to evaluate the dynamics parameters related to the rocking (e.g. by varying the MAS rate). The detailed investigation of this feature is however not the scope of the present study. The results of CEST experiments are depicted in Figure 6.4c and d. The presence of a single dip in the profiles signals the absence of slow ms-s exchange between conformations. Noteworthy, the 15 N R 1ρ relaxation dispersion and the 15 N CEST profiles determined for the 28 other residues possess the same features than described for Alanine-77 and Arginine-78 (Figure E.2 and E.3 in Appendix E). Altogether, these results support a scenario where AlkL pore is not in slow exchange between two defined conformations but indeed has a more flexible behavior. In order to further investigate the mechanism of transport, we prepared a sample of AlkL with a molecular substrate.

AlkL in presence of carvone 6.4.1 Chemical shift perturbation

AlkL is potentially able to transport a large range of molecules [244]. In particular, it was shown to increase the hydroxylation efficiency of (S)-limonene by P. putida [245]. We therefore used the small terpenoid carvone, structurally similar to (S)-limonene, as a model substrate of AlkL. We first measured chemical shift perturbations (CSP) upon addition of carvone. The completeness of resonance assignment as well as the high resolution observed in the 3D (H)CONH spectra allowed us to monitor the chemical shift changes of the three backbone nuclei 1 H N , 15 N and 13 C'. Chemical shift perturbation values are calculated with the expression (∆δ 1 H N ) 2 + (∆δ 15 N/10) 2 + (∆δ 13 C'/2.5) 2 . In Figure 6.5, CSPs are mapped onto the structure of AlkL. Highest perturbations are found for residues in the extracellular part, consistently with the putative funnel role of this part of the protein. In particular, CSPs greater than 0.2 ppm are observed for residues and Phenylalanine-172. Interestingly, those residues are either glycine or have side-chains pointing outward of the funnel. All of them thus create a significant space in the hydrophobic pore for the transit of carvone. Since CSPs are determined for backbone nuclei, enhanced CSP should in principle be observed at the positions where carvone can directly interact with AlkL backbone, so where the side-chain do not have a shielding effect. Our results are therefore consistent with the passage of the molecule inside the extracellular pore.

Effects of carvone uptake on backbone flexibility

We then investigated the effect of carvone uptake on the backbone dynamics of AlkL. We focussed on the slow regime probed by 15 N R 1ρ relaxation since it corresponds most likely to the range of timescales at which the transit of carvone through the pore may occur. The 15 N R 1ρ relaxation rates measured for AlkL in presence of carvone are shown in Figure 6.6. For most residues, relaxation rates are unaffected by the addition of carvone. Most important changes are found at the extracellular part of AlkL, in agreement with the CSP. For the residues of this part of the protein, 15 N R 1ρ are higher in presence of carvone, suggesting either an enhancement of the structural dynamics induced by the substrate or transient protein-ligand interactions. Intriguingly, highest differences in 15 N R 1ρ (∆R 1ρ ) and in chemical shifts for the two samples do not correlate. If the increase of 15 N R 1ρ is due to interactions between the ligand and AlkL backbone, we expect at the contrary a correlation of the CSP and ∆R 1ρ values. When scrutinizing with more attention the results, one can observe that most of the residues with highest difference in 15 N R 1ρ have sidechains pointing inward the pore. For example, residues I49, F81 and V191 form a hydrophobic ring reducing the pore size to less than two water molecules at their height. It is therefore no surprise that increased dynamics is observed when the substrate passes through the pore.

Conclusions

In summary, the MAS NMR structure of AlkL in lipids is similar to its homology model. An hydrophobic pore links the extracellular environment to the central part of the barrel. However, the potential lateral exit sites of AlkL emerging from the structure are located in a different part of the molecule (while in close proximity) than what has been proposed for OmpW and OprG. We showed that a continuum of dynamics exists in the extracellular part with enhanced flexibility observed for residues at the threshold and at the possible exit of the tunnel.

The affinity of the tunnel of OmpW for hydrophobic molecules was deduced from the presence of a detergent molecule in the X-ray structure [240] and by analogy with the transport mechanism of FADL [246]. For AlkL, we directly monitored the passage of the substrate carvone by chemical shift mapping. We demonstrated that carvone is indeed present inside the extracellular part and does not cross the whole barrel. The comparison of 15 N R 1ρ relaxation rates between the samples with and without carvone suggests that the passage of the substrate increases backbone dynamics. We highlighted three residues important for the transit of carvone. Indeed, Leucine-32, Phenylalanine-81 and Valine-191 sidechains are pointing inward the barrel, reducing the width of the pore and have enhanced µs-ms timescale dynamics when carvone is present. In general, it seems very likely that the flexibility of some parts of the protein smoothen the molecular transit, which may be an explanation for the low specificity of AlkL [245].

Conclusions and perspectives

The work presented in this thesis have been made possible by the development of methodologies adapted to the use of MAS above 60 kHz and high magnetic fields. I participated to the design of new correlation schemes for backbone resonance assignment, structure elucidation, and investigation of backbone and side-chain dynamics. These strategies permit a reduction of the requirements in terms of experimental time, sample quantities and isotopic labeling, and were benchmarked on different preparations such as microcrystals, sedimented proteins, amyloid fibrils and membrane proteins. Hopefully, the coming new generations of MAS probes and high field spectrometers will extend the horizon of MAS NMR even further by increasing both spectral resolution and sensitivity.

I then exploited the advantages provided by the versatility of MAS NMR to address two types of biological problems:

(i) Amyloid fibrils formation. The work on the amyloidogenic protein D76N β2m highlights the interest of characterizing both structural extremities of a folding/aggregation pathway, i.e. the native fold of the protein and the amyloid fibrils. The characterization of the native ensemble of conformations has proven to be primordial for understanding the reactivity at the earliest steps of fibrillization. In the other hand, the structure and dynamics of the fibrils themselves help to bring light on the mechanism of fibril formation and on the role of the fibrils in the pathology. A first characterization of the fibrillar state of D76N β2m was performed. While these results are already highly informative on the aggregation pathway, they most importantly pave the way for structural determination of the amyloid fibrils.

(ii) Functional mechanisms of membrane proteins. Two outer membrane proteins were investigated in lipid bilayers. The results on KpOmpA showed that the high protein concentration in MAS NMR samples constrains the overall motion of the protein while internal dynamics related to the protein function are preserved. Following this, the mechanism of molecular transport of AlkL in lipid bilayers was studied, highlighting the specific role of dynamics in molecular uptake. The results highlight signatures of the molecular transport. It is very likely that extending the analysis to side-chains and titrating the substrate will enrich our comprehension of the process.

Overall, this work was performed in a context of scientific excitement in the field of MAS NMR. We believe that the new methodologies presented herein and their applications to biological problems contribute to the extension of the capacities of MAS NMR.

The advantages of ssNMR as compared to its solution counterpart, in particular the theoretical absence of molecular weight limit, are more and more commonly used for studying large and highly symmetric protein assemblies such as viral capsids or membrane proteins [94, 110-112, 114, 247]. These systems possess a high degeneracy of the resonances and therefore provide high sensitivity and low signal overlap. Many biological systems however result from the interaction between several different partners and therefore possess many magnetically non equivalent chemical sites. A possible route to tackle this challenge is to reconstitute the system while labelling a single type of molecule (for example a protein in a non-labelled lipid environment). Because the labelled material is diluted, such studies strongly benefit from the possibility to enhance spectral sensitivity with 1 H detection. This has been demonstrated by recent investigations of intermolecular complexes in which the molecule of interest can represent as low as 1% of the total sample quantity [START_REF] Lamley | Intermolecular Interactions and Protein Dynamics by Solid-State NMR Spectroscopy[END_REF][248][249][250].

A step forward in understanding the molecular basis of protein function is to directly investigate their structure and dynamics in their native context such as cell membranes or directly whole cells. The complexity of these environments further increases the requirements in terms of sensitivity [251], in particular if the protein is expressed at a native concentration. In this case, 1 H detection is not sufficient to observe correlations in a reasonable amount of time and other sensitivity enhancement methods have to be used. Notably, Dynamic Nuclear Polarization (DNP) spectacularly increases NMR sensitivity by transferring the polarization of single-electron spins to the nuclei, enabling detection of proteins in highly diluted conditions [250,252]. However, the cryogenic temperature and the presence of paramagnetic species in the DNP samples significantly broaden the NMR lines. We therefore hope that adaptating high field and fast MAS technologies to DNP [253] together with the use of higher temperatures will enable high-resolution investigation of proteins in a next future. 56 65,25 W60 28,14 176,74 S61 7,93 111,87 57,11 67,18 170,84 F62 8,80 125,98 56,28 43,56 172,50 Y63 9,44 126,10 56,60 41,73 174,20 L64 8,80 124,63 54,18 44,45 176,65 L65 8,80 125,92 55,12 37,16 178,33 Y66 7,35 120,17 61,99 38,30 in which e NH is the time-independent set of coordinates representing the NH vector in the instantaneous peptide plane frame and D

(2)

ml are Wigner rotation matrix elements.

The correlation function therefore reads: e -i π 2 (k-k ) e -ik α(t)+ikα(0) e lβ(t)-ilβ(0) e im γ(t)-imγ(0) d

G l (t) =
(2) The GAF order parameter is: The GAF correlation function is not amenable to analytical Fourier transformation. The spectral density is therefore obtained by (i) fitting the correlation function for each order and for each β angles to a monoexponential function (ii) performing the Fourier transform of the approximate formulae.

k l ( π 2 )d (2) mk ( π 2 )d
S 2 = lim
Different variants of this approach are found in literature. In a combined NMR and MD investigation, Lienin et al. showed that backbone dynamics of ubiquitin is dominated by fluctuations around the e γ axis (Figure D.1) [175]. A simplest 1D-GAF model therefore assumes that the peptide plane only fluctuates around this axis. The parameters of this model reduces to σ γ and τ GAF . 3D-GAF was also used to describe collective motions of proteins. In this case, the orientation of the vector e NH is expressed in a fixed molecular frame.

Figure 1 . 1 :

 11 Figure 1.1: The mechanism of function of ATPases characterized by two orthogonal methods. (a) Model of F-ATPase (left, side view; right, cross-section at the level of the fluorophore) where a flurophore (FRET donor, green) is bound to the rotor part of the assembly and another fluorophore (FRET acceptor, red) is bound to the stator. (b) Fluorescence intensities of the donor (F D , green) and of the receptor (F A , red) monitored over time. The ratio (F D /F A , orange) shows a three steps process with a 1-3-2-1 sequence (the numbering corresponds to the positions depicted in panel A). (c) Rotational states of V-ATPase determined by cryo-electron microscopy. On the left are shown cross-sections from the map of rotational state 1. The scale bar is 25Å.The figures are reproduced with permissions from[START_REF] Diez | Proton-powered subunit rotation in single membrane-bound F 0 F 1 -ATP synthase[END_REF] and[START_REF] Zhao | Electron cryomicroscopy observation of rotational states in a eukaryotic V-ATPase[END_REF].

Figure 1 . 2 :

 12 Figure 1.2: Heuristic representation of relaxation processes where each individual nuclear spin is represented as a vector precessing in a cone either in the up state (upper part of the cone) or down state (lower part of the cone). Transverse relaxation:During an NMR experiment, the spins are synchronously precessing around the external magnetic field. Because of local fluctuations of the experienced magnetic field, each spin will have a stochastically modulated precession rate, leading to a global dephasing and a subsequent loss of transverse magnetization M x/y . Longitudinal relaxation: In a case where the system is out of equilibrium with randomly oriented spins, there is no net macroscopic magnetization. Reorientations of the local magnetic field induces transitions between the Zeeman energy states. After a certain amount of time, spins are polarized in the (energetically favored) sense of the external magnetic field, resulting in a macroscopic magnetization M z along the magnet axis. The polarization depends on the temperature and the energy gap between the Zeeman levels. For illustration purposes only, the bottom right panel represents a fully polarized system.

