Lorraine A Malaspina

Erna K Wieduwilt

Justin Bergmann

Florian Kleemiss

Benjamin Meyer

Manuel F Ruiz-López

Rumpa Pal

Emanuel Hupf

Jens Beckmann

Ross O Piltz

Alison J Edwards

Simon Grabowsky

Alessandro Genoni

Fast and Accurate Quantum Crystallography: from Small to Large, from Light to Heavy

The coupling of the crystallographic refinement method Hirshfeld Atom Refinement (HAR) with the recently constructed libraries of extremely localized molecular orbitals (ELMOs) gives rise to the new quantum-crystallographic method HAR-ELMO. This method is significantly faster than HAR but as accurate and precise, especially concerning the free refinement of hydrogen atoms from X-ray diffraction data, so that the first fully quantumcrystallographic refinement of a protein is presented here. However, the promise of HAR-ELMO exceeds large molecules and protein crystallography. In fact, it also renders possible electron-density investigations of heavy elements in small molecules, such as mercury as presented here, and facilitates to detect and isolate systematic errors from physical effects.

Structure determination via single-crystal X-ray diffraction is undoubtedly the most important technique for the experimental elucidation of the connectivity and three-dimensional structure of small as well as large molecules. Famous examples in the history of science are the X-ray structure determinations of biological molecules such as vitamins [START_REF] Hodgkin | Structure of Vitamin B12[END_REF] , DNA 2 or proteins [START_REF] Jaskolski | A brief history of macromolecular crystallography, illustrated by a family tree and its Nobel fruits[END_REF] . The method is so successful and widespread that it gave rise to about 30 Nobel Prizes in chemistry, physics and medicine. [START_REF] Jaskolski | A brief history of macromolecular crystallography, illustrated by a family tree and its Nobel fruits[END_REF] Despite its paramount importance, it is sometimes forgotten that the refinement of the diffraction data is based on a crude approximation. In fact, the electron densities of the atoms are estimated as spheres (so-called Independent Atom Model, IAM [START_REF] Compton | The Distribution of Electrons in Atoms[END_REF] ), which yields accurate atomic positions for all atoms except hydrogen, but no direct information on chemical bonding and lone pairs, energies or other properties of the molecules or materials under investigation. Moreover, if hydrogen atom positions are not attainable, intermolecular interactions such as hydrogen bonding or agostic interactions cannot be characterized adequately, [START_REF] Braga | Agostic interactions in organometallic compounds. A Cambridge Structural Database study[END_REF] thermodynamic properties cannot be estimated, [START_REF] Hoser | Dynamic quantum crystallography: lattice dynamical models refined against diffraction data. II. Applications to L-alanine, naphthalene and xylitol[END_REF] and protonation states of enzymes remain unknown [START_REF] Fisher | Protonation-state determination in proteins using high-resolution X-ray crystallography: effects of resolution and completeness[END_REF] . If electron deformation due to chemical effects is neglected, bond properties in materials or in metalloproteins are in doubt. 8 To solve such problems, on the experimental side, an enormous technical and financial effort has been made to obtain increasingly higher quality and resolution X-ray diffraction data sets, especially in protein crystallography. More and more dedicated end stations at synchrotron sources around the world provide a rapidly increasing number of high-resolution structures of proteins. [START_REF] Petrova | Protein crystallography at subatomic resolution[END_REF][START_REF] Ogata | Hydrogens detected by subatomic resolution protein crystallography in a [NiFe] hydrogenase[END_REF] On the other hand, method development is trying to keep up by providing tools to replace the IAM with refinement techniques that take into account electron deformation and can potentially locate hydrogen atoms. For small molecules, multipole modeling has become a mature technique [START_REF] Coppens | Charge Density Come of Age[END_REF] and provides information on chemical bonding, [START_REF] Stalke | Electron Density and Chemical Bonding I[END_REF] although refining hydrogen [START_REF] Hoser | Towards the best model for H atoms in experimental charge-density refinement[END_REF][START_REF] Zhurov | Importance of the consideration of anharmonic motion in charge-density studies: a comparison of variabletemperature studies on two explosives, RDX and HMX[END_REF][START_REF] Dittrich | Accurate Bond Lengths to Hydrogen Atoms from Single-Crystal X-ray Diffraction by Including Estimated Hydrogen ADPs and Comparison to Neutron and QM/MM Benchmarks[END_REF] and heavy atom [START_REF] Iversen | Accurate charge densities in days -use of synchrotrons, image plates and very low temperatures[END_REF]17 parameters is only possible in special cases. Unfortunately, for large molecules, multipole modeling is not feasible due to limited computer resources. However, fixed multipole parameters are stored in properly designed databanks [START_REF] Domagała | An Improved Experimental Databank of Transferable Multipolar Atom Models -ELMAM2. Construction Details and Applications[END_REF][START_REF] Dominiak | A Theoretical Databank of Transferable Aspherical Atoms and Its Application to Electrostatic Interaction Energy Calculations of Macromolecules[END_REF][START_REF] Dittrich | The Generalized Invariom Database (GID)[END_REF] and successfully transferred to the protein structures of choice to perform accurate refinement of coordinates and atomic displacement parameters (ADPs). [START_REF] Elias | Hydrogen atoms in protein structures: high-resolution X-ray diffraction structure of the DFPase[END_REF][START_REF] Pröpper | Invariom refinement of a new monoclinic solvate of thiostrepton at 0.64 Å resolution[END_REF] Introducing quantum-chemical information into X-ray refinements to replace the IAM is an important aspect of quantum crystallography. 23 , 24 Within this framework, Hirshfeld Atom Refinement (HAR) [START_REF] Jayatilaka | X-ray structure refinement using aspherical atomic density functions obtained from quantum-mechanical calculations[END_REF][START_REF] Capelli | Hirshfeld atom refinement[END_REF] has been shown to be able to locate hydrogen atoms for small molecules from routine X-ray data with the same accuracy and precision as from neutrondiffraction experiments [START_REF] Woińska | Hydrogen atoms can be located accurately and precisely by X-ray crystallography[END_REF] and now is accessible in standard crystallographic refinement software. [START_REF] Fugel | Probing the accuracy and precision of Hirshfeld atom refinement with HARt interfaced with Olex2[END_REF] Since it is based on regularly updated full wavefunction calculations, it is too slow to be applied to proteins or molecules containing heavy elements, the two compound classes that have been addressed above and that bear the most pressing problems of crystallography.

For proteins, quantum-based refinements have been realized by utilizing restraints calculated quantum mechanically. [START_REF] Zheng | quantum-based refinement[END_REF][START_REF] Ryde | Quantum Chemical geometry optimizations in proteins using crystallographic raw data[END_REF][START_REF] Yu | Refinement of protein crystal structures using energy restraints derived from linear scaling quantum mechanics[END_REF] In this work, we introduce a new method of quantum crystallographic refinement (HAR-ELMO) that combines HAR with the recently constructed libraries of extremely localized molecular orbitals (ELMOs) that rapidly provide wavefunctions of large and complicated molecules [START_REF] Meyer | Libraries of Extremely Localized Molecular Orbitals. 3. Construction and Preliminary Assessment of the New Databanks[END_REF] . In fact, it has been shown previously that ELMOs for bonds and functional groups exhibit the same transferability from model compounds to proteins (or any other organic framework) as multipoles do, but with the significant advantage of allowing the reconstruction of full wavefunctions for the target systems. 33,[START_REF] Meyer | Libraries of Extremely Localized Molecular Orbitals. 2. Comparison with the Pseudoatoms Transferability[END_REF] Here, we apply the new HAR-ELMO strategy to the amino acid L-alanine (L-Ala) [START_REF] Destro | A low-temperature (23 K) study of L-alanine[END_REF] and the dipeptide glycyl-L-alanine (Gly-L-Ala) [START_REF] Capelli | Hirshfeld atom refinement[END_REF][START_REF] Capelli | Glycyl-L-alanine: a multitemperature neutron study[END_REF] to validate it against the original HAR technique and against suitable neutron-diffraction studies. Subsequently, we apply it to the polypeptide Leu-enkephalin, [START_REF] Pichon-Pesme | Modeling fragments for the ab initio determination of electron density in polypeptides. An experimental and theoretical approach to the electron distribution in Leu-enkephalin trihydrate[END_REF] the fibril-forming segment of the human prion protein [START_REF] Sawaya | Atomic structures of amyloid cross-β spines reveal varied steric zippers[END_REF] and to crambin [START_REF] Jelsch | Accurate protein crystallography at ultra-high resolution: valence electron distribution in crambin[END_REF] to provide the first full and complete multipole-free quantum crystallographic refinement of a protein. This opens the door for a new research stream in protein crystallography. To demonstrate the potential applicability of HAR-ELMO in the field of coordination chemistry (e.g., organometallic compounds and metalloproteins), we apply it to the two coordination compounds (Ph3P)2Hg(NO3)2 and (Ph3P)2Hg(CN)2, which are the first two fully quantum crystallographic refinements of molecules containing heavy elements.