Figure 1 . 4 :

 14 Figure 1.4: Energy diagram for two coupled spins, I and S. The ket notations indicate the spin states of I and S, in this order.

Figure 1 . 5 :

 15 Figure 1.5: Site-specific measurement of 15 N R 1 relaxation rates. (a-f) 1 H-15 N correlation spectra acquired with variable z-storage delay(50, 210, 370, 530, 690, and 850 ms, respectively). Adapted with permission from reference[START_REF] Kay | Backbone dynamics of proteins as studied by nitrogen-15 inverse detected heteronuclear NMR spectroscopy: application to staphylococcal nuclease[END_REF].

Figure 1 . 6 :

 16 Figure 1.6: Observation of the change of dynamics of Calbindin upon (Ca 2+ ) 2 binding by NMR. (a) Structures of the apo and (Ca 2+ ) 2 -bond forms (PDB codes: 2BCA and 1CLB, respectively). (b)The order parameters obtained from the model-free analysis of 15 N relaxation rates are plotted versus the sequence for the apo (white) and bond (black) forms. High increase in order in site II where the ion first binds was observed compared to more modest changes in site I. The position of the α-helix as well as the binding sites are shown above the panel. Adapted with permission from reference[START_REF] Koerdel | Backbone dynamics of calcium-loaded calbindin D9k studied by twodimensional proton-detected nitrogen-15 NMR spectroscopy[END_REF]

Figure 1 . 7 :

 17 Figure 1.7: Schematic representation of the frequencies sampled by 15 N R 1 , R 2 and 1 H-15 N cross relaxation rates (σ = γ S γ I R 1 η IS )

Figure 1 . 8 :

 18 Figure 1.8: Conformational exchange in NMR. (a-e) Exchange lineshapes for skewed populations (p 1 = 0.75, p 2 = 0.25) calculated with k ex = 0 s -1 (a), 50 s -1 (b), 100 s -1 (c), 500 s -1 (d) and 5000 s -1 (e). Basic pulse sequences for the acquisition of CPMG (f), R 1ρ (g) and CEST (h) experiments. Starting magnetization is on z. (i) and (j), respectively CPMG and R 1ρ dispersion curves calculated using equation 1.13 with p 1 = 0.9, p 2 = 0.1 and, respectively k ex = 1000 s -1 and 10000 s -1 . (k) CEST profile simulated with equation 1.17 with p 1 = 0.95, p 2 = 0.05 and k ex = 50 s -1 . The spin lock is set at a frequency of 10 Hz during 0.5 s. In all simulations, R • 2 is set to 10 Hz and ∆Ω/2π to 100 Hz. Green and orange dotted lines in pannels (a-e) and (k) correspond, respectively, to the main and second resonances. Narrow and broad black rectangles indicateπ/2 and π pulses, respectively, and white rectangles continuous irradiations.

Figure 1 . 9 :

 19 Figure1.9: Series of rotations of coordinate systems used to describe relaxation under MAS using the example of 15 N 1 H dipolar coupling.

Figure 1 .

 1 Figure 1.10: 15 N R 1 relaxation rate in function of the 15 N-1 H bond orientation in the rotor frame. The rates are calculated with a model in which the bond diffuses in a cone of semi-angle 11,2• with a diffusion time of 66 ns. Reprinted with permission from reference[START_REF] Giraud | Quantitative analysis of backbone dynamics in a crystalline protein from nitrogen-15 spin-lattice relaxation[END_REF]

Figure 1 . 11 :

 111 Figure 1.11: Site-specific 15 N longitudinal relaxation rates (R 1 , a) and longitudinal relaxation rates in the rotating frame (R 1ρ , b) for microcrystalline Cu + , Zn-SOD. The secondary structure elements are depicted in top of the plot. Dynamical parameters obtained from 1D-GAF model analysis; in (c) timescales of internal motion and in (d) order parameters are depicted on the dimeric structure of SOD (PDB code: 1SOS).Readapted with permission from[START_REF] Knight | Structure and backbone dynamics of a microcrystalline metalloprotein by solid-state NMR[END_REF].

Figure 1 .

 1 Figure 1.12: 15 N R 1ρ relaxation dispersion curves measured on perdeuterated ubiquitin. (a) Dispersion profiles at low spin-lock fields and 39.5 kHz MAS fitted to a two-sites fast exchange model using equation 1.13. (b)15 N R 1ρ relaxation is enhanced at the rotary resonance condition: 15 N R 1ρ rates (red) obtained at 20 kHz MAS with simulated curves for an exchange rate of 2900 s -1 and a minor state relative population of 0.093. Reprinted with permission from reference[START_REF] Ma | Probing Transient Conformational States of Proteins by Solid-State R11 Relaxation-Dispersion NMR Spectroscopy[END_REF]

Figure 1 .

 1 Figure 1.13: 1 H-detected 1 H-15 N dipole recoupling sequences used in this PhD work. (a) 1 H-detected REDOR sequence. (b) Time evolution of the REDOR signal calculated for a 1 H-15 N spin pair with equation 1.26. (c) 1 H-detected CP-VC sequence.(d,e) Time evolution of the CP-VC signal calculated for a 1 H-15 N spin pair with the equations given in[START_REF] Park | Resolution and measurement of heteronuclear dipolar couplings of a noncrystalline protein immobilized in a biological supramolecular assembly by proton-detected MAS solid-state NMR spectroscopy[END_REF] (d) and its Fourier transform (e). For sake of clarity, we omit here the 13 C channel which is used for decoupling during indirect acquisition.

Figure 1 . 14 :

 114 Figure 1.14: Site-specific measurement of motion-averaged anisotropic informations. (a) Model of mature HIV-1 capsid (left) with the 3D structure of the monomeric unit (center). On the right, the representation of a 15 N CSA tensor. (b) Experimental (grey) and simulated (red) recoupled 15 N CSA line shapes for the flexible G89 and the rigid E98 residues. (c) 2D 2 H-13 C plane from a 3D 2 H-13 C-13 C correlation experiment acquired on [ 2 H, 13 C, 15 N]-GB1. Selected 2 H experimental (black) and simulated (red) line shapes are depicted together with the reiduals (blue). (d) C Q values for Gln and Glu residues are shown onto GB1 structure. Red corresponds to higher flexibility and blue to more rigidity. Reproduced with permissions from [101] and [102]

Figure 1 . 15 :

 115 Figure 1.15: Summary of the timescale of molecular dynamics in proteins and the associated MAS NMR probes treated in this section. NMR observables are classified into four categories: averaged anisotropic interactions (purple), longitudinal relaxation based rates (red), transverse relaxation rates (orange) and exchange contributions to relaxation rates (blue).

Figure 2 . 2 :

 22 Figure 2.2: 15 N-1 H CP-HSQC spectra measured at a 1 H frequency of 1 GHz and MAS rate of 60 kHz on five perdeuterated [100%-1 H N ]-labeled protein samples. (a) Microcrystalline α-spectrin SH3 domain, (b) microcrystalline β-2 microglobulin (b2m), (c) sedimented viral nucleocapsid of Acinetobacter phage 205 (AP205), (d) conductance domain from influenza A M2, (e) outer membrane protein G (OmpG) from Gramnegative bacteria. Reprinted from [118].

Figure 2 . 3 :

 23 Figure 2.3: Pulse schemes for the six 1 H-detected 13 C-15 N-1 H 3D correlation experiments, with an illustration of the coherence transfers and spins involved: (A) interresidue (H)CONH experiment (CO i-1 N i H i ), (B) intraresidue (H)CO(CA)NH experiment (CO i N i H i ), (C) intraresidue (H)CANH experiment (CA i N i H i ), (D) interresidue (H)(CO)CA(CO)NH experiment (CA i-1 N i H i ), (E) intraresidue (H)(CA)CB(CA)NH experiment (CB i N i H i ), and (F) interresidue (H)(CA)CB(CACO)NH experiment (CB i-1 N i H i ).Orange boxes indicate crosspolarization, and gray boxes indicate heteronuclear decoupling. The MISSISSIPPI sequence for solvent signal suppression is indicated by the striped box. Spin-echoes involved in the transfer between C and Cα are highlighted in blue and red, when the coherence is present on the C and Cα, respectively, and spin-echoes involved in transfer between Cα and Cβ are highlighted in green. Specific CP to or from13 C may require a frequency shift, and in this case, the phases of RF during the two13 C CP periods are aligned respectively at the end and at the beginning of the contact time. In the diagrams, filled circles denote spins for which the evolution of chemical shifts is performed. Arrows represent magnetization transfers between nuclei. Reproduced from reference[128]

Figure 2 . 4 :

 24 Figure 2.4: Illustration of the 1 H N -detected resonance assignment procedure of D76N β2m crystals (a) and fibrils (b) for residues S33-V37. Strip plot of inter-and intra-residue Cα, Cβ, and CO resonances for each N-H pairs (the 15 N resonance frequency in parentheses) . Each panel is 1 ppm wide in the 1 H dimension.

Figure 2 .

 2 Figure 2.5 shows two variants of a new 3D pulse sequence, (H)N(CO)(CA)NH and (H)N(CA)(CO)NH, that correlate the backbone amide 15 N-1 H pair of a residue i with the amide 15 N of residues i + 1 and i -1, respectively. Five coherence transfers are used: one 13 C-13 C scalar based transfer, and 4 cross polarization

O

Figure 2 . 5 :

 25 Figure 2.5: Pulse sequence schemes corresponding to the two variants of 3D 1 H-15 N-15 N correlation spectra. Each is used to correlate an amide 1 H-15 N pair with either the (a) preceding or (b) subsequent amide 15 N. Heteronuclear transfers via cross polarization (CP) are shown in yellow, while the homonuclear13 Cα-13 C transfer is highlighted in green. We set the carrier frequency of the13 C channel to about 100 ppm, and the offset of13 C on resonance with either 13 C or13 Cα for CP. Bell shapes represent selective inversion pulses, here 350 ms Q3 pulses[149] with the frequency offset applied at the center of the range for either 13 C or13 Cα. Excitation and inversion pulses are shown as narrow and wide black rectangles, respectively. Grey rectangles represent heteronuclear decoupling. Water suppression [150] is shown as purple and grey rectangles on the 1 H channel, and was typically applied for 100 ms. Delays δ 1 and δ 2 were set to 3.5 and 3.0 ms, respectively.

Figure 2 . 6 :

 26 Figure 2.6: Sequential linking using 15 N-15 N-1 H correlations in microcrystalline ubiquitin. Sequential connections for residues D39 to E51 are shown by blue arrows in the 15 N-15 N projection of the (H)N(CA)(CO)NH spectrum (a). In (b) the double linking in the pair of 15 N-15 N-1 H spectra is illustrated, as compared with single linking in a pair of 13 C-15 N-1 H spectra. In (c) double linking is shown with a superposition of (H)N(CO)(CA)NH (red) and (H)N(CA)(CO)NH (green) spectra. Spectra are recorded at ω 0H /2π = 800 MHz.

Figure 2 .Figure 2 . 7 :

 227 Figure 2.7 shows the RF irradiation scheme of two experiments. The first one (Figure2.7a, H(CO)NH) correlates the chemical shift of the 1 H atom involved in the H-bond, to the shifts of amide 15 N and 1 H nuclei of the residue subsequent ((i + 1) th ) to the one containing the H-bonded13 C atom (i th ). The second experiment (Figure2.7d, (H)CO(H)NH) shows correlations of the shift of H-bonded13 C atom with 15 N and 1 H shifts of the H-bonded amide group. In H(CO)NH, we chose to perform indirect detection on 15 N i+1 instead of 13 C i because of the better resolution of the 15 N dimension.