Results. Before applying the novel HAR-ELMO technique to proteins, we validated it on smaller systems, namely on Gly-L-Ala at five different temperatures (12, 50, 100, 150 and 295 K) and on L-Ala at three different temperatures (23, 100 and 150 K). The results clearly indicate that approximations introduced by the ELMO approach do not reduce the accuracy and precision of the HAR results (see Figure 1 and Supporting Information). In fact, unlike IAM, C-H and N-H bond distances obtained at HAR-ELMO level are as accurate (average values agree within one sample standard deviation (ssd) as shown in Figure 1a) and as precise (ssd values depicted as error bars in Figure 1a approximately agree) as the HAR and neutron results for all temperatures. This also holds for intermolecular hydrogen-bond parameters (Tables S14 andS15 in the Supporting Information). This means that the hydrogen atoms could be located successfully, but using a significantly lower CPU time compared to traditional HAR (see Table 1

below).

HAR-ELMO even allows obtaining hydrogen Anisotropic Displacement Parameters (ADPs) with precision and accuracy similar to that observed for the bond lengths. ADPs for hydrogen atoms are known to be important for the experimental determination of thermodynamic properties. [START_REF] Hoser | Dynamic quantum crystallography: lattice dynamical models refined against diffraction data. II. Applications to L-alanine, naphthalene and xylitol[END_REF] From visual inspection of Figure 1b, there is a qualitative agreement between the hydrogen and non-hydrogen ADPs obtained through neutron, HAR and HAR-ELMO refinements of Gly-L-Ala at 150 K. This is confirmed quantitatively by extensive statistical analyses detailed in the Supporting Information for both compounds at all investigated temperatures. After validation against amino acids and dipeptides, we applied HAR-ELMO to two polypeptides (Leu-enkephalin and the fibril-forming segment of the human prion protein) as well as to the protein crambin at two different resolutions (d=0.54 and 0.73 Å). All the refinements successfully converged, yielding atomic coordinates and atomic displacement parameters for hydrogen atoms. Only in crambin, the constraint of some atomic parameters mainly belonging to the outermost and flexible loop of the protein was required because of disorder (see the Methods section and the Supporting Information for more details). In Figure 2a the HAR-ELMO refined crambin structure is depicted, constituting the first quantum crystallographic structure of a protein. In Figure 2b, we show deformation density maps for a peptide bond fragment and a disulphide bond of crambin, which highlights the capability of the new HAR-ELMO method to describe bonding and lone-pair valence electron density distributions accurately. Residual and deformation densities for other regions of crambin and of the other two polypeptides can be found in the Supporting Information. As a consequence of the improved modelling of bonding effects, the novel HAR-ELMO method is able to determine the positions of hydrogen atoms precisely and accurately, also in situations where they could not be detected previously. This can be seen in Figure 3, where, for the fibril-forming segment of the human prion protein, we show the model of a hydrogen bond interaction in the deposited PDB file (2OL9, Figure 3a) and upon HAR-ELMO refinement (Figure 3b). In the deposited model, all the hydrogen atoms are missing although residual densities clearly indicate their presence (and the presence of related hydrogen bonds), while, after detecting them, we could refine them freely exploiting the IAM and HAR-ELMO models. Modeled by IAM, the indicated N-H bond length is too short, whereas, modeled by HAR-ELMO, it matches the reference neutron bond length of 1.03 Å. [START_REF] Allen | Bond lengths in organic and metal-organic compounds revisited: X-H bond lengths from neutron diffraction data[END_REF] Determination of accurate X-H bond lengths allowed us to calculate intermolecular interaction energies 41 between the fibril strands, which are a key to the investigation of the biological relevance of fibrils. [START_REF] Sawaya | Atomic structures of amyloid cross-β spines reveal varied steric zippers[END_REF] Parallel fibril strands are held together with a total interaction energy of 169 kJ/mol, whereby the electrostatic component and the dispersion component are about equal in size. This means that hydrogen bonding does not dominate the interaction inside the fibril. Overall, average bond lengths involving hydrogen atoms in the two polypeptides and crambin are significantly elongated with respect to the IAM refinements, and for all C-H bonds and most N-H and O-H bonds they agree with bond lengths derived from neutron diffraction 40 within a single sample standard deviation (see Figure 3c). For O-H and N-H bonds in crambin, the agreement is more problematic, which is independent of the resolution, but only depending on the data quality of protein crystallographic measurements (compare with Figure 1a for smallmolecule crystallography).

The success of the new HAR-ELMO technique with respect to hydrogen atom localization derives from the fact that hydrogen has only one electron, which is always involved in bonding interactions and it cannot be properly described with spherical atomic electron densities, as usually implemented in IAM. Overall, accounting for valence bonding deformations ensures a better agreement of the model with the measured diffraction pattern as long as the quality of the X-ray measurements includes this information. Table 1 shows, in terms of crystallographic R values, that this is the case for Gly-L-Ala, L-Ala, Leu-enkephalin and the mercury complexes that will be discussed below. For Leu-enkephalin, the improvement is significant (from RIAM=0.0557 to RHAR-ELMO=0.0430), and Figure S87 (see Supporting Information) shows that this is due to residual electron density in the bonding and lone-pair regions after IAM. For crambin and the fibril forming segment of the human prion protein, the quality of the measurements is not sufficient to observe valence electron density (see Figures 3a and3b), so there is no improvement in R-values, but only in refined structural parameters and in electronic properties contained in the wavefunction. The more sensitive 𝜒 2 values in Table 1 show that HAR is always in slightly better agreement with the measured diffraction pattern than HAR-ELMO because of the approximations used in the ELMO procedure, but this difference does not reduce the accuracy of geometrical and derived properties as shown above. This is important since, for crambin, and consequently every larger molecules, HAR is impossible or impractical because it is too computationally demanding requiring the recalculation of the wavefunction after every refinement step (see Table 1). Therefore, HAR-ELMO renders quantum crystallographic refinement possible for protein crystallography. The CPU time depends on the size of the molecule (total number of atomic orbitals used in the calculation) and on the number of structure factors used in the least-squares refinement. For crambin, the number of reflections in the higher-resolution data set (d=0.54Å, 96139 refl.) is more than twice as high as the number in the lower-resolution data set (d=0.73Å, 45265 refl.). The HAR-ELMO CPU time reduces from almost 10 days to about 6 days (see Table 1) from the higher to the lower resolution data set.

The reduction of CPU time from HAR to HAR-ELMO for the two coordination compounds (Ph3P)2Hg(NO3)2 and (Ph3P)2Hg(CN)2 is significant (22-and 13-fold reduction, Table 1). Figure 4b also shows that the hydrogen atom positions and ADPs in the phenyl groups of (Ph3P)2Hg(NO3)2 are accurately determined, even with the heavy mercury dominating the scattering. A comparison of refinement results with isotropic hydrogen atom displacement parameters and with (Ph3P)2Hg(CN)2 is given in the Supporting Information.