Figure 2 . 8 :

 28 Figure 2.8: SIMPSON simulations of the transfer efficiency during 1 H-13 C transfer. (a) Illustration of the spin system used for the simulations: four 1 H atoms with distances from a 13 C atom of 2.0 Å, 2.7 Å, 2.9 Å and 4.0 Å (not shown in the figure), characteristic for anti-parallel β-sheet secondary structure. Only dipolar couplings are taken into account. (b) Relative intensity of transferred magnetization in function of the mixing time. The colors correspond to panel (a) and purple is used for the transfer between the 1 H and 13 C spaced of 4 Å.

Figure 2 . 9 :Figure 2 .

 292 Figure 2.9: (a) Strip from H(CO)NH experiment (above) and 15 N-1 H plane from the (H)CO(H)NH experiment at the 15 N frequency of residue T51 (red). Experiments were run on [100% 1 H N , 2 H, 15 N, 13 C]-labelled microcrystalline GB1 at 60 kHz MAS and ω 0,H = 700 MHz. For an easier interpretation, peaks are superimposed onto the (H)NH 2D spectrum (in purple). (b) Strip from H(CO)NH experiment at the frequency of residue G9 showing side-chain-to-backbone contact. (c) Structure of the residues 8 and 9 from[START_REF] Andreas | Structure of fully protonated proteins by proton-detected magic-angle spinning NMR[END_REF]

Figure 2 . 11 :

 211 Figure 2.11: Strips from the proton-detected H(CO)NH 3D experiment recorded for [100%-1 H N , 2 H, 13 C, 15 N]-labelled microcrystals (a) and amyloid fibrils (b) of D76N β-2 microglobulin on a 800 MHz 1 H Larmor frequency spectrometer at 60 kHz MAS.The diagonal peaks are indicated in each strip by black circles, the "trivial" contacts by blue circles, and H-bond correlations by green circles. The width of the strips is set to 2 ppm in the 15 N dimension.

Figure 3 . 1 :

 31 Figure 3.1: MAS NMR structures of GB1 (PDB ID code 5JXV; Top) and AP205CPCP (PDB ID code 5JZR; Bottom). Ribbon diagram of the 10 lowest-energy conformers for GB1 (a) and dimeric AP205CPCP (e), with monomers colored in cyan and tan. The measured long-range proton-proton contacts are depicted as dark blue lines onto the lowest-energy conformers of the two proteins: contacts between amide protons (b and f), contacts among amide and ILV-methyl-labeled protons (c and g), and all proton-proton contacts (d and h). Reproduced from reference [94].

Figure 3 . 2 : 3 .

 323 Figure 3.2: (a) Pulse sequence for 1 H-detected 1 H-13 C CP-VC experiment. Narrow and broad black rectangles indicate π/2 and π pulses, respectively. Orange boxes indicate cross-polarization, and gray boxes indicate heteronuclear decoupling. The MISSISSIPPI sequence [150] is indicated by the striped box. φ 1 = 1 1 1 1 3 3 3 3, φ 2 = 1 1 3 3, φ 3 = 1 3, φ rec = 1 3 3 1 3 1 1 3. The length of the second CP is varied during CP-VC experiments. (b) Experimental 1 H-13 C CP matching conditions at 110 kHz MAS (with courtesy of Dr. Stanek). The ZQ, DQ and Second Order (SO) CP conditions are highlighted by arrows on the right side of the plot.

Figure 3 . 3 :

 33 Figure 3.3: Data processing procedure of the CP-VC curves. Typical experimental build-up curve and its Fourier transform (resp. a and b), after baseline correction (resp. c and d) and zero-filling (resp. d and e). The grey box indicates the frequency range which is not taken into account in the fitting procedure.

Figure 3 . 4 :

 34 Figure 3.4: 1 H-detected CP-VC experiments on fully protonated microcrystalline GB1 at 110 kHz MAS. (a) 1 H-13 C CP-HSQC of GB1. (b) Dipolar lineshapes extracted from the 3D-CPVC experiments for different sites. The corresponding residue types and numbers as well as the calculated dipolar coupling parameters are given above each plot.

Figure 3 . 5 :

 35 Figure 3.5: Dynamics parameters as derived from the results of CP-VC experiments for α and β sites in GB1. (a) Order parameters calculated from13 Cα-1 Hα dipolar coupling constants. For comparison, the order parameters obtained by the analysis of15 N R 1 and R 1ρ relaxation rates are depicted in the same plot (blue line). (b) Asymmetry of the13 Cα-1 Hα dipolar coupling. (c) Order parameters calculated from13 Cβ-1 Hβ dipolar coupling constants.

Figure 3 . 6 :

 36 Figure 3.6: Comparison between the 1 H-13 C dipolar coupling parameters obtained with CP-VC and the 2 H quadrupolar constants reported in[174]. (a) Correlation plot. Residue N35 is out of trend (colored in red) and was not taken into account in the linear regression. Its behavior can be explained by the changes observed between the structures obtained from reference[START_REF] Andreas | Structure of fully protonated proteins by proton-detected magic-angle spinning NMR[END_REF] with the same preparation used in this work (b) and from reference [176] (c).

Figure 3 . 7 :

 37 Figure 3.7: Methyl dynamics in GB1. (a) 1 H-13 C correlation spectra acquired on a protonated (blue lines) and ILV-methyl labelled sample (red lines). (b) REDOR curves obtained for the CH group of residues I6 and V39 of ILV-GB1. (c) CPVC profiles for the CH 3 group of the same residues in protonated GB1. (d) Order parameters for the 7 labelled methyls determined by REDOR (red) and CPVC (blue).

)Figure 3 . 8 :

 38 Figure 3.8: Evolution of 13 C R 1 relaxation rates with MAS frequency. (a-b) Sitespecific R 1 relaxation rates of Cα (a), Cβ (b) obtained at 80 (blue), 90 (purple), 100 (red) and 110 (black) kHz MAS. (c-e) Mean values of R 1 relaxation rates of Cα (a), Cα (b) and the other aliphatic carbons (c) plotted versus MAS rate.

Figure 3 . 9 :

 39 Figure 3.9: Correlations between site-specific13 Cα R 1 relaxation rates and the amplitude of motion (1 -S 2 ). Best linear fits are shown in red. Corresponding MAS rates and correlation coefficient R 2 are shown given the plots.

Figure 3 .

 3 Figure 3.10: 1 H N and 15 N R 1ρ relaxation rates measured on a sample of [100% 1 H N , 2 H, 13 C, 15 N]-labelled GB1. (a) 1 H N R 1ρ relaxation dispersion measured at 100 kHz MAS on a 1 GHz 1 H Larmor frequency spectrometer. The grey zones correspond to the HORROR and rotary conditions. (b) 15 N R 1ρ relaxation rates plotted versus the primary sequence. Data were measured on the same spectrometer at 60 kHz MAS with 10 kHz spin-lock power (ω 1 /2π). (c) 1 H R 1ρ relaxation rates measured with ω 1 /2π = 80 kHz. (d) Difference between 1 H R 1ρ relaxation rates measured with ω 1 /2π = 55 kHz and ω 1 /2π = 80 kHz. The secondary structure of GB1 is depicted above the plots. Arrows, the white rectangle and lines represent, respectively, βstrands, the α-helix and loop regions.

Figure 4 . 1 :

 41 Figure 4.1: Schematic (non exhaustive) representation of competing folding (green) and amyloid fibril formation (red) pathways for a globular protein. During conversion from the unfolded (a) to the native folded conformation (c), the protein populates intermediate states (c) which have different degrees of folding.At room temperature, some of these partially folded states, generally structurally close to the native fold, are populated. These conformations may have higher amyloid formation propensity than the protected native state and be able to form oligomers (f) which will evolve into protofibrils (g) and then to mature fibrils (h).

Figure 4 . 2 :

 42 Figure 4.2: (a) Superposition of the structures of wt β2m (2D4F, magenta) and of the D76N (4FXL, light blue) variant. The residues in position 25, 32, 76 and 80 are shown as sticks. (b,c) Zoom-in of the EF loop of the structures shown in A. H-bonds are displayed as dash lines and electron density is clipped around the EF loop.

Figure 4 . 3 :

 43 Figure 4.3: Assignment of microcrystalline [100%-1 H N , 2 H, 13 C, 15 N]-wt and D76N β2m. The color code red for wt and blue for D76N β2m is used for all the figures presented here. (a,b) Assigned 1 H-15 N CP-HSQC spectra of respectively wt and D76N β2m. (c-g) Comparison of the 1 H, 13 C and 15 N chemical shifts obtained for the two proteins.

Figure 4 . 4 :

 44 Figure 4.4: (a,b) Pulse sequences used for 15 N R 1 (a) and 15 N R 1ρ (b) relaxation rate measurement. Narrow and wide rectangles indicate respectively 90 • and 180 • pulses. Cross-Polarization (CP) transfers between 1 H and 15 N (orange boxes)were performed with a linearly ramped pulse from 90% to 100% of a maximum frequency 100 kHz on 1 H and a rectangular pulse on 15 N of 40 kHz. CP lengths were set to respectively 900 and 600 µs for 1 H to 15 N and 15 N to 1 H transfers. Water suppression was achieved by storing15 N magnetization along the z axis and applying the MISSISSIPI pulse sequence with a 19 kHz irradiation during 300 ms. Grey rectangles indicate heteronuclear decoupling using WALTZ-16 with 10 kHz RF power. (c,d) Examples of relaxation decay curves for wt (red) and D76N (blue) β2m: (c)15 N R 1 (d)15 N R 1ρ at 20 kHz spin-lock field.

Figure 4 . 5 :

 45 Figure 4.5: Detection of fast local motions by 15 N R 1 and R 1ρ relaxation experiments.The color code red for wt and blue for D76N β2m is used for all the figures presented here. Site-specific 15 N R 1 (a) and R 1ρ (b) relaxation rates of the two proteins are used to calculate the order parameters S 2 (c) and the correlation time (d) with the modelfree approach. The results of a 50 ns MD trajectory are plotted together with the experimental results in (c) (red and blue bundles). β2m secondary structures are plotted on top of the panels. The analysis of the 15 N relaxation was performed with a home-made Matlab program.

Figure 4 . 6 :

 46 Figure 4.6: Conformational exchange of β2m probed by relaxation dispersion experiments. Selected examples of 15 N R 1ρ relaxation dispersion curves for wt (red) and D76N (blue) β2m (a). The purple, orange and blue rectangles show respectively residues located at the end of the A-strand, in the CD-loop and EF-loop. The difference between 15 N R 1ρ rates at 3 kHz and 20 kHz spin-lock field are depicted versus the sequence in (b).

Figure 4 .Figure 4 . 7 :

 447 Figure 4.7: Conformational ensembles for wt and D76N β2m. (a,b) Free energies surfaces for the wt (a) and D76N (b) ensembles. The surfaces are shown as a function of the side chains rotameric state, AB, and the antiparallel β-structure content (antiβ) with a set of representative conformations for the major and minor substates. (c,d) β-structure content per residue (within each ensemble) for the wt and D76N variants, respectively. The major state is shown in black while the minor in cyan. (e) Perturbation of the energy network upon D76N mutation. The network shows the interactions between couples of amino acids that are strengthened (red) or weakened (blue) upon mutation as calculated over the two ensembles of structures.

Figure 4 . 8 :

 48 Figure 4.8: Proposed mechanism for the first steps of D76N aggregation. Under native conditions and in the crystal lattice both wt and D76N are in equilibrium between a low-and a high-energy conformations. While wt high-energy conformation has a contained aggregation propensity, the D76N excited state is less structured, highly dynamic and it exposes highly aggregation prone regions shown in orange. Thus modest energies such as shear forces present in the circulation system are sufficient to further unfold D76N β2m termini (model) destabilizing completely β2m fold and leading to amyloid formation.

Figure 4 .Figure 4 . 9 :

 449 Figure 4.9: (a) ThT fluorescence profile for D76N β2m fibril formation at 37 • C under stirring conditions (phosphate buffer saline pH 7). (b) Cryo-EM picture of D76N b2m reprinted from reference [159].