In the IAM refinement (Figure 4b), there is a significant amount of unmodelled physically meaningful residual density in the C-C, C-P, N-O, Hg-O and Hg-P bonds as well as in the Hg core region. The maximum unmodelled electron density peak in the Hg core is +2.1 eÅ -3 . The HAR-ELMO refinement accounts for the deformation of the electron density caused by bonding, so unmodelled features are absent from the residual-density map after HAR-ELMO (Figure 4c). The maximum unmodelled core Hg density reduces to +1.6 eÅ -3 , a significant reduction showing the importance of modelling the aspherical core deformation of heavy mercury. Similar, but less pronounced, features can be seen for (Ph3P)2Hg(CN)2, described in the Supporting Information (maximum residual density for this compound being +1.3 eÅ -3 (IAM) and +1.1 eÅ -3 (HAR-ELMO)). HAR gives very similar results in terms of residual electron density (+1.7 and +1.1 eÅ -3 for (Ph3P)2Hg(NO3)2 and (Ph3P)2Hg(CN)2, respectively). The residual density distribution after HAR-ELMO shown in Figure 4c is not free of systematic effects. Unmodelled effects remain at the Hg and P cores, and there are two ghost peaks that are not due to aspherical bonding deformation. The HAR-ELMO treatment allows examination of these additional effects. The ghost peaks may be Fourier truncation errors that we could not remove in the data reduction procedure. The features at the Hg core are physical effects that the HAR-ELMO model (including anomalous dispersion and absorption correction) did not resolve. The most obvious problems in our theoretical ansatz are the neglect of electron correlation and relativistic effects. Therefore, we performed fully relativistic HARs at the B3LYP-IOTC level [START_REF] Bučinský | Importance of Relativistic Effects and Electron Correlation in Structure Factors and Electron Density of Diphenyl Mercury and Triphenyl Bismuth[END_REF] , which is currently not possible for HAR-ELMO. This refinement took 8 days and 19 hours, and only reduced the positive maximum residual density to 1.15 eÅ -3 , still leaving very similar systematic features at the Hg core (see Figure S97, Supporting Information). Other possible effects could be anharmonicity, insufficiently modelled absorption or radiation damage. All these effects can be investigated in detail in future projects because the fast HAR-ELMO refinements allowed us to separate deformation due to bonding from all the other important effects.

Discussion. Considering all the results shown above, the HAR-ELMO structural model includes chemically meaningful bonding and non-bonding features, in contrast to IAM. As long as the quality of the diffraction experiment is high enough to detect such features, the new HAR-ELMO refinement method leads to significantly improved agreements between the model and the measured diffraction data, as enumerated by reduced values of figures of merit (R values, 𝜒 2 , residual density distributions). When the quality of the measurements is not high enough, the intrinsic quantum-mechanical nature of the underlying model nevertheless allows the derivation of accurate experimental geometric parameters even in cases where ADPs become non-positive definite in free HAR and free HAR-ELMO refinements, and can be used to obtain properties that are not restricted to those associated with the electron density of the system under consideration. Accessible properties are, for example, bonding descriptors such as the Electron Localizability Indicator (ELI-D) [START_REF] Kohout | A measure of electron localizability[END_REF] , electrostatic potentials (ESPs), noncovalent interaction (NCI) plots [START_REF] Johnson | Revealing noncovalent interactions[END_REF][START_REF] Contreras-García | NCIPLOT: a program for plotting noncovalent interaction regions[END_REF][START_REF] Arias-Olivares | NCI-ELMO: A New Method To Quickly and Accurately Detect Noncovalent Interactions in Biosystems[END_REF] , or inter-and intra-molecular interaction energies, all of which are important for rational design. [START_REF] Morra | Molecular Recognition and Drug-Lead identification: What Can Molecular Simulations Tell Us?[END_REF][START_REF] Genoni | Ligand selection from the analysis of protein conformational substates: new leads targeting the N-terminal domain of Hsp90[END_REF]49 Figure 5 shows the deformation density, ESP and the ELI-D for the entire crambin molecule, calculated rapidly on graphics cards using our own GPU-accelerated software cuQCT, based on the wavefunction and geometry from the last HAR-ELMO step.

For all the mentioned properties, accurate hydrogen atom positions are imperative.

This study shows that HAR-ELMO can detect hydrogen atoms and locate them accurately and precisely, even for those large systems for which a fully quantum crystallographic refinement was previously considered prohibitive.

Notwithstanding the promise of the current results, the major drawback associated with protein refinements is solvent and intrinsic disorder. At the moment, there is no refinement engine underlying original quantum crystallography methods such as HAR that can treat this problem in a proper crystallographic way. It will be of utmost importance to develop and implement a novel strategy to refine disorder in HAR-ELMO or couple HAR-ELMO to existing software for protein refinement that already provides the solution.

Although nowadays the number of exceptionally high-resolution protein structures is increasing, our study shows that the results of the HAR-ELMO strategy do not depend on the highest resolution (0.85 Å is sufficient), but, rather, on the quality of the data. To improve the current version of the HAR-ELMO technique, on the one hand, we have already envisaged the implementation of a parallelized version of the presented HAR-ELMO procedure, also introducing linear-scaling options for the computation of one-electron density matrices. This will result in a further and drastic reduction of the computational cost.

On the other hand, we are also developing a multi-scale HAR-QM/ELMO technique. It will consist in coupling HAR with the QM/ELMO method, [START_REF] Macetti | Quantum Mechanics/Extremely Localized Molecular Orbital Method: a Fully Quantum Mechanical Embedding Approach for Macromolecules[END_REF] a recently developed embedding strategy that describes the most chemically/biologically relevant part the system (e.g., the active site of a protein or the heavy element in coordination compounds) at a very high QM level (e.g., including dispersion for weak intermolecular interactions in enzyme pockets, or core excitations and relativity for heavy elements), while treating the rest of the system with fixed ELMOs transferred from the databanks. This will increase the accuracy of the results for the regions of particular interest while keeping the computational cost within reasonable limits.

The HAR-ELMO refinement captures aspherical electron-density features, such as bonds, lone pairs and the core deformation of heavy elements, in a rapid and automated way. This allows the separation of such physical effects from unmodelled systematic effects, which are smaller but still significant. These unmodelled effects can be of two different natures: i) effects that are related to the adopted model (e.g., relativity, basis-set dependency and anharmonicity); ii) effects related to the measurement and data reduction strategy. To tackle problem (i) we propose to apply the above-mentioned HAR-QM/ELMO strategy. Furthermore, we can exploit the speed of HAR-ELMO and HAR-QM/ELMO in order to use them as debugging tools to address problem (ii). In this way, changes in data-reduction routines (for instance, absorption correction or mitigation of radiation damage) can directly be mapped on the electron density.

While traditional refinement strategies normally place hydrogen atoms according to valence considerations and not according to measured electron-density peaks and while the pioneering and innovative QM/MM refinements of proteins [START_REF] Zheng | quantum-based refinement[END_REF][START_REF] Ryde | Quantum Chemical geometry optimizations in proteins using crystallographic raw data[END_REF][START_REF] Yu | Refinement of protein crystal structures using energy restraints derived from linear scaling quantum mechanics[END_REF] are still based on the use of tabulated spherical atomic form factors, the new HAR-ELMO technique introduces for the first time the explicit quantum mechanical treatment of the electron density asphericity in the computation of structure factors for the refinement of macromolecular crystal structures. This adds unprecedented insights into protein structure determination. Moreover, for the chemicalcrystallographic community, a major advantage of the new approach also lies in the future possibility of extending the ELMO libraries to very different classes of compounds, such as protein inhibitors or ligands that regularly occur in syntheses of, for example, metal and covalent organic frameworks (MOFs and COFs). In particular, if ligands are sterically demanding or contain heavy elements in a closed-shell state, resorting to such ELMO libraries will render quantum crystallographic refinements possible for high-throughput investigations of chemical bonding features.

In conclusion, two classes of compounds can now be modelled by quantum crystallography thanks to the developments presented here: proteins and molecules containing heavy elements. For large molecules, the treatment of the lightest atoms is crucial; for small molecules, the treatment of the heaviest elements is the biggest challenge. HAR-ELMO is today the only method that allows fully quantum-crystallographic refinements of both such very different compound types on a reasonable timeframe. This reduction of computational cost derives from the introduction of the ELMO approximation in the procedure, but, despite this, accuracy and precision are only marginally affected. In future work, we will exploit the new possibilities in macromolecular chemistry to investigate bonding situations in metalloproteins and for enzyme-ligand binding as well as in chemical crystallography to accurately model bonding to heavy elements and understand unusually close intermolecular contacts, e.g. in halogen and hydrogen bonding motifs. using the Laue diffraction technique. [START_REF] Piltz | First results from the KOALA neutron Laue instrument[END_REF] KOALA is equipped with a large cylindrical image plate, so that a resolution of 0.50Å was reached in the wavelength range 0.750 to 1.500 Å, refined to R-values R(F)= 0.033/0.038. Other than in the Supporting Information, more details are in the Cambridge Structural Database (CSD) under nos. CCDC-1917810-1917812, 1917816-1917818, 1917820-1917822, 1917823-1917837 (X-ray) and CCDC-1917685, 1917813 (neutron).

Protein refinement. The experimental data of crambin were taken from the literature. [START_REF] Jelsch | Accurate protein crystallography at ultra-high resolution: valence electron distribution in crambin[END_REF] In the available PDB file, there was an unspecified and unresolved amount of disordered water molecules. Since disorder for HAR, HAR-ELMO or any other refinement belonging to the original definition of quantum crystallography is currently an unsolved problem, we used the SQUEEZE routine [START_REF] Spek | SQUEEZE: a tool for the calculation of the disordered solvent contribution to the calculated structure factors[END_REF] to calculate the scattering contributions of the solvent molecules and excluded them from the original structure factors. These structure factors were used for fullmatrix least-squares IAM and HAR-ELMO refinements in TONTO at two resolutions, 0.54 and 0.73Å. Parameters of atoms belonging to lower occupancy regions in the reported crambin structure were removed and only the major component atoms were included at full occupancy. Positional and displacement parameters were freely refined for all atoms but those in the disorder regions and those of some atoms belonging to the outermost flexible loop of crambin. For those atoms, positions and displacement parameters were kept fixed at values obtained from an initial unconstrained IAM refinement.

The original structure in the 2OL9 PDB file for the fibril-forming segment of the human prion protein is resolved up to 0.85 Å, contains ordered oxygen sites of water solvent molecules, but no hydrogen atoms. [START_REF] Sawaya | Atomic structures of amyloid cross-β spines reveal varied steric zippers[END_REF] Before the initial IAM refinement, all hydrogen atoms were geometrically placed, and then refined freely in the subsequent HAR and HAR-ELMO refinements. The experimental data of Leu-enkephalin 37 are resolved to d=0.43Å, and all atoms are present and ordered in the original file. The reference IAM refinement was performed in the software TONTO, and followed by both HAR and HAR-ELMO treatments.

The files associated with the polypeptide refinements correspond to CSD depositions CCDC-1917596-1917598, 1917730-1917732. Small-molecule refinement involving heavy elements. The two coordination compounds (Ph3P)2Hg(NO3)2 and (Ph3P)2Hg(CN)2 are known from literature, 55 but were re-synthesized according to the literature procedures, crystallized and measured at the same X-ray setup described above to resolutions of 0.50 and 0.55Å, respectively. IAM, HAR and HAR-ELMO refinements were performed in TONTO. For HAR and HAR-ELMO, two models were adopted, one with anisotropic and one with isotropic hydrogen displacement parameters.

(Ph3P)2Hg(NO3)2 is the only compound in this study with half a molecule in the asymmetric unit (Z'=0.5), whereas for all the others Z'=1.

ASSOCIATED CONTENT

Supporting Information

Theoretical bases of the HAR-ELMO technique, details about the validation of the HAR-ELMO method, details and further results of the polypeptides and protein refinements, details and further results of the refinements of the coordination compounds. (PDF)

Crystallographic information files (cif files) and files containing experimental and calculated structure factors (fco files) associated with all the refinements (all the considered models) for all the compounds in this study. (ZIP) This material is available free of charge via the Internet at http://pubs.acs.org/. acknowledge Dylan Jayatilaka (University of Western Australia) for helpful discussions, Benoît Guillot (University of Lorraine) for providing experimental data of the Leu-enkephalin polypeptide and Lukas Bučinský (Slovak University of Technology) for providing assistance with the relativistic refinement.

Figure 1 .

 1 Figure 1. a) Accuracy and precision of bond distances involving hydrogen atoms (X-H) for Gly-L-Ala at all temperatures (note that there is no neutron diffraction experiment at 100 K). Estimated accuracy is expressed by a match of the averaged bond lengths with the reference neutron experiment. Estimated precision is expressed by the size of the error bar, which represents the sample standard deviation upon averaging. b) Visual comparison of refinement results of Gly-L-Ala at 150K for neutron, HAR and HAR-ELMO refinements. The remaining comparisons are depicted in the Supporting Information. Mean absolute differences (MADs) relative to the neutron experiment for Gly-L-Ala at 150K: MAD(dX-H,IAM) = 0.128Å, MAD(dX-H,HAR-ELMO) = 0.013Å, MAD(dX-H,HAR) = 0.008Å; MAD(Uij,H,HAR-ELMO) = 0.0079Å 2 , MAD(Uij,H,HAR) = 0.0078Å 2 .

Figure 2 .

 2 Figure 2. a) Refined protein structure of crambin at d=0.54 Å, not showing the disordered regions. For clarity, all H atoms are drawn with fixed spheres of 0.3 Å radius. Different representations for crambin at both selected resolutions and for the polypeptides are given in the Supporting Information. b) Deformation density maps of crambin in a peptide region (C=O in Leu25) and a disulphide bond (between Cys4 and Cys32). Contour interval: 0.05 eÅ -3 , blue=positive, red=negative, green=zero. For a) and b) ellipsoids at 50% probability.

Figure 3 .

 3 Figure 3. Residual electron density of the investigated fibril forming segment (original PDB structure: 2OL9) in a region of intermolecular interaction between two parallel strands (namely, hydrogen-bond N2-H2... O2=C3 [according to the deposited CIF labels] between residues Asn2 and Ser1): a) IAM without hydrogen atoms as deposited in the original PDB, b) HAR-ELMO with refined hydrogen atoms, including the IAM and HAR-ELMO refined N-H distances. For (a) and (b), contour interval=0.05 eÅ -3 , blue=positive, red=negative, green=zero. For a) and b) ellipsoids at 50% probability. c) Average values and associated standard deviations of C-H, O-H and N-H bond lengths in IAM, HAR (if possible) and HAR-ELMO structures of the fibril-forming segment, Leu-enkephalin and crambin compared to average values from neutron diffraction. 40

Figure 4 .

 4 Figure 4. Refined structures of a) (Ph3P)2Hg(NO3)2 with residual electron density distribution and minimum/maximum value b) after IAM and c) after HAR-ELMO. Isovalue = ± 0.2 eÅ -3 , green=positive, red=negative. ADPs at 50% probability level. The two nitro and triphenylphosphine groups are symmetry-equivalent with their counterpart leading to six-fold coordination of Hg. Distances after HAR-ELMO: Hg-P = 2.4502(5)Å, Hg-O = 2.4692(7), Hg-O' = 2.8446(9).

Figure 5 .

 5 Figure 5. cuQCT GPU-accelerated calculations of molecular properties for crambin based on the final HAR-ELMO geometry and wavefunction: a) isosurfaces of deformation density at ±0.04 eÅ -3 (blue = positive, red = negative); b) Electrostatic potential in a range from -0.1 to +0.1 e bohr -1 mapped on an electron-density isosurface of 0.001 e bohr -3 ; c) ELI-D at an isovalue of 2.2 for the main picture and 2.0 for the magnified details. All grids were calculated within 2 hours on a normal desktop computer with a grid-point separation of 0.1 bohr, except for the ESP grid which took about 500 hours at a 0.6 bohr separation. The program cuQCT is a software for the rapid calculation of large and dense grids of molecular properties on GPUs developed by some of us at the Universities of Bremen and Bern.

Table 1 .

 1 CPU wall-clock timing and refinement statistics. The values in brackets refer to the IAM Rvalues obtained after the refinement with the software TONTO. For Gly-L-Ala and L-Ala, only the 150K measurements are given as examples (complete details are given in TableS11of the Supporting Information).