Figure 4 . 10 :

 410 Figure 4.10: Assigned 1 H-15 N 2D projection of (H)CANH correlation spectrum (a, dark blue), and 13 C-15 N 2D projections of (H)CONH (b, purple), (H)(CA)CB(CA)NH (c, orange) and (H)CANH (c, green) correlation spectra of [ 2 H, 13 C, 15 N] D76N β2m fibrils with 100% reprotonation at the exchangeable sites.

Figure 4 . 11 :

 411 Figure 4.11: Comparison of the backbone chemical shifts of native and fibrillar states of D76N β2m. (a) Overlay of the 1 H-15 N correlation spectra of D76N β2m in solution (red), in crystals (black) and in amyloid fibrils (light blue). Resonance assignment for two selected stretches of residues is depicted on the spectra for comparison. (bd) Secondary chemical shifts (∆δ) [208] of Cα (b), Cβ (c) and C (d) for native (in solution, red bars) and fibrillar (light blue) D76N β2m. Black lines represent the -1 (b,d) or +1 (c) ppm threshold beyond which the secondary chemical shifts are indicative of β-strand elements.

Figure 4 .

 4 Figure 4.12: (a) Secondary structure prediction of D76N β2m fibrils by TALOS-N (black arrows correspond to β-strands, straight lines to loops and dashed lines to an absence of prediction). The β-strands of the native fold are depicted above the primary sequence. The secondary structures of the excited state obtained with the RAM simulations are also shown for comparison. (b) Backbone dihedral angles phi and psi predicted by TALOS-N. Yellow boxes highlight residues for which TALOS-N predictions are ambiguous. Error bars are based on the 10 best TALOS-N database matches. (c) Prediction of D76N β2m amyloid fibril formation propensity by Zyggregator based on the sequence [216]. The residues for which Zseq score is greater than 1 are predicted to be aggregation prone.

Figure 4 .

 4 Figure 4.13: 2D projections of the (a) (H)NHH and (b) (H)N(H)(H)NH spectra. The assignment of cross peaks is shown on the spectra. All assigned correlations correspond to contacts between residues separated of less than 3 residues in the sequence.

Figure 4 .

 4 Figure 4.14 shows a comparison of the secondary structure elements predicted for D76N and wt [206] b2m fibrils. The secondary structure of the two proteins share some similar features. For example, native C -D and E-F loops are part of the β-structure in both forms. In contrast, in their study, Su et al. highlight the role of native BC-DE-and FG-loops in the formation of the amyloid core [206]. For the D76N variant, we do not observe secondary structure elements in those parts of the fibrils either because of the lack of assigned resonances (DE-and FGloops) or because the backbone chemical shifts are more indicative of random-coil structure (BC-loop).

Figure 4 . 14 :

 414 Figure 4.14: Comparison of the secondary structure elements of wt[206] and D76N fibrils. The secondary structure of native wt and D76N β2m is determined from their crystal structure (PDB code: 2d4f and 4fxl, respectively). The sequence is shown on top of the figure with the proteolysis sites of D76N β2m fibrils in red, and residue N76 in blue. The rigid core of the fibrils, protected from proteolysis (residues S20 to C80), is highlighted with a red rectangle.

Figure 5 . 1 :

 51 Figure 5.1: (a) 15 N-1 H correlation spectrum of [U-1 H N , 2 H, 13 C, 15 N]-labeled KpOmpA reconstituted in E. coli polar extract liposomes, recorded at 1 GHz spectrometer and 60 kHz MAS frequency and 15 • C. Assigned residues are annotated. (b) Two regions of (A) containing glycine resonances. Glycine residues at the strandto-loop transition (i.e., G26 and G53) are weaker compared to G22 and G58 placed in the same strand and G71 in β-strand 3. (c) Topological representation of assigned residues in N-KpOmpA illustrating residue-specific mobility within distinct protein segments and Arg and Lys potential cleavage sites for trypsin. Residues within βsheet and random coil regions are represented by squares and circles, respectively.Color code: assigned residues are highlighted in green, residues that exhibit intermediate time scale motions are colored in orange, Arg and Lys residues are colored in red, and the unique cleavage site (i.e., R134) accessible to trypsin is highlighted and annotated in red while the surrounding residues within loop 3 are in black.

Figure 5 . 2 :

 52 Figure 5.2: Residue-specific characterization of dynamics in [U-1 H N , 2 H, 13 C, 15 N]labeled KpOmpA reconstituted in E. coli polar extract liposomes with a lipid to protein ratio of 0.5 (w/w) and at 25 • C. Representation of (a) a full rotation of Kp-OmpA around the cylinder axis due to axial diffusion along the bilayer normal and (c) of rocking motion with the amplitudes σ = 6 • and σ ⊥ = 10 • . (b,d) 15 N-1 H bond order parameters <S> as measured by REDOR (red dots) and calculated for the full rotation (b, blue bars) and the rocking motion (d, blue bars). (e) Contour plot of the reduced chi square function (χ 2 red ) calculated as a function of the amplitudes of the rocking motion. The dots correspond to different pairs of amplitudes used to calculate 15 N R 1 and R 1ρ relaxation rates. Above them are depicted the correlation time (τ GAF ) obtained from the fits. The value of χ 2 red corresponding to the thicker contour line is 2. (f) and (g), respectively, 15 N R 1 and R 1ρ relaxation rates determined experimentally (red dots) and calculated with σ = 6 • , σ ⊥ = 10 • , and τ GAF = 216 ns (blue bars). Residues for which the agreement is poor (G67, A98, T112, W151) are subject to local motion.

Figure 5 . 3 :

 53 Figure 5.3: Dynamics analysis with 3D collective GAF model. REDOR order parameters were first fit to obtain the motion amplitudes.15 N R 1 and R 1ρ relaxation rates were fit afterward by fixing these angular fluctuations in order to extract the fluctuation timescale.

Figure 5 . 4 :

 54 Figure 5.4: Results of the fit of 15 N R 1 (a,b) and R 1ρ relaxation rates (c,d) for different amplitudes of motion. The values of amplitudes (σ and σ ⊥ ) and the GAF timescale (τ GAF ) obtained by fitting are shown on the upper left corner of each plot. The calculated values are represented as blue bars and the experimental values by red dots.

Figure 6 . 1 :

 61 Figure 6.1: AlkL structure determined in lipid bilayers. (a) NMR structure represented by the 20 lowest energy conformations. (b) Non polar and non charged residues are colored in purple on the NMR structure. Sidechains of Tyr and Trp residues on the surface of the lipid bilayer are depicted. The surface of the molecule is shown with transparency. The surface of the transmembrane region is highly hydrophobic. For the extracellular part, the hydrophobic residues are generally pointing inward so that the solvent-exposed surface remains hydrophilic. (c) The hydrophobic channel of AlkL found with V3 software [242] is shown with green mesh structure. The sidechains of the residues closing the channel are shown as well as the sidechains of P76, P100 and A101. (d) Zoom-in view of the bottom of the channel. All the representations are "side-views".

Figure 6 .

 6 Figure 6.1 a shows the structure of AlkL recently obtained by 1 H-detected NMR experiments performed on [ 1 H N , 2 H, 13 C,15 N]-and [ 1 H, 13 C,15 N]-labelled samples of AlkL in bilayers with 60 and 111 kHz MAS, respectively. As predicted by the homology model[239], the structure features a transmembrane barrel composed of eight β-strands with hydrophobic membrane-exposed and inner surface. The extra-cellular part of the protein is an extension of the betabarrel with hydrophobic residues pointing inside, forming a hydrophobic tunnel (Figure6.1b). The inner volume of the pore is shown in Figure6.1c. A cluster of residues blocks the tunnel close to the extracellular-membrane interface, strengthening the lateral exit hypothesis. Among them, Arginine-168 and Phenylalanine-72 form the previously described "hydrophobic gate"[240]. The three proline residues surrounding the putative exit hole in OmpW correspond to residues Proline-76 and -100, and Alanine-101 in AlkL. The NMR structure does not contain evidence of the formation of such a pore. Instead, putative exits are found on both sides of the nearby loop connecting strands β 1 EC and β 2 TM (for

Figure 6 . 2 :

 62 Figure 6.2: 1 H-detected (H)CONH spectrum of [ 1 H N , 2 H, 13 C, 15 N]-labeled AlkL recorded on a 800 MHz 1 H Larmor frequency spectrometer at 60 kHz MAS: (a) 2D 1 H-15 N projection, (b) 3D representation. 30 peaks are isolated in the 2D spectrum compared to 80 in the 3D version.

Figure 6 . 3 : 6 . 3 . 2

 63632 Figure 6.3: AlkL dynamics probed by site-specific 15 N relaxation. (a,b) 15 N R 1 (a) and R 1ρ (b) relaxation rates plotted versus the primary sequence. β-strands are shown above the plots by black arrows. Parts of the strands in the trans-membrane region are highlighted with purple rectangles and extracellular parts in orange. Purple and orange dashed lines indicate the mean relaxation rate values for those two regions, respectively. (c,d) 15 N R 1 (c) and R 1ρ (d) relaxation rates depicted onto the MAS NMR structure represented in "front view". Larger balls and more red indicate higher relaxation rates. Residues for which assignment is lacking are colored in black. Residues for which peaks are two weak in the relaxation experiments to extract reliable parameters are colored in red. For both categories, high dynamics is expected. All experiments were recorded on [ 1 H N , 2 H, 13 C, 15 N]-labeled AlkL at 1 GHz 1 H Larmor frequency and 60 kHz MAS

Figure 6 . 4 :

 64 Figure 6.4: Investigating conformational exchange in AlkL. (a,b) 15 N R 1ρ relaxation dispersion for residues Alanine-77 and Arginine-78. Straight dashed line is the mean 15 N R 1ρ value for all spin-lock field strengths except 20 kHz and represent the relaxation value if there is no exchange contribution. (c,d) CEST profiles obtained for residues Alanine-77 and Arginine-78, respectively. Both types of experiment suggest the absence of conformational exchange on a µs-s timescale. All experiments were recorded on [ 1 H N , 2 H, 13 C, 15 N]-labeled AlkL at 1 GHz 1 H Larmor frequency and 60 kHz MAS

Figure 6 . 5 :

 65 Figure 6.5: Chemical shift perturbation of AlkL upon addition of carvone. (a) CSP values calculated as (∆δ 1 H N ) 2 + (∆δ 15 N/10) 2 + (∆δ 13 C'/2.5) 2 plotted versus the primary sequence. Secondary structure elements are shown on the top of the pannel. (b) CSP mapped onto the MAS NMR structure of apo-AlkL (left: "front view", right: "side view"). The values highly correlate with the position of the hydrophobic tunnel (depicted with a green mesh). Sidechains of residues with CSP value higher than 0.2 ppm are depicted. They orient toward the exterior of the barrel, allowing inner space for the passage of carvone. All experiments were recorded on [ 1 H N , 2 H, 13 C, 15 N]-labeled AlkL at 1 GHz 1 H Larmor frequency and 60 kHz MAS

Figure 6 . 6 :

 66 Figure 6.6: AlkL dynamics enhanced upon addition of carvone. (a) 15 N R 1ρ relaxation rates in presence (blue) or in absence (red) of carvone plotted against the primary sequence. (b) Difference in 15 N R 1ρ relaxation rates depicted as colored worms on the structure of apo-AlkL (left: "front view", right: "side view"). More red represent larger rates in presence of carvone. Sidechains of residues with a difference in 15 N R 1ρ higher than 5 s -1 are shown. (c) Cross-eye stereo view of the extracellular part of AlkL. All experiments were recorded on [ 1 H N , 2 H, 13 C, 15 N]-labeled AlkL at 1 GHz 1 H Larmor frequency and 60 kHz MAS

  the Wigner matrix elements detailed in [255], the correlation function becomes:G l (t) = 2 k,k ,m,m =-2

2 )

 2 are the reduced Wigner matrix elements. For gaussian axial fluctuations, the correlation functions in brackets can be expressed as:e im γ(t)-imγ(0) = exp -1 2 σ 2 γ m 2 + m 2 -2mm e (-Dt/σγ ) 2

F72 5 Figure E. 2 :Figure E. 3 :

 523 Figure E.2: 15 N R 1ρ relaxation dispersion curves for AlkL acquired on a 1 GHz spectrometer. Error on the relaxation rates was evaluated from 1000 Monte-Carlo simulations.