	Structure	Method	CPU wall time	R1(F)	χ²
			dd:hh:mm:ss
		HAR	00:00:12:57	0.0161	2.336
	Gly-L-Ala (150K)	HAR-ELMO	00:00:03:36	0.0168	2.725
				(0.0251)
		HAR	00:00:05:55	0.0202	3.152
	L-Ala (150K)	HAR-ELMO	00:00:02:25	0.0210	3.523
				(0.0279)
		HAR	00:09:52:00	0.0422	0.505
	Leu-Enkephalin	HAR-ELMO	00:01:44:17	0.0430	0.545
				(0.0557)
		HAR	01:07:00:00	0.0436	9.903
	Fibril forming segment	HAR-ELMO	00:00:22:56	0.0474 11.913
				(0.0446)
		HAR	Impossible	-	-
	Crambin (d=0.54Å)	HAR-ELMO	09:23:47:53	0.0715	5.004
				(0.0704)
		HAR	Impossible	-	-
	Crambin (d=0.73Å)	HAR-ELMO	06:00:15:16	0.0624	7.672
				(0.0618)
		HAR	00:11:00:00	0.0162	4.232
	(Ph3P)2Hg(NO3)2	HAR-ELMO	00:00:31:47	0.0162	4.400
				(0.0204)
		HAR	00:13:30:00	0.0187	2.603
	(Ph3P)2Hg(CN)2	HAR-ELMO	00:01:09:00	0.0187	2.622
				(0.0215)

ACKNOWLEDGMENTS S. Grabowsky thanks the German Research Foundation (Deutsche Forschungsgemeinschaft DFG) for funding within the projects GR 4451/1-1 and GR 4451/2-1, J. Beckmann within BE 3716/7-1 and E. Hupf for a research fellowship HU 2512/1-1. A. Genoni acknowledges the French Research Agency (ANR) for financial support of the Young Researcher Project QuMacroRef through Grant No. ANR-17-CE29-0005-01. The neutron-diffraction experiments of L-alanine were performed on the KOALA beamline of the OPAL reactor under the Australian Centre for Neutron Scattering proposal no. 6049. The authors

DATA AVAILABILITY

The best-quality models for all compounds obtained in the present study (except those for crambin) have been deposited at the Cambridge Crystallographic Data Centre (CCDC) under the deposition numbers indicated in the methods section. This data can be obtained free of charge via www.ccdc.cam.ac.uk/data_request/cif. All models for all refinements (including the inferior models and including the protein crambin) plus the measured structure factors have been deposited with this journal in addition to the CCDC deposition to ensure reproducibility of the study. The lamaGOET GUI can be obtained free of charge for registered users at http://www.tinyurl.com/lamaGOET. All other programs developed for the present study are available from the corresponding authors of the paper upon motivated request.

COMPUTATIONAL AND EXPERIMENTAL METHODS

In this section, all the computational and experimental methods used in the present investigation are briefly presented. More details are given in the Supporting Information.

HAR-ELMO implementation. The HAR-ELMO technique was implemented interfacing the HAR functionalities in the software TONTO [START_REF] Jayatilaka | Tonto: A Fortran Based Object-Oriented System for Quantum Chemistry and Crystallography[END_REF] with the ELMOdb program associated with the ELMO libraries. [START_REF] Meyer | Libraries of Extremely Localized Molecular Orbitals. 3. Construction and Preliminary Assessment of the New Databanks[END_REF] The interface between the programs is the lamaGOET graphical user interface (GUI), which reads the initial geometry (PDB, CIF or XYZ file) from a previous IAM refinement and provides it to ELMOdb for the automatic transfer of molecular orbitals.

After the transfer, the wavefunction and first-order density matrix of the system are computed and written to a formatted checkpoint file, which is afterwards passed to TONTO by lamaGOET together with the ADPs of the IAM refinement, in order to perform the first HAR iteration. The updated geometry and ADPs are used as input for iteratively repeating the procedure until convergence. Although interfaced, lamaGOET and ELMOdb are currently two independent software. The former is available free of charge at the link: http://www.tinyurl.com/lamaGOET (source code is also available on https://github.com/lomalaspina/lamaGOET). On the contrary, at the moment, the ELMO libraries and the associated ELMOdb program are not publicly available, but they can be obtained upon motivated request to A.G. (Alessandro.Genoni@univ-lorraine.fr). The construction of a dedicated webserver is already envisaged. was performed with the DZP-DKH basis-set. [START_REF] Bučinský | Importance of Relativistic Effects and Electron Correlation in Structure Factors and Electron Density of Diphenyl Mercury and Triphenyl Bismuth[END_REF] All refinements were performed in TONTO 18.12.17 github v. 2b3dd9c.

ELMO libraries. The used ELMO libraries

Validation of HAR-ELMO.

The experimental X-ray [START_REF] Capelli | Hirshfeld atom refinement[END_REF] and neutron 36 data of Gly-L-Ala were obtained from the literature. For L-Ala, the X-ray data at 23K were taken from Destro et al. [START_REF] Destro | A low-temperature (23 K) study of L-alanine[END_REF] while the data at 100K and 150K were measured using a Bruker Venture D8 diffractometer

Notes

The authors declare no competing financial interests.


domain: phys spi spi.fluid

text:
J.-M Stanislas

Foucaut

S Srinath

J C Vassilicos

C Cuvier

J. -P Laval

M Stanislas

J. -M Foucaut

Attached flow structure and streamwise energy spectra in a turbulent boundary layer

Keywords: Turbulent Boundary Layers, wall-attached flow structures, Particle Image Velocimetry 1

HAL is

I. INTRODUCTION:

In the past forty years, the turbulence spectrum of velocity fluctuations in wall turbulence has received considerable attention as it gives valuable insight into the behaviour of wallbounded flows by indicating the distribution of energy across scales. Spectral scaling laws built on ideas initiated by Townsend [START_REF] Townsend | The structure of turbulent shear flow[END_REF] , in particular the attached eddy hypothesis, have seen consistent development over the years (see Refs. Perry & Chong [START_REF] Perry | On the mechanism of wall turbulence[END_REF], Perry et al. [START_REF] Perry | A theoretical and experimental study of wall turbulence[END_REF],

Perry & Li [START_REF] Perry | Experimental support for the attached-eddy hypothesis in zero-pressure-gradient turbulent boundary layers[END_REF], Marusic et al. [START_REF] Marusic | Similarity law for the streamwise turbulence intensity in zero-pressure-gradient turbulent boundary layers[END_REF] and Marusic & Kunkel [START_REF] Marusic | Streamwise turbulence intensity formulation for flat-plate boundary layers[END_REF]). Perry & Abell [START_REF] Perry | Asymptotic similarity of turbulence structures in smooth-and rough-walled pipes[END_REF] and Perry et al. [START_REF] Perry | A theoretical and experimental study of wall turbulence[END_REF] showed how Townsend's attached eddy hypothesis implies that the energy spectrum E 11 (k x ) of the turbulent streamwise fluctuating velocity at a distance y from the wall scales

as E 11 (k x ) ∼ U 2 τ k -1
x in the range 1/δ k x 1/y where U τ is the friction velocity and δ is the boundary layer thickness. Nickels et al. [START_REF] Nickels | Evidence of the k -1 1 law in a high-Reynolds-number turbulent boundary layer[END_REF] stressed the use of overlap arguments to deduce the -1 power law behaviour. That is, a k -1

x region in the spectra would exist where the inner scaling (based on y and U τ ) and outer scaling (based on δ and U τ ) are simultaneously valid over the same wavenumber range. Nickels et al. [START_REF] Nickels | Some predictions of the attached eddy model for a high Reynolds number boundary layer[END_REF] stated that it is necessary to take measurements surprisingly close to the wall to observe a k -1

x behaviour and thought this was the reason why Morrison et al. [START_REF] Morrison | Scaling of the streamwise velocity component in turbulent pipe flow[END_REF] and McKeon & Morrison [START_REF] Mckeon | Asymptotic scaling in turbulent pipe flow[END_REF] did not observe any -1 region in their spectra as their measurements were not close enough to the wall. However, recent experiments by Vallikivi et al. [START_REF] Vallikivi | Spectral scaling in boundary layers and pipes at very high Reynolds numbers[END_REF] do not show an overlap region and these authors infer that the k -1

x region cannot be expected even at very high Reynolds numbers.

The present work looks at the basis for the k -1

x range in flat plate turbulent boundary layers from a new perspective. Using Particle Image Velocimetry (PIV) and a simple model which can in principle be applied to various wall-bounded turbulent flows, we show how, in the turbulent boundary layer, a power-law spectral range exists but is not a Townsend-Perry

k -1
x range and how it can be accounted for by taking only streamwise lengths and intensities of wall-attached structures into account. This paper is organized as follows. In sections II and III we provide a model for the streamwise energy spectrum. The experimental set-up of the flat plate boundary layer is presented in section IV. Our data set is validated in V A and the method for educing the wall-attached flow structures relevant to our model is described in section V B. The main results of the paper are in V C and V D followed by a discussion in V E. We conclude in section VI.