  15

(α,β,γ) Powder average (-ω r t,θ m ,0) Magic Angle spinning

  2 .

	Ω(t)
	Model-dependent
	H
	N
	Principal axis system	Molecular/crystal	Rotor frame	Laboratory frame
	(PAS)	frame

Table 1 . 2 :

 12 Published quantitative analysis of backbone dynamics with spin relaxation. *GAF = Gaussian Axial Fluctuations; SMF = Simple Model Free; EMF = Extended Model Free; C = diffusion in a Cone. EAS stands for explicit averaged sum:

	Protein (form)	Proton content	MAS (kHz) Method of analysis*	First author (year)	Ref.
	Crh (microcrystalline)		10	C-EAS	Giraud (2005)	[45]
			10	3D-GAF-EAS	Lewandowski (2010) [64]
	SH3 (microcrystalline)	10% H N	20	EMF, C	Chevelkov (2009)	[58]
	Ubiquitin (microcrystalline)	30% H N	45	EMF	Schanda (2010)	[59]
		50% H N	39.4	EMF	Haller (2013)	[50]
	GB1 (microcrystalline)	100% H	60	C-EAS, 1D-GAF-EAS	Lewandowski (2011) [56]
		100% H	60	EMF	Lamley (2015)	[65]
	(in complex with IgG)	100% H	45/52/60 3D-GAF	Lamley (2015)	[62]
	SOD (microcrystalline)	100% H	60	C-EAS, 1D-GAF-EAS, SMF Knight (2012)	[47]
	ASR (membrane embedded)	100% H	8/12/50	SMF	Good (2014)	[66]
		100%H	20/60	EMF, 3D-GAF	Good (2017)	[67]
	HET-s (fibrils)	100% H N /25%H (iFD)	40/60	EMF	Smith (2016)	[63]
	OmpA (membrane embedded) 100% H N	60	3D-GAF-EAS	Saurel (2017)	[68]
	β2m (microcrystalline)	100% H N	60	SMF	Le Marchand (2018) [69]

Table 1 .

 1

	1 H-13 C dipolar coupling	δ D /2π	22.5 kHz
	1 H-15 N dipolar coupling	δ D /2π	11 kHz
	2 H quadrupolar coupling	C Q	200 kHz
	15 N CSA	δ σ	160 ppm †

3: List of important anisotropic interactions in a labelled protein with corresponding typical values for the anisotropic component of the interaction tensor.

  kHz Labelling MAS (kHz) B 0 (MHz) 1 H T 2 ' (ms)1 H LW (Hz) 15 N ( 13 C) T 2 ' (ms) 15 N ( 13 C) LW (Hz) Source

			(a)				(b)				(c)
	SH3		10% H N	60		1000	95 ± 3.1	39 ± 10		146.9 ± 6.7	[123]
			10% H N	24		1000	62.6 ± 9.5			93.0 ± 7.5
			100% H N 60		1000	19 ± 0.7	49 ± 17		31.1 ± 1.9
			100% H N 40		1000	12.7 ± 0.3	61 ± 23		15.8 ± 0.4
			100% H N 24		1000	3.4 ± 0.1	117 ± 55	5.7 ± 0.4
			100% H	60		1000	1.5 ± 0.1	253 ± 123	5.6 ± 0.3
			10% H N	24		400		60.6		19.0 ± 3.3	75.8	11.3 ± 5.4	[124]
			60% H N	24		400		29.8		32.5 ± 10.3	23.6	25.1 ± 13.5
			100% H N 24		400		11.4		58.3 ± 20.5	9.6	41.2 ± 14.4
			10% H N	8-24		600				17-35			[125]
			5% RAP	20		600				25-60 (aliphatic)	(110)	[126]
	Ubiquitin 100% H N 100		850		13.5		19-73			30-68	[127]
	SOD		(d) 100% H N 60		500 MHz 1000	(e)	100	800 MHz	(f)	1 GHz	[128]
	186 (A83) 100% H	60		800				220			[120]
	SSB M2	N) / ppm δ( 15	100% H N 60 100% H 40 60 100% H N 60		1000 1000 800 800 1000	10 0.85 1.5		180 100 350 ± 75 225 ± 75 100	(12.5-40)	[120] [118]
	bR		10% H N	∼20		700				>30			>9	[129]
	OmpG	30% H N	20		850		9.2		101 ± 31	11.1	53 ± 23	[129]
			100% H N 60		1000			130-180			[118]
	Aβ		25% H N	20		600		8.2		170 ± 47	13.5	38 ± 10	[129]
				10	9	8	7 6		10	9	8	7 6	10	9	8	7 6
				δ( 1 H) / ppm		δ( 1 H) / ppm	δ( 1 H) / ppm
			(g)				(h)	20	30	MAS (kHz) 40	50	60
		δ( 15 N) / ppm						1 H Line width (Hz)	600 400 200	T108	A64	Bulk 1 H T 2 '	1.6 1.2 0.8 H T 2 ' / ms 1
													0.4
									0
				10	9	8	7 6					500	800	1000
				δ( 1 H) / ppm			External magnetic field (MHz)
	Figure 2.1: 1 H-15 N correlation spectra of SSB as a function of spinning frequency
	(a-c) at a H Larmor frequency (ω 0,H /2π) of 800 MHz (external magnetic field of 18.8
	T), and as a function of ω 0,H /2π (d-f) at 60 kHz MAS. In (a-f), the sample is fully
	protonated. In (g), the protein is perdeuterated with back-exchange in 100% H 2 O.
	The spectrum in (g) is recorded at 60 kHz MAS on a ω 0,H /2π = 1 GHz spectrometer.

In (h)

are plotted the MAS-rate dependence at ω 0,H /2π = 800 MHz (red, dashed lines) and the field strength dependence at 60 kHz MAS (blue, dotted lines) of 1 H NMR linewidths for two signals (A64 and T108) of [ 13 C,

15

N]-SSB. In the same plot, bulk 1 H T 2 's are depicted (red circles, dashed lines) as a function of MAS-rate at ω 0,H /2π = 800 MHz. Readapted with permission from reference

[120]

. Sample

Table 2 .

 2

1: Linewidths and experimental conditions found for various proteins under conditions used for proton-detected NMR spectroscopy. Reproduced from reference

[117]

  Spin-echoes involved in the transfer between C and Cα are highlighted in blue and red, when the coherence is present on the C and Cα, respectively, and spin-echoes involved in transfer between Cα and Cβ are highlighted in green. Specific CP to or from13 C may require a frequency shift, and in this case, the phases of RF during the two13 C CP periods are aligned respectively at the end and at the beginning of the contact time. In the diagrams, filled circles denote spins for which the evolution of chemical shifts is performed. Arrows represent magnetization transfers between nuclei.

	Reproduced from reference [128]

and (F) interresidue (H)(CA)CB(CACO)NH experiment (CB i-1 N i H i ). Orange boxes indicate crosspolarization, and gray boxes indicate heteronuclear decoupling. The MISSISSIPPI sequence for solvent signal suppression is indicated by the striped box.

this set of experiments. Experimental efficiencies, measured by comparing the first FID to the CP-HSQC spectrum, typically range from about 5% for (H)(CA)CB(CA)(CO)NH to more than 30% for the most sensitive (H)CANH and (H)CONH experiments. Depending on the sample conditions, the whole set of experiments is acquired in few days to more than a week in the worst cases.

Table 2 .

 2 2: Experimental performance of the 13 C-linking assignment strategy for [100% 1 H N , 2 H, 15 N,13 C]-labelled microcrystalline and amyloid fibrils of D76N β2m. (114.0 ppm) (120.5 ppm) (121.9 ppm) (123.8 ppm) (123.8 ppm)

	Experiment	Parameter	D76N β2m	D76N β2m
			(microcrystals) (fibrils)
	(H)CANH	1 H Larmor frequency (MHz)	800	1000
		Number of increments in 13 C dimension 90	128
		Number of transients	4	8
		Experimental time	5 h	18 h
		Number of assigned peaks	71	48
	(H)(CO)CA(CO)NH	1 H Larmor frequency (MHz)	800	1000
		Number of increments in 13 C dimension 90	100
		Number of transients	8	16
		Experimental time	10 h	1 d 1 h 30 min
		Number of assigned peaks	67	49
	(H)CONH	1 H Larmor frequency (MHz)	800	1000
		Number of increments in 13 C dimension 40	68
		Number of transients	4	8
		Experimental time	4 h 30 min	9 h
		Number of assigned peaks	69	40
	(H)CO(CA)NH	1 H Larmor frequency (MHz)	800	n/a
		Number of increments in 13 C dimension 40	n/a
		Number of transients	16	n/a
		Experimental time	8 h 52	n/a
		Number of assigned peaks	64	n/a
	(H)(CA)CB(CA)NH	1 H Larmor frequency (MHz)	800	1000
		Number of increments in 13 C dimension 130	128
		Number of transients	8	8
		Experimental time	14 h 30 min	18 h
		Number of assigned peaks	68	45
	(H)(CA)CB(CACO)NH 1 H Larmor frequency (MHz)	800	700
		Number of increments in 13 C dimension 150	110
		Number of transients	8	112
		Experimental time	20h	5 d 7 h
		Number of assigned peaks	68	46

(114.4 ppm) (119.0 ppm) (116.0 ppm) (127.3 ppm) (127.1 ppm)

(H)(CA)CB(CA)NH (H)(CA)CB(CA)(CO)NH (H)CANH (H)(CO)CA(CO)NH (H)CO(CA)NH (H)CONH

Table 2 .

 2

	Protein	GB1 D76N β2m (microcrystals) D76N β2m (fibrils)
	H(CO)NH
	1 H-13 C CP contact time (ms)	4	5	10
	1 H Larmor frequency (MHz)	700	800	800
	Experimental time	13h	1d 21h	2d 7h
	Number of trivial contacts*	108	122	73
	Number of "H-bond" contacts	33	33	? ‡
	Number of other contacts	62	28	7
	(H)CO(H)NH
	13 C -1 H CP contact time (ms)	4
	1 H Larmor frequency (MHz)	700
	Experimental time	17h
	Number of trivial contacts*	104
	Number of "H-bond" contacts	28
	Number of other contacts	47

3: Experimental parameters of (H)CO(H)NH and (H)CO(H)NH. *Trivial contacts are here defined as intraresidue contacts or contacts between H i+1 and C i . ‡ In the case of parallel in-register amyloid fibrils, the intermolecular β-sheet organization is such that contacts between strands would appear as correlations between H i and C i . "H-bond" contacts are therefore undistinguishable from intraresidue ones unless mixed labelling schemes are employed.

  1 H T 2 ' (ms)1 H LW (Hz)1 Hα LW (Hz) 15 N T 2 ' (ms)13 Cα T 2 ' (ms)

	GB1	100% H N 60	11	87 ± 14	n/a	114	32
		iFD	60	2.3	187 ± 32	146 ± 19	97	33
		100% H	60	1.6	247 ± 34	195 ± 47	60	14
		100% H N 111	23	81 ± 13	n/a	178	47
		iFD	111	6.4	105 ± 14	102 ± 12	152	67
		100% H	111	4.6	117 ± 21	120 ± 21	133	46
	AP205CP iFD	100	3.3	112 ± 17	99 ± 30
		100% H	100	3.0	139 ± 31	119 ± 31

Table 3 .