II. A SIMPLE MODEL FOR THE SPECTRAL SIGNATURE OF THE TOWNSEND-PERRY ATTACHED EDDY RANGE OF WAVENUMBERS

As already mentioned in the introduction, Perry & Abell [START_REF] Perry | Asymptotic similarity of turbulence structures in smooth-and rough-walled pipes[END_REF], Perry & Chong [START_REF] Perry | On the mechanism of wall turbulence[END_REF] and Perry et al. [START_REF] Perry | A theoretical and experimental study of wall turbulence[END_REF] showed how Townsend's attached eddy hypothesis implies E 11 (k x ) ∼ U 2 τ k -1

x in the range 1/δ k x 1/y. Perry et al. [START_REF] Perry | A theoretical and experimental study of wall turbulence[END_REF] also developed a flow structure model for this spectral range in terms of specific attached eddies of varying sizes randomly distributed in space and with a number density that is inversely proportional to size. In this paper we attempt to distill such a type of model to its bare essentials. These bare essentials are that flow structures are primarily objects with clear spatial boundaries. In section V we model these boundaries with on-off functions in the expectation that the spectral signature in the attached eddy wavenumber range is dominated by these sharp gradient, effectively onoff, behaviours. The concomitant expectation is that the additional superimposed velocity fluctuations fill the content of a predominantly higher frequency spectral range. In this section we show that the streamwise energy spectrum's k -1

x spectral range can be captured by simple on-off representations of elongated streaky structures of varying sizes as long as their number density has a space-filling power law dependence on size.

We therefore assume that the attached eddies responsible for the k -1

x spectral range have a long streaky structure footprint on the 1D streamwise fluctuating velocity signals at a distance y from the wall. We also assume that these streaky structures can be modeled as simple on-off functions and that it is sufficient to represent the streamwise velocity fluctuations u(x) at a given height y from the wall as follows

u(x) = n,m a nm Π(ξ) (1)
where Π(ξ) = 1 if -1 < ξ < 1 with ξ = x-xnm λn and Π(ξ) = 0 otherwise. The on-off function Π(ξ) is our cartoon model of a streaky structure. Streaky structures of length λ n are centred at random positions x nm and their intensity is given by the coefficients a nm . For each subscript n, the subscript m counts the spatial positions where cartoon structures of size λ n can be centred in a given realisation. The sum in ( 1) is over all structures lengths

λ n and all their positions x nm .

The energy spectrum of u

(x) is E 11 (k x ) = (2π) 2 Lx |û(k x )| 2
where L x is the length of the record, û(k x ) is the Fourier transform of u(x), and the overbar signifies an average over realisations. The Fourier transform of Π(

x-xnm λn ) being Π(k x , λ n , x nm ) = 2ik -1 x e ikxxnm sin(k x λ n ), it follows that û(k x ) = 2ik -1 x nm
a nm e ikxxnm sin(k

x λ n ) (2)
which implies that the energy spectrum is given by

E 11 (k x ) = 4 (2π) 2 L x k -2
x nm

a nm e ikxxnm sin(k

x λ n ) pq a pq e -ikxxpq sin(k x λ p ). (3)
We introduce two assumptions which were also used by Perry et al. [START_REF] Perry | A theoretical and experimental study of wall turbulence[END_REF] in their more intricate model. The first assumption is that the positions and amplitudes of our cartoon stuctures are uncorrelated and that different positions are not correlated to each other either, i.e. e ikxxnm e ikxxpq = δ pn δ qm . As a result, the expression for the energy spectrum simplifies as follows:

E 11 (k x ) = 4 (2π) 2 L x k -2 x nm (a nm ) 2 sin 2 (k x λ n ). (4)
Let us say that there is an average number N n of cartoon stuctures of size λ n centred within an integral scale along the x-axis. The expression for E 11 (k x ) simplifies even further:

E 11 (k x ) = 4 (2π) 2 L x k -2 x n a 2 n N n sin 2 (k x λ n ) (5)
where a 2 n ≡ (a nm ) 2 is the same irrespective of position x nm . We now consider a continuum of different structure sizes λ rather than discrete lengthscales λ n and the previous expression for E 11 (k x ) must therefore be replaced by

E 11 (k x ) = 4 (2π) 2 L x k -2 x dλa 2 (λ)N (λ) sin 2 (k x λ) (6)
in terms of easily understandable notation. At this point we introduce a generalised form of the second assumption which was also used by Perry et al. Vassilicos & Hunt [START_REF] Vassilicos | Fractal dimensions and spectra of interfaces with application to turbulence[END_REF] proved that, if 0 ≤ D ≤ 1, then the set of points defining the edges of the on-off functions Π(ξ) is fractal and D is effectively the fractal dimension of this set of points. The case where this fractal dimension is D = 1 is the case where these points are space-filling. The population density assumption of Perry et al. [START_REF] Perry | A theoretical and experimental study of wall turbulence[END_REF] corresponds to D = 1 which is also the choice we make in this work. We now show that this choice can lead to

E 11 (k x ) ∼ k -1 x in the range 1/λ o k x 1/λ i .
We calculate the energy spectrum by carrying out the integral in [START_REF] Marusic | Streamwise turbulence intensity formulation for flat-plate boundary layers[END_REF]. This requires a model for a 2 (λ) which, in this section, we chose to be as simple as possible and therefore independent of λ in the relevant range, i.e. a 2 (λ) = A 2 /δ for λ i < λ < λ o where A 2 is a constant. Using our models for N (λ) and a 2 (λ) and the change of variables λk x = l, (

becomes

E 11 (k x ) = A 2 δ(C o (k x δ) -2+D -C M (k x δ) -2 ) (7)
where

C o = 4(2π) 2 N o δ L x λokx λ i kx dl sin 2 (l)l -1-D
and

C M = 4(2π) 2 N M δ L x (k x δ) -1 λokx λ i kx dl sin 2 (l)
which is bounded from above by N M Lx λo-λ i δ . In the attached eddy range 1/λ o k x 1/λ i ,

C o ≈ 4(2π) 2 No Lx ∞ 0 dl sin 2 (l)l -1-D which means that C o is approximately independent of k x in this range.
Substituting the value D = 1 in equation ( 7), we get

E 11 (k x ) = A 2 (C o k -1 x -C M δ -1 k -2
x ) which is well approximated by

E 11 (k x ) ≈ C o A 2 k -1 x (8) for wavenumbers k x δ C M /C o (i.e. C o k -1 x C m δ -1 k -2 x ). Note that C M /C o is
A 2 ∼ U 2 τ .

III. A STRAIGHTFORWARD GENERALISATION

It is worth generalising the previous section's model by assuming that a 2 (λ) is not constant but varies with λ in the range λ i < λ < λ o , for example as a 2 (λ) = (A 2 /δ)(λ/δ) p where p is a real number with bounds which we determine below. The arguments of the previous section can be reproduced till equation ( 6) which now becomes

E 11 (k x ) = A 2 δ[c o (k x δ) -2+D-p -c M (k x δ) -2 ] ( 9
)
where

c o = 4(2π) 2 N o δ L x λokx λ i kx dl sin 2 (l)l -1-D+p and c M = 4(2π) 2 N M δ L x (k x δ) -1-p λokx λ i kx dl l +p sin 2 (l) which is bounded from above by Nm (1+p)Lx [( λo δ ) 1+p -( λ i δ ) 1+p ]. In the attached eddy range 1/λ o k x 1/λ i , c o ≈ 4(2π) 2 No Lx ∞ 0 dl sin 2 (l)l -1-D+p which means that c o is approximately independent of k x in this range if 0 < D -p < 2.
Substituting the value D = 1 in (9), we obtain the following leading order approximation in the parameter range -1 < p < 1:

E 11 (k x ) ≈ c 0 A 2 δ(k x δ) q (10)
where

p + q = -1 (11)
for wavenumbers

k x δ (c M /c o ) 1 1-p . Note that c M /c o is much smaller than 1 if p is not too close to 1 because N M is much smaller than N o .
The spectral shape [START_REF] Morrison | Scaling of the streamwise velocity component in turbulent pipe flow[END_REF] and Re θ = 20600 respectively.

V. RESULTS AND DISCUSSION

A. Validation of experimental data is present in the range 100 < y + < 300 for our higher Reynolds number case. Close to the wall, the u + values obtained from our PIV are slightly underestimated, in particular for Re θ = 20600, demonstrating some filtering of the PIV at this resolution (Foucaut et al.

[16]). To compute from PIV the energy spectra used in this paper, we used the method of Foucaut et al. [START_REF] Foucaut | PIV optimization for the study of turbulent flow using spectral analysis[END_REF]. As seen in figure 2 The raw instantaneous streamwise velocity fields are affected by noise so that single structures in figure 3(a) appear split in many little parts. To smooth out these structures without modifying their shape and statistics we used a two-dimensional Gaussian filter. This filtering operation was found to be sufficient to capture and connect the structures while To educe on-off functions such as the ones required by our model we apply a threshold u th on the gaussian-filetered u * to obtain binary images which distinguish between u * < u th and u * > u th . Effects of the threshold on the statistics of educed structures were investigated in the range 0.1u 300 + < |u th | < u 300 + where u 300 + is u at y + = 300.

A threshold u th equal to -0.4u 300 + was finally chosen to detect low momentum structures in the present study as it corresponds to the value that leads to least threshold-dependency of our statistics for a negative u th (for example, u th equal to -0.2u 300 + or -0.6u 300 + return results with no significant difference, see Appendix A). This thresholding operation leads to of lengths λ at various wall distances. The most probable length λ lies between 0.3δ and 0.5δ and lengths λ longer than 3.5δ occur very rarely.

We tested for finite size effects of the field of view by computing the PDF on smaller domains, namely 3.5δ and 3δ long in the streamwise direction but same in the wall normal direction. As shown in figure 5 there is no significant differences caused by the three fields of view except that the smallest field returns a slightly more noisy PDF. Indeed, a reduced field of view leads to a smaller number of detected wall-attached elongated binary structures and therefore to reduced statistical convergence.

Figures 5 and6 show a power law dependence on λ between about 0.5δ and 2δ with

D. Energy spectra

Figure 7 shows log-log plots of premultiplied energy spectra of streamwise fluctuating velocities u(x) which have been obtained from our PIV data at various normalised wall distances y + for both Reynolds numbers. These plots might suggest that

E 11 (k x ) ∼ U 2 τ k -1
x in a range of wavenumbers 2π/(4δ) k x 0.63/y for y + larger than about 88 and smaller than the value of y + where this range of wavenumbers no longer exists. The apparent k -1

x wavenumber range is close to a decade long at y + = 90 for Re θ = 20600 and shorter for higher wall normal distances and for the lower Reynolds number (Re θ = 8100). One would be justified to conclude that this is indeed experimental support for the Townsend-Perry

k -1
x spectrum if the only available theoretical glasses through which to look at these spectral plots were those of the Townsend-Perry attached eddy model. However the situation is subtler and, in effect, quite different.

A closer look at the spectra in the lin-log plot of figure 8 suggests the possibility for small corrections to this conclusion, particularly at the lower of the two Re θ values, but the result ( 10)-( 11) of our model in section III may pave the way for a significantly different interpretation. This model leads to As an aside for the moment, note that the large-scale motions (LSMs) and very large-scale motions (VLSMs), which have been found to exist in the logarithmic and lower wake regions of a turbulent boundary layer (see Kovasznay et al. [START_REF] Kovasznay | Large-scale motion in the intermittent region of a turbulent boundary layer[END_REF], Brown & Thomas [START_REF] Brown | Large structure in a turbulent boundary layer[END_REF], Hutchins & Marusic [START_REF] Hutchins | Evidence of very long meandering features in the logarithmic region of turbulent boundary layers[END_REF], Dennis & Nickels [START_REF] Dennis | Experimental measurement of large-scale threedimensional structures in a turbulent boundary layer. Part 1. Vortex packets[END_REF] and Lee & Sung [START_REF] Lee | Very-large-scale motions in a turbulent boundary layer[END_REF]) generally refer to elongated regions of streamwise velocity fluctuations having a streamwise extent from about 2δ to 3δ for LSMs and larger than 3δ for VLSMs (see Kim & Adrian [23], Guala et al. [START_REF] Guala | Large-scale and very-large-scale motions in turbulent pipe flow[END_REF] and Balakumar & Adrian [START_REF] Balakumar | Large-and very-large-scale motions in channel and boundary-layer flows[END_REF]). The LSMs near the wall and the VLSMs have been interpreted as being responsible for the k -1

E 11 (k x ) ∼ (k x δ) q with p + q = -1 if D = 1. Support
x scaling range of the turbulence spectrum (Smits et al. [START_REF] Smits | High-reynolds number wall turbulence[END_REF]). The range of scales we concentrate on, in figures 9 to 12, just about includes some LSMs at its upper range.

Returning now to figures 9 to 12, we have included best fits of power law curves in the plots of a 2 versus λ/δ and of E 11 versus k x . These best fits are indicated in the inserts of each plot and provide an estimation of the exponents p and q in a 2 ∼ (λ/δ) p and E 11 (k x ) ∼ k q

x . Figure 13 summarizes the information with plots of p, q and p + q as functions of y + . It is perhaps remarkable that p + q is very close to -1 (see figure 13) as predicted by ( 10)- [START_REF] Mckeon | Asymptotic scaling in turbulent pipe flow[END_REF] for all examined values of y + and for both Reynolds numbers Re θ . Whereas this subsection's initial interpretation in terms of the Townsend attached eddy model is limited to y + larger or equal to 88 (based on the log-log plots of figure 7), the lin-lin plots of figures 9 to 12 present a different and consistent picture which covers both Reynolds numbers and all our y + positions, including y + smaller than 88. This picture is confirmed by Hot Wire Anemometry (HWA) data from a turbulent boundary layer in the same wind tunnel by Tutkun et al. [START_REF] Tutkun | Two-point correlations in high reynolds number flat plate turbulent boundary layers[END_REF] and from the recent Direct Numerical Simulations (DNS) of a turbulent channel flow at Re τ = 5200 by Lee & Moser [START_REF] Lee | Direct numerical simulation of turbulent channel flow up to Re τ 5200[END_REF]. Indeed, these HWA and DNS data show the same variation of the spectral exponent q with y + that we found from using ( 10)- [START_REF] Mckeon | Asymptotic scaling in turbulent pipe flow[END_REF] on our PIV data; see figure 14, and also figure 13(b) where we collected the values of q from different data. The HWA data, in particular, provide a confirmation of our PIV results because they extend to a wider range at the lower end of wavenumbers (see figures 14(a) and also figure 13(b) where it is shown that the HWA's extended wavenumber range returns effectively same values of q). The much higher Reynolds number measurements of Vallikivi et al. [START_REF] Vallikivi | Spectral scaling in boundary layers and pipes at very high Reynolds numbers[END_REF] did not find support for the Townsend-Perry k -1 x spectrum either. However, these authors did find some agreement with the k -1

x log(8π/k x y) spectrum model of del Álamo et al. [START_REF] Del Álamo | Scaling of the energy spectra of turbulent channels[END_REF]. This approximate agreement was found in a range of wall-normal distances where we find positive values of p, i.e. in a region where the spectrum scales as k -1

x k -p x with values of p above but close to 0. It is quite difficult to distinguish between such a weak power law and log(8π/k x y), so the two models qualitatively agree in this range of wall-normal distances. However, the model of del Álamo et al. [START_REF] Del Álamo | Scaling of the energy spectra of turbulent channels[END_REF] cannot account for the scaling of the energy spectrum at closer distances to the wall where we find p ≤ 0, whereas our model fits the data in this region too.

E. Discussion

It is important to stress that the support for (10)- [START_REF] Mckeon | Asymptotic scaling in turbulent pipe flow[END_REF] in figures 9 to 12 cannot be obtained without the crucial last step of our structure detection algorithm in subsection V B which discards structures that are not attached to the wall. The structures which do not touch the wall are in fact less elongated and less intense (i.e. smaller a 2 ) on average. We have checked that if we only consider them, we do not find anything close to p + q = -1, i.e. [START_REF] Mckeon | Asymptotic scaling in turbulent pipe flow[END_REF].