 3

1: Linewidths and experimental conditions found for proton-detected NMR spectroscopy reported in

[163]

.

  15 N R 1ρ relaxation dispersion for residues Alanine-77 and Arginine-78. Straight dashed line is the mean 15 N R 1ρ value for all spin-lock field strengths except 20 kHz and represent the relaxation value if there is no exchange contribution. (c,d) CEST profiles obtained for residues Alanine-77 and Arginine-78, respectively. Both types of experiment suggest the absence of conformational exchange on a µs-s timescale. All experiments were recorded on [ 1 H

N

, 2 H, 13 C,

15

N]-labeled AlkL at 1 GHz 1 H Larmor frequency and 60 kHz MAS

  Residue δ 1 H N δ 15 N δ 13 Cα δ 13 Cβ δ 13 C

	I1		60,62	37,55 175,48
	Q2	8,32	123,05 55,27	28,37 175,86
	R3	9,37	122,59 55,02	33,04 175,13
	T4	8,37	119,79 58,75	70,14
	P5			175,94
	K6	9,18	124,27 55,32	31,28 176,16
	I7	8,36	124,19 60,84	40,31 174,87
	Q8	8,89	126,53 54,38	31,97 173,75
	V9	9,02	125,02 60,07	32,63 175,17
	Y10	8,33	122,29 55,66	38,14 173,31
	S11	9,28	116,36 55,79	64,17 174,69
	R12	8,75	122,84 58,61	31,88 175,31
	H13	8,23	114,51 52,02	29,29
	P14		63,53	30,21 176,95
	A15	9,01	126,73 52,62	18,58 176,89
	E16	8,82	123,82 54,63	31,28 175,07
	N17	8,72	122,65 53,78	36,25 177,48
	G18	8,80	108,89 44,82	173,27
	K19	7,79	121,09 53,61	32,25 175,35
	S20	8,33	119,62 59,18	62,18 173,39
	N21	8,86	126,74 50,82	41,19 173,80
	F22	10,30 119,52 57,11	41,68 173,39
	L23	8,88	126,53 52,75	40,28 173,13
	N24	8,03	121,01 51,12	40,58 173,41
	C25	9,52	120,21 53,35	40,31 171,49

Table A .1 continued from previous page

 A

	Y26	9,63	129,37 55,70	39,93 174,44
	V27	8,73	128,37 59,12	32,07
	S28			66,68 174,36
	G29	8,08	107,79 46,10
	S33	8,23	111,69 60,21	61,56 176,73
	D34	7,16	119,73 54,46	39,39 173,77
	I35	7,86	123,32 60,38	39,75 174,00
	E36	7,91	125,03 54,33	31,13 171,77
	V37	7,88	124,69 60,19	32,61 174,18
	D38	8,72	121,95 52,15	45,13 173,39
	L39	8,96	120,62 53,09	42,30 174,26
	L40	8,87	119,59 52,36	44,48 174,54
	K41	8,71	120,45 53,69	32,66 175,46
	N42	9,62	128,26 53,78	36,18 176,29
	G43	8,82	102,20 44,77	175,58
	E44	7,70	120,52 53,74	30,44 172,92
	R45	8,65	124,12 57,11	29,38 175,70
	I46	8,73	128,38 61,31	37,63 176,31
	E47	8,44	125,91 57,88	29,39 176,00
	K48	8,03	118,79 55,36	29,29 177,12
	V49	7,79	122,16 60,64	33,24 174,85
	E50	8,33	125,19 53,73	31,44
	Y63		55,83	39,48 174,18
	L64	9,05	119,37 55,24	45,28 172,30
	L65	8,04	121,41 53,48	45,13 175,21
	Y66	9,03	126,50 56,39	40,67 174,99
	Y67	8,80	116,14 55,36	40,04 173,65
	T68	8,19	111,67 59,60	69,50 171,34
	E69	8,46	129,85 56,13	28,37 175,74
	F70	8,60	125,33 54,76	39,77 171,69
	T71	8,19	116,96 57,94	69,07
	P72		62,48	30,64 175,09
	T73	7,88	110,86 59,52	71,59 175,03
	E74	8,98	119,61 58,24	28,30 177,28
	K75	7,89	114,33 56,01	31,66 176,31
	N76	7,16	116,28 52,45	39,54 174,04
	E77	8,37	121,88 54,29	31,20 174,89

Table A .1 continued from previous page

 A

	Y78	9,46	123,20 56,26	41,30 175,17
	A79	8,65	121,01 50,56	22,71 173,55
	C80	9,01	119,37 52,45	42,43 171,10
	R81	9,30	128,00 53,09	32,27 173,96
	V82	8,96	127,67 59,72	34,60 173,29
	N83	8,93	123,02 50,61	40,54 172,76
	H84	7,63	122,25 56,48	34,80 175,17
	V85	7,96	124,73 63,92	29,98 175,37
	T86	7,41	110,50 62,70	69,05 174,56
	L87	7,98	123,10 52,83	41,30
	P90		63,02	30,59 175,84
	K91	8,68	125,34 54,50	33,96 174,57
	I92	8,49	125,11 59,77	37,32 175,82
	V93	9,01	129,52 60,96	32,69 175,15
	K94	8,76	127,35 56,04	32,25 176,23
	W95	8,60	121,89 56,18	27,07 174,44
	D96	8,40	131,11 52,45	40,54 175,01
	R97	7,44	120,74 56,01	28,53 175,66
	D98	8,12	119,70 54,12	40,15 175,21
	M99	7,48	124,89 56,78	32,66

Table A .

 A 2: 1 H N ,15 N and13 C chemical shifts in ppm for crystalline D76N β2m, referenced to DSS, without isotope correction.Residue δ 1 H N δ 15 N δ 13 Cα δ 13 Cβ δ 13 C

	Table A.2 continued from previous page Table A.2 continued from previous page
	F22 F70	9,05 9,19	121,40 55,56 123,46 54,85	40,81 174,06 39,82 173,20
	L23 T71	8,48 8,08	126,00 53,04 113,71 57,53	38,46 173,81 69,53 172,68
	N24 P72	8,33	121,03 51,64 62,26	41,08 172,75 31,50 176,72
	C25 T73	9,47 7,77	121,18 53,77 111,67 59,49	41,26 171,72 70,78 175,83
	Y26 E74	9,61 9,04	128,19 54,33 119,88 57,86	39,68 174,26 28,34
	V27 K75	9,61 7,83	132,56 60,44 114,64 56,57	31,46 175,05 32,20 175,78
	S28 N76	9,65 7,25	119,29 56,69 114,28 51,97	66,75 174,53 41,56 173,27
	G29 E77	6,57 8,51	109,30 46,65 120,54 54,39	173,62 32,11 175,19
	F30 Y78	7,51 9,37	111,58 54,50 124,58 56,41	41,44 174,80 41,17 175,22
	H31 A79	8,55 8,91	117,64 57,47 121,66 50,80	174,75 23,49 173,98
	P32 C80	9,20	62,26 119,37 52,65	34,80 176,10 43,68 171,64
	S33 R81	8,65 9,43	114,01 59,04 127,61 53,49	60,37 174,43 32,98 174,35
	D34 V82	8,14 9,11	120,54 54,50 128,03 59,83	40,18 175,09 35,00 173,81
	I35 N83	8,32 9,16	121,93 61,90 123,14 51,28	39,02 171,32 41,08 173,03
	E36 H84	7,86 7,99	123,84 54,39 122,94 56,80	32,29 174,99 35,25 175,49
	V37 V85	8,42 8,59	123,77 60,11 125,72 64,43	34,62 173,51 30,61 174,02
	D38 T86	9,11 7,47	123,12 53,29 107,08 62,13	45,93 175,24 69,17 175,96
	L39 L87	8,66 8,10	120,09 53,49 122,75 52,98	43,59 174,95 41,53 176,81
	L40 S88	9,19 8,79	120,01 52,59 118,57 59,30	44,85 175,74 61,54 173,81
	K41 Q89	9,00 7,25	120,90 53,99 116,78 52,42	33,36 176,35 29,14 172,43
	N42 P90	9,74	127,91 54,05 63,38	36,64 175,59 30,64 176,16
	G43 K91	8,97 8,78	102,62 45,36 125,33 54,78	172,98 33,90 174,92
	E44 I92	7,83 8,57	120,28 53,88 125,45 59,97	30,85 176,06 37,76 175,78
	R45 V93	8,90 9,08	123,35 57,19 129,25 60,89	29,69 176,90 33,03 175,49
	Q2 I46 K94 R3 L54 W95 T4 P5 K6 I7 Q8 V9 Y10 S11 R12 H13 N21 E69 T68 Y67 Y66 L65 L64 Y63 F62 S57 F56 S55 D96	8,71 8,81 8,89 9,76 8,72 9,29 8,66 8,49 8,94 8,98 8,60 10,56 119,88 55,96 120,90 54,39 126,02 61,68 127,37 56,18 123,45 54,50 53,55 122,77 56,35 122,92 59,66 62,03 123,54 55,56 123,07 60,72 125,53 54,50 124,02 60,28 123,11 55,71 8,70 120,87 53,33 8,83 119,34 51,02 51,58 8,65 127,40 56,85 8,09 111,17 60,00 8,49 122,01 55,51 8,82 131,77 55,84 9,11 128,79 53,66 9,83 125,83 53,38 8,84 115,66 57,41 57,41 9,10 115,25 55,79 8,93 113,09 55,73 9,72 121,94 57,41 8,64 131,96 52,93	37,30 176,15 32,53 176,45 29,95 174,78 44,08 175,36 27,96 175,22 34,35 174,89 44,08 173,88 29,43 176,13 28,16 172,68 70,69 171,79 35,74 174,13 40,99 173,02 65,94 174,82 41,46 173,89 38,77 172,90 43,64 173,81 33,19 175,21 46,48 173,59 33,24 174,03 40,54 174,13 40,99 175,17 40,42 174,41 30,95 176,40 65,49 31,41 176,18 40,00 174,72 69,71 173,59 66,93 172,18 40,95

Table A .

 A 3: 1 H N ,15 N and13 C chemical shifts in ppm for amyloid fibrils of D76N β2m, referenced to DSS, without isotope correction.Residue δ 1 H N δ 15 N δ 13 Cα δ 13 Cβ δ 13 C

	Table A.3 continued from previous page
	C25	8,31	119,56 55,57	39,33
	Y26	7,62	120,53 56,79	40,45 172,82
	V27	8,43	127,79 60,33	33,49
	S28			175,74
	G29	8,82	110,59 47,84
	F30	8,63	117,73 51,77	38,98
	H31	7,66	125,33 53,08	30,54
	P32		60,59	32,09 174,95
	S33	8,40	114,37 55,93	65,83 173,36
	D34	7,70	118,99 53,12	37,25 176,37
	I35	7,07	116,05 57,78	41,07 174,10
	E36	8,22	127,33 53,42	31,73 173,68
	V37	8,42	127,05 60,66	33,63 174,95
	D38	9,01	118,58 51,76	41,73 172,29
	L39	8,19	122,48 52,49	45,69 173,97
	L40	8,67	124,78 55,41	38,89
	K41			177,48
	N42	10,40 122,94 54,16	36,65 175,19
	G43	7,89	105,84 45,33
	E47		53,40	36,52 175,21
	K48	8,40	119,69 56,84	28,11 175,29
	V49	7,76	120,76 60,07	33,71 174,71
	E50	8,93	122,61 53,07	33,62
	H51	7,98	122,66 57,13	174,56
	S52	7,95	110,12 58,30	67,75 170,84
	D53	8,45	122,26 52,77	42,72 173,57
	L54	9,30	124,96 54,53	45,44 176,37
	S55	8,35	114,80 56,71	65,72 174,23
	F56	7,82	117,09 59,92	37,37 175,04
	S57	8,09	113,04 55,
	N17		53,78	32,25 175,49
	G18	8,78	106,80 43,85
	S20	7,90	112,80 55,93	65,54
	F22	8,24	121,58 54,21	39,23 173,22
	L23	7,38	114,76 54,34	38,49 174,86
	N24	7,74	108,91 50,87	41,33 176,64

Table A .3 continued from previous page

 A

		T68	8,67	126,60 61,49	70,29 172,30
	E69 F70 Appendix B	8,52 9,54	126,70 54,09 130,38 56,82	29,93 172,79 39,32 172,67
		T71	8,98	119,23 55,60	69,17 175,84
	T73 E74 Relaxation rates and dipolar coupling 8,58 117,76 61,85 69,86 173,53 8,93 124,87 54,45 33,54 175,28
	data	K75 N76	8,00 8,38	129,54 55,08 121,01 50,74	31,73 173,64 42,33 174,67
		E77	7,55	118,00 53,96	32,16 174,42
		Y78	8,87	119,14 57,30	41,17

Table B .

 B 1:15 N R 1 and 15 N R 1ρ relaxation rates obtained for microcrystalline wt β2m on an 18.8 T spectrometer.Residue 15 N R 1 (10 -3 s -1 ) 15 N R 1ρ (s -1 )

	Table B.1 continued from previous page Table B.1 continued from previous page
	D34 L87	44.5 27.9	3.5 2.7	1.90 1.17	0.18 0.14
	I35 Q89	25.9 105.4	1.0 3.1	0.74 8.31	0.05 0.27
	E36 K91	22.1 33.6	1.1 1.2	1.24 0.87	0.07 0.06
	D38 I92	20.7 60.5	0.8 2.0	0.92 3.18	0.05 0.12
	L39 V93	25.4 38.2	1.3 1.9	1.84 1.02	0.11 0.10
	L40 K94	25.4 41.8	2.4 2.7	3.18 1.55	0.24 0.14
	K41 W95	21.8 73.5	2.2 5.3	1.20 4.31	0.17 0.35
	N42 D96	37.2 67.7	2.0 21.1	2.44 30.08 15.61 0.14
	G43	39.2	2.8	5.77	0.45
	E44	48.6	2.4	2.05	0.13
	R45	85.7	3.1	3.87	0.16
	I46	42.3	2.2	1.48	0.11
	L54	65.8	3.7	8.88	0.52
	S55	24.8	1.7	2.63	0.17
	F56	mean 17.4	std 1.8	mean std 2.70 0.20
	Q2 Y63	49.1 8.6	2.3 1.8	5.71 1.25	0.27 0.13
	R3 L64	29.8 15.0	2.8 2.3	1.38 2.00	0.17 0.19
	T4 L65	43.5 21.0	1.6 1.9	3.87 1.84	0.15 0.16
	K6 Y66	28.3 23.0	1.4 1.6	2.21 1.68	0.11 0.12
	I7 Y67	47.7 22.2	1.7 1.1	1.71 3.24	0.09 0.13
	Q8 T68	24.4 35.7	1.1 1.2	0.98 7.51	0.07 0.25
	V9 E69	30.5 103.7	0.7 11.3	1.91 16.35 1.63 0.06
	Y10 F70	21.8 56.0	1.5 6.7	4.23 5.36	0.25 0.71
	S11 T71	47.2 78.0	9.8 6.5	5.97 4.64	1.16 0.40
	R12 T73	35.5 158.7	4.8 32.8	4.72 14.19 2.25 0.65
	H13 E74	47.1 220.2	2.4 22.6	6.22 14.69 1.22 0.37
	F22 K75	44.2 251.1	2.1 35.4	2.89 13.88 1.72 0.16
	L23 D76	51.0 208.1	2.2 14.8	2.80 9.77	0.17 0.63
	N24 E77	29.9 81.5	5.0 8.3	2.28 3.03	0.42 0.37
	C25 Y78	14.0 21.0	1.0 2.1	1.53 3.88	0.09 0.28
	Y26 A79	9.7 18.8	0.9 1.4	1.30 1.72	0.07 0.11
	V27 C80	15.7 14.9	1.4 1.1	0.89 1.45	0.10 0.10
	S28 R81	21.9 14.9	1.0 1.5	0.69 0.80	0.07 0.10
	G29 V82	25.6 20.8	5.2 1.5	1.70 1.32	0.37 0.11
	F30 N83	32.6 22.5	2.6 0.7	2.33 1.82	0.18 0.06
	H31 H84	13.0 22.8	2.9 1.8	1.37 1.54	0.24 0.13
	S33 T86	26.3 16.8	2.5 3.4	1.73 2.55	0.18 0.35

Table B .

 B 2:15 N R 1 and 15 N R 1ρ relaxation rates obtained for microcrystalline D76N β2m on an 18.8 T spectrometer.Residue 15 N R 1 (10 -3 s -1 ) 15 N R 1ρ (s -1 )

	Table B.2 continued from previous page Table B.2 continued from previous page
	E36 Q89	21,3 99,2	0,5 2,3	0,51 9,27	0,08 0,28
	V37 K91	24,7 54,3	0,4 1,8	0,37 1,43	0,03 0,11
	D38 I92	21,2 48,1	0,3 0,6	1,19 0,74	0,02 0,04
	L39 V93	29,8 41,3	0,6 0,5	2,40 1,17	0,06 0,04
	L40 K94	23,2 39	0,7 0,7	2,57 0,05	0,10 0,04
	K41 W95	26,6 46,2	0,9 2,5	1,88 7,07	0,08 0,61
	N42	34,1	0,6	1,13	0,06
	G43	41,9	1,8	5,61	0,28
	E44	47,7	0,9	2,48	0,07
	R45	58,2	0,7	2,66	0,04
	I46	78,1	1,9	3,76	0,17
	S55	25,1	1,4	3,44	0,17
	F56 I1 S57 R3 Y63 T4 L64 K6 L65 I7 Y66 Q8 Y67 V9 T68 Y10 E69 S11 F70 R12 T71 H13 T73 F22 E74 L23 K75 N24 D76 C25 E77 Y26 Y78 V27 A79 S28 C80 G29 R81 F30 V82 H31 H84 S33 T86 D34 L87 I35 S88	mean 15,8 57,6 34,9 23,9 8,6 36,2 14,8 32,1 35,4 54,5 31,2 21,6 23,5 26,5 27,4 25,5 165,9 43,5 54,7 43,9 92,6 38,7 468,9 43,5 165,5 22,7 137,6 13,2 129,3 10,4 53,8 8,9 55,5 14,6 18,3 21,1 8,8 27,2 13,4 35,8 16,5 8,8 20,2 24,9 16,4 37,7 22,3 21,6 136,7	std 1,0 1,2 1,6 0,8 0,8 0,8 1,1 1,6 0,7 1,1 1,4 0,6 0,5 1,3 0,6 1,4 7,4 5,1 5,5 2,3 6,4 1,3 37,7 0,9 14,7 0,6 17,7 1,1 32,8 0,5 1,6 0,6 2,5 0,8 0,9 0,7 0,5 1,8 0,6 0,7 0,5 1,0 0,6 1,0 1,3 1,5 0,8 0,4 11,5	3,46 mean std 0,18 6,22 8,24 0,48 0,21 0,80 1,36 0,06 0,06 3,00 1,27 0,14 0,13 0,68 1,94 0,06 0,09 0,48 1,99 0,16 0,04 0,36 5,04 0,13 0,05 0,32 6,31 0,19 0,09 3,42 13,01 0,77 0,28 18,46 1,91 11,37 1,25 10,11 0,85 11,80 0,72 6,05 18,83 1,59 0,27 3,51 23,71 2,06 0,09 2,64 32,65 4,44 0,09 1,50 33,37 10,81 0,13 1,06 6,51 0,18 0,05 0,37 3,98 0,21 0,06 0,15 1,73 0,09 0,06 0,19 0,52 0,06 0,05 3,17 0,27 0,05 0,26 2,35 0,18 0,05 0,08 1,39 1,07 0,06 0,10 0,89 2,55 0,14 0,05 1,26 2,81 0,10 0,11 0,71 0,03 32,51 4,13

Table B .

 B 3: Order parameters, and15 N R 1 and 15 N R 1ρ relaxation rates obtained for KpOmpA on an 23.5 T spectrometer.

		Table B.3 continued from previous page
	V118	0.87	0.05	5.69	1.4	0.0128 0.0069
	R120	0.94	0.04 13.28	1.4	0.0233 0.0010
	G147	0.82	0.05 11.84	1.4	0.0297 0.0056
	G148	0.85	0.06 11.44	1.8	0.0206 0.0049
	E150	1	0.06 14.36	1.3	0.0223 0.0019
	W151	0.8	v0.04 22.01	3.2	0.0255 0.0063
	R155	0.87	0.05 19.84	7.1	0.0234 0.0112
	A158	0.87	0.05 14.18	2.4	0.027 0.0077
	T159	0.84	0.03 10.86	1.1	0.0284 0.0034
	Residue L161 E162 A21 Y163 G22 L184 G23 S185 K24 G187 G26 S189 G53 Y190 G55 R191	<S> 0.86 0.07 14.99 15 N R 1 (10 -3 s -1 ) 1.8 mean std mean std 0.87 0.03 11.42 2.1 0.82 0.04 7.78 1.4 0.91 0.02 9.84 2.1 0.82 0.05 11.84 1.4 0.82 0.03 13.62 1.7 0.82 0.03 13.01 2.1 0.86 0.03 9.38 1.6 0.88 0.04 6.22 1.1 0.9 0.02 2.47 0.6 0.87 0.05 13.67 3.1 0.97 0.09 14.1 3.1 0.87 0.04 12.84 1.8 0.93 0.03 10.17 1.2 0.82 0.03 7.29 1.1 0.82 0.03 14.4 1.8	0.0225 0.0079 15 N R 1ρ (s -1 ) mean 0.0334 0.0110 std 0.0187 0.0076 0.0115 0.0050 0.0244 0.0067 0.0297 0.0056 0.0321 0.0040 0.0203 0.0053 0.0116 0.0041 0.0158 0.0111 0.0203 0.0089 0.0276 0.0074 0.0184 0.0019 0.0249 0.0013 0.0087 0.0042 0.0212 0.0125
	G58	0.84	0.06	7.05	1.1	0.0161 0.0035
	G59	0.87	0.04 11.74	1.2	0.0313 0.0054
	Y60	0.88	0.03	12.5	2.2	0.0225 0.0082
	Q61	0.88	0.04 10.68	1.7	0.0203 0.0078
	V62	0.84	0.04 17.36	3.2	0.0305 0.0132
	G67	0.87	0.08 31.44	6.0	0.0198 0.0076
	M70	n.d.		11.29	1.5	0.0187 0.0021
	G71	0.91	0.05	7.04	1.4	0.0309 0.0133
	D73	0.87	0.05 14.18	2.4	0.027 0.0077
	Q95	0.93	0.04	4.63	1.8	0.0186 0.0040
	T97	0.89	0.03	6.23	0.8	0.011 0.0039
	A98	0.9	0.03 23.34	5.0	0.021 0.0041
	K99	0.97	0.02 12.66	1.7	0.0162 0.0026
	L100	0.87	0.04 15.11	1.8	0.0151 0.0021
	G101	0.91	0.06	4.47	1.6	0.022 0.0117
	T112	0.97	0.11 34.52	7.2	0.0321 0.0118
	G115	0.93	0.03	9.89	1.8	0.0109 0.0019
	G116	0.99	0.06 14.34	3.8	0.0203 0.0047
	M117	0.9	0.07 14.55	4.8	0.0148 0.0131

Table B .

 B 4:15 N R 1 and15 N R 1ρ relaxation rates obtained for AlkL on an 23.5 T spectrometer.Residue 15 N R 1 (10 -3 s -1 )15 N R 1ρ (s -1 )In order to take into account the reorientations of the peptide plane in the GAF model, one can use the transformation properties of the spherical harmonics:

	Table B.4 continued from previous page Table B.4 continued from previous page
	I59 V123	14.6 17.8	2.2 3.8	n.d. 5.68	0.05
	A60 G124	13.5 18.1	4.1 3.8	10.32 0.13 10.79 0.11
	Y61 R125	n.d. n.d.		5.85 9.92	0.06 0.12
	F62 L127	12.5 16.1	3.1 7.2	7.79 n.d.	0.06
	V63 D130	25.3 33.6	6.7 5.8	15.02 0.25 n.d.
	N66 T132	n.d. 27.6	3.6	108.35 3.77 7.74 0.08
	I67 G134	n.d. 30.4	5.1	11.47 0.09 11.36 0.08
	A68 A135	12.6 29.6	1.6 6.4	7.66 6.65	0.04 0.08
	V69 S138	12.1 15.8	2.1 2.7	5.85 15.25 0.12 0.03
	D70 F139	14.3 n.d.	2.1	7.32 15.47 0.10 0.04
	F72 I141	10.4 n.d.	2.7	7.97 6.00	0.06 0.06
	V75 K142	28.2 41.5	9.5 9.5	12.04 0.14 8.74 0.10
	A77 D143	n.d. 44.9	5.2	27.99 0.67 7.68 0.04
	R78 K144	39.9 n.d.	9.0	28.23 0.67 9.43 0.12
	A79 W145	39.9 n.d.	9.0	14.63 0.18 9.34 0.21
	F81 A146	n.d. n.d.		13.16 0.18 9.81 0.12
	G83 A148	23.1 n.d.	4.5	n.d. 8.32	0.06
	E84 Q150	n.d. 9.3	2.1	12.35 0.07 9.05 0.05
	K85 V151	36.8 10.6	3.8 1.8	12.80 0.08 7.86 0.04
	S88 L153	90.5 22.3	29.1 2.7	13.35 0.21 7.66 0.05
	S89 M162	40.2 24.9	9.8 8.4	13.53 0.16 10.43 0.11
	G91 S165	74.2 28.8	10.8 2.6	11.60 0.09 6.64 0.04
	R92 D166 Q14 V93 V167 A19 S94 T174 S20 V96 D175 F21 D97 G178 N22 G99 T179 F23 A101 T187 E31 I102 K188 L32 L103 V191 G33 S104 L196 D46 L105 S197 I49 Y107 G199 L55 Y109 A200 T56 G120 S201 F57 V121 Y202	mean 51.7 27.1 46.9 28.6 14.1 18.5 16.7 52.9 n.d. n.d. 89.7 n.d. n.d. 18.1 14.1 46.9 25.8 24.3 29.3 39.7 92.8 14.5 n.d. 39.5 7.6 n.d. n.d. 14.5 14.1 n.d. 13.0 21.1 43.0 9.9 9.9 n.d. 6.1 15.7 19.2 18.6 16.0 23.5 12.7 18.7	std 5.1 5.9 3.7 4.6 1.8 4.8 1.9 7.4 15.3 1.9 3.1 13.1 2.8 4.8 8.1 9.2 45.4 4.4 13.5 2.0 2.3 2.4 2.4 5.0 10.2 1.8 2.6 2.4 3.7 2.7 3.0 4.5 5.7 3.7 3.6	10.89 0.06 9.33 0.09 mean std 7.79 0.08 7.09 0.04 12.03 0.06 7.69 6.37 0.04 10.48 0.10 0.12 9.17 9.54 0.07 8.91 0.07 0.11 8.57 12.62 0.26 6.78 0.04 0.09 13.97 0.27 8.05 0.05 10.43 0.10 n.d. 11.10 0.15 n.d. 7.85 0.07 13.04 0.18 17.68 0.26 5.50 0.04 14.69 0.17 22.79 0.57 10.05 0.06 6.11 0.06 11.74 0.21 n.d. 10.97 0.23 14.29 0.16 6.17 0.04 8.96 0.07 16.25 0.21 9.63 0.07 7.23 0.07 16.32 0.30 9.61 0.08 8.35 0.09 n.d. 9.42 0.11 8.97 0.08 9.56 0.09

RAM simulations and all the calculations were performed by Dr. C. Camilloni from the University of Milan, Italy.

In this work, all the biochemistry has been done by Dr. O. Saurel and coworkers in Toulouse and the NMR experiments by Dr. L. Andreas in Lyon. I contributed to this project by the analysis of the solid-state NMR results.

Remerciements

List of publications

Research articles

-Le Marchand, T., de Rosa, M. , Salvi N., Sala B., Andreas, L. B., Barbet-Massin, E., Sormanni, P., Barbiroli, A., Porcari, R., Sousa Mota, C., de Sanctis, D., Bolognesi, M., Emsley, L., Bellotti, V., Blackledge, M., Camilloni, C., Pintacuda, G. & Ricagno, S. Conformational dynamics in crystals reveal the molecular bases for D76N beta-2 microglobulin aggregation propensity, Nat. Comm., 9, 1658Comm., 9, (2018)).

List of publications

Review articles

-Schubeis, T., Le Marchand, T., Andreas, L. B. & Pintacuda, G. 1 H magicangle spinning NMR evolves as a powerful new tool for membrane proteins, J. Magn. Reson., 287, 140-152 (2018).

-Andreas, L. B., Le Marchand, T., Jaudzems, K., Pintacuda, G. Highresolution proton-detected NMR, J. Magn. Reson., 253, 36-49 (2015). The purification and refolding step were performed as described previously [189] in non-deuterated water allowing proton back-exchange at every amide site. Briefly, inclusion bodies were washed and dissolved in 6 M guanidinium hydrochloride (GdHCl) buffer containing 10 mM Tris at pH 8.0. β2m was purified with size exclusion chromatography under denaturating conditions in the same buffer. The denatured protein was refolded in three consecutive dialysis steps against a buffer (10 mM Na 2 PO 4 at pH 7.4) containing decreasing concentration of NaCl (150 mM, 75 mM and 35 mM). The buffer was then exchanged to 20 mM Tris HCl at pH 8.3 by a 24 h dialysis. Last purification step was an ion exchanged chromatography. The purity and successful refolding of the proteins was monitored with SDS PAGE.

List of abbreviations

Crystallization and fibrillization

Crystals were grown using the sitting drop vapor diffusion technique (10 mg/mL protein concentration, crystallization solution: 22% w/v PEG, 20% v/v glycerol, 0.1 mol.L-1 sodium acetate, pH 5.5). After two weeks, microcrystals of D76N and wt β2m were filled into 1.3 mm NMR rotor by centrifugation (14000 rpm).

D76N amyloid fibrils were grown under agitation at pH 7 and at 37 • C as described previously [159]. Fibril formation was checked by TEM and Congo red test.

C.2 KpOmpA Protein Expression and Purification

N-KpOmpA recombinant protein lacks the native N-terminal signal peptide for outer membrane localization and therefore accumulate as inclusion bodies in the cytoplasm. The expression/purification procedure for the triple isotopically labeled N-terminal domain of the outer membrane protein from Klebsiella pneumoniae, [U-1 H N , 13 C, 15 N, 2 H] N-KpOmpA, was described previously [236]. Briefly, exponentially growing E. coli BL21 cells harboring the pET26b-N-KpOmpA plasmid were induced with 1 mM IPTG for 4 h at 37 • C and then collected. Upon cell lysis, the released inclusion bodies were washed and dissolved in 6 M guanidine/HCl buffer containing 25 mM Tris (pH 8.5) and 5 mM EDTA. The denatured protein was refolded in 15× excess volume of 1% (w/v) Zwittergent 3-14 buffer containing 25 mM Tris (pH 8.5) and 150 mM NaCl. The detergent concentration was then reduced to 0.1% ZW3-14 via dialysis (12-14 kDa cutoff), and the protein was purified by affinity chromatography with elution step of 400 mM imidazole. The imidazole was removed by a second dialysis and the final protein stock (in 0.1% ZW3-14, 25 mM Tris (pH 8.5), 150 mM NaCl) was stored at 4 • C.

Proteoliposomes Preparation

Proteoliposomes containing N-KpOmpA were prepared as described previously [237]. Briefly, E. coli Polar Lipids Extract (PLE) was dissolved in chloroform, dried under nitrogen stream and then in a vacuum chamber overnight. This lipid film was dissolved at 10 mg/mL in 2% n-octyl-β-D-glucopyranoside, 20 mM Tris (pH 8.5), and 100 mM NaCl and sonicated in a water bath several times. Purified N-KpOmpA was bound to nickel-chelating resin, extensively washed with 2% n-octyl-β-D-glucopyranoside solution containing 20 mM Tris (pH 8.5) and 100 mM NaCl, and eluted with the same buffer supplemented with 400 mM imidazole. Mixed micelles were formed by adding the E. coli Polar Lipids Extract solution to the protein eluate at the desired lipid-to-protein ratio (LPR), ranging from 0.5 to 5 w/w (equivalent to ∼15-150:1 mol/mol) in the different experiments, and NMR samples were finally prepared at a LPR of 0.5 w/w. This mixture was incubated for 10 min at 4 • C under gentle mixing and then dialyzed at 37 • C against detergent-free buffer, until complete removal of the detergent and formation of large unilamellar vesicles. During the dialysis, the NaCl was gradually removed in 25 mM steps. The final proteoliposomes sample is thus found in only 20 mM Tris buffer (pH 8.5).

C.3 AlkL Protein Expression and Purification

AlkL from Pseudomonas putida GPo1 (Uniprot Q00595) with a Cterminal His-8 tag was expressed in E. coli in a triple labelled ( 2 H, 13 C, 15 N) M9 medium (D 2 O, d 7 -13 C-glucose and 15 N-ammonium chloride). The detailed protocol has been described previously [254]. After cell lysis, inclusion bodies were washed and dissolved (50 mM Tris/HCl pH 8, 500 mM NaCl, 20 mM imidazole, 10% glycerol, 6 M guanidine HCl). The protein was purified by affinity chromatography with elution with a linear imidazole gradient from 20 to 500 Refolding into Lauryldimethylamine oxide (LDAO) micelles was performed by dilution into a 20 mM Tris/HCl, 1 mM EDTA buffer at pH 8.

Proteoliposomes Preparation

The detergent was exchanged to 2% octyl glucoside (OG) for reconstitution into lipids. DMPC was added to AlkL in OG at a lipid to protein ratio of 1:2 (w/w). Detergent was removed by dialysis against NMR buffer. The white precipitate was packed into 1.3 mm rotor (Bruker) using an ultracentrifuge packing device (Giotto Biotech). To investigate the effect of the presence of a hydrophobic small molecule a drop of Carvone was added to a solid state NMR sample after recording a reference spectrum and reopening the NMR rotor.

Appendix D

Gaussian axial fluctuations analyses

The so-called gaussian axial fluctuation (GAF) model was developed in order to take into account the anisotropy of motions into the analysis of spin relaxation [START_REF] Bremi | Locally anisotropic internal polypeptide backbone dynamics by NMR relaxation[END_REF][START_REF] Lewandowski | Site-Specific Measurement of Slow Motions in Proteins[END_REF]. In this model, the vector representing the peptide plane undergoes gaussian fluctuations around three perpendicular axes ( e α , e β and e γ , as shown in Figure D.1) described by the standard deviation of the fluctuations σ α , σ β and σ γ , and an effective timescale τ GAF . In the following, we will broadly describe the mathematical expressions used in a GAF analysis for 1 H N -15 N dipolar coupling. Other interactions are subjected to an identical treatment. where Y 2l are spherical harmonics of rank 2 with l = -2, -1, 0, 1, 2 and Ω(t) is the time-dependent orientation of the NH vector in an arbitrary but fixed molecular frame.

Appendix E

Relaxation dispersions and CEST profiles