The attached eddy concept introduced by Townsend [START_REF] Townsend | The structure of turbulent shear flow[END_REF] is therefore important for explaining E 11 (k x ) but the results of our analysis suggest that the Townsend-Perry model does x for the present PIV data, the HWA turbulent boundary layer data of Tutkun et al. [START_REF] Tutkun | Two-point correlations in high reynolds number flat plate turbulent boundary layers[END_REF] (see figure 14(a) and the DNS of turbulent channel flow data of Lee & Moser [START_REF] Lee | Direct numerical simulation of turbulent channel flow up to Re τ 5200[END_REF] (see figure 14 (c) p + q versus y + . All these fits are obtained over the range of scales investigated in figures 9 to 12 (except for the HWA case in (b) where we have also included a fit over a range of length-scales extended up to 7δ). The resulting exponents are plotted with the 95% confidence intervals for these fits. not hold without some significant corrections because the turbulent kinetic energy content in these wall-attached flow structures does not just scale with U τ . (If it did, a 2 would scale with U τ and p would be uniformly 0.) At different y inside such a structure, the level of tur-bulent kinetic energy depends both on U τ and on the streamwise length of the structure at that height. Furthermore, this dependence varies with height: a 2 decreases with increasing λ/δ very close to the wall, in the buffer layer, and increases with increasing λ further up.

As a 2 transits smoothly from one dependence to the other, a particular height exists where a 2 is independent of λ and therefore depends only on U τ . At that very particular height,

E 11 (k x ) ∼ k -1
x . However, strictly speaking, this is not a Townsend-Perry spectrum, it is just the spectrum at that particular distance from the wall where the turbulent kinetic energy inside the streaky structures transits from a decreasing to an increasing dependence on the length of these structures. Our conclusion agrees with Nickels et al. [START_REF] Nickels | Some predictions of the attached eddy model for a high Reynolds number boundary layer[END_REF] in their statement that it is necessary to take measurements close to the wall to observe a k -1

x behaviour, in fact at y + between 100 and 200 as they also found. However, these authors were not in possession of ( 10)-( 11) and therefore did not measure a 2 at various heights and for various values of λ/δ which now allows us to see that the k -1

x behaviour at the edge of the buffer layer is not the Townsend-Perry spectrum but just a transitional instance of a more involved spectral structure. In fact, the spectral picture which emerges from our analysis is a unified picture which brings together the buffer and inertial layers in a seemless way.

In figure 15 we plot examples of measured streamwise velocity fluctuations and the on-off signals with which we model them at various heights from the wall. Our model on-off signals are clearly a drastic simplification of the data but one gets the impression from these plots that they capture the sharpest gradients in the signal and therefore much of its spectral content at the length-scales considered here. The lengths of the non-zero parts of the model signals correspond to λ and the actual values of the on-off signal in these non-zero parts correspond to the average value α of the streamwise fluctuating velocity component inside each part. We stress that it is enough that our on-off model agrees with the data in the way it does in figures 9 to 12 for a certain range of thresholds u th (see subsection V B). Our model does not need to work for any arbitrary threshold; it only needs to work for those thresholds which effectively capture the spatial boundaries of the flow structure objects simulated by our on-off functions as mentioned at the end of the first paragraph of section II.

It is clear that a wider range of Reynolds numbers needs to be examined to establish the scalings of the lower and upper bounds of the range of wavenumbers where (10)-( 11 Our work has shed some new light on the streamwise turbulence spectra of wall turbulence by revealing that some of the inner structure of wall-attached eddies is reflected in the scalings of these spectra via p(y + ). An important implication of this structure is that the friction velocity is not sufficient to scale the spectra. Future work must now further probe the inner structure of wall-attached eddies, attempt to explain it and extend our analysis to higher Reynolds numbers so as to establish with certainty the ranges of the power laws (exponents p and q in (10)-( 11)) discussed in this paper. When this will be done, a be reproduced for this positive threshold u th = 0.4u 300 + and show the exact same trend with p increasing while q is decreasing with increasing y + . However, whereas p takes values similar to those for u th = -0.4u 300 + in the lower Re θ case, it does not do so in the higher Re θ case. As a result p + q is quite close to -1 in the lower Re θ case but less so, and in fact closer to -1.1 on average, for the higher Re θ (see figure 18).

[ 3 ]

 3 : we assume a power-law form for N (λ) in the range λ i < λ < λ o where λ i ∼ y and λ o ∼ δ, and N (λ) = 0 outside this range for simplicity. This power law form is N (λ) = (-N M + N o (λ/δ) -1-D ) where N M and N o are positive dimensionless numbers which increase propotionally to L x so as to keep number densities constant. The number N M is introduced to allow for the possibility of an upper bound on streaky structure size given by N (λ o ) = 0, i.e. N M = N o (λ o /δ) -1-D which should be small given that LSM and VLSM streaky structures have been observed with lengths greater than δ [see Smits et al. 13].

  36δ and 3.85δ, for Re θ = 8100 and 20600 respectively) and 0.3m high (≈ 0.86δ and 1δ) for Re θ = 8100 and 20600 respectively). Nikon lenses of 50mm focal length were set on the cameras and the magnification obtained was 0.05. The Software HIRIS was used to acquire the images of the four cameras simultaneously. A total of 22500 and 29500 samples were recorded at the highest and lowest Reynolds numbers respectively. The flow was seeded with 1µm Poly-Ethylene glycol and illuminated by a double pulsed NdYAG laser at 400mJ/pulse. The modified version by LML of MatPIV toolbox, was used under Matlab to process the acquired images from the 2D2C PIV. A multipass software was used with a final pass of 28x28 pixels (with a mean overlap of 65%) corresponding to 4mm x 4mm i.e. 33x33 wall units for Re θ = 8100 and 100x100 wall units for Re θ = 20600. Image deformation was applied at the final pass. The final grid had 766 points along the wall and 199 points in the wall-normal direction with a grid spacing of 1.5mm corresponding to 11 wall units and 35 wall units for the test cases at Re θ = 8100

Figure 1 FIG. 1 :

 11 Figure 1 shows profiles of the mean streamwise velocity U and the rms streamwise fluctuating velocity u obtained from PIV at Re θ = 8100 and Re θ = 20600 and compared with the hot-wire anemometry results of Carlier & Stanislas [15]. The mean velocity profiles are in good agreement with the hot-wire data and are well resolved from y + ≈ 30 and y + ≈ 90 upwards for Re θ = 8100 and 20600 respectively. Comparisons of the profiles of u + (u scaled with inner variables) show a fairly good match with the hot-wire data. A plateau of u +

  for the particular case of wall distance y + = 200 at Re θ = 20600, the agreement between the spatial spectrum from the PIV and the temporal spectrum from the hot-wire anemometry of Carlier & Stanislas [15] is good up to wavenumbers k x ≈ 500 corresponding to length-scales of 2mm.

FIG. 2 :

 2 FIG. 2: Comparison of the streamwise energy spectra obtained from PIV and hot-wire anemometry at y + = 200 for Re θ = 20600. E 11 (k x ) is in m 3 /s 2 and k x is in rad/m. The hot-wire anemometry was made by Carlier & Stanislas [15] at 19.6m from wind tunnel inlet in the same wind tunnel.

Figure 3 (

 3 Figure 3(a) shows a sample field of instantaneous streamwise fluctuating velocity components u. The existence of well-defined elongated and tilted wall-attached regions of relatively high (positive or negative) u values is clear. It is these regions that we need to target in relation to the elongated streaky structures of our model.

1 FIG. 3 :

 13 FIG. 3: Wall-attached elongated streaky structure eduction method applied on a sample instantaneous streamwise velocity field at Re θ = 20600. From top: (a) Raw instantaneous streamwise fluctuating velocity component field (b) The same field after application of a Gaussian filter (c) Binary image obtained after thresholding negative streamwise fluctuating momentum regions. (d) Final image obtained after cleaning as described in subsection V B.

FIG. 4 :

 4 FIG. 4: Figure 3(d) reproduced in the top plot, with, in the bottom plot, the average streamwise fluctuating velocity α and the streamwise length λ of the detected wall-attached structures at y/δ = 0.03.

figure 3 ( 2 FIG. 5 :

 325 figure 3(c) when applied to figure 3(b). The white structures in figure 3(c) correspond to u * < u th . One more step is required before comparing with our model. White structures which cut through the vertical borders of the figure are discarded because their streamwise extent is unknown; and white structures which are not attached to the bottom wall (at y = 0 but in

2 FIG. 6 :

 26 FIG. 6: PDFs of streamwise lengths λ of wall-attached structures (see figure 4) at selected wall distances for Re θ = 8100 (top) and Re θ = 20600 (middle). The fits shown in these top and middle plots are for y + = 198 at Re θ = 8100 and y + = 195 at Re θ = 20600. The bottom plot is a log-log reproduction of the middle plot's data.

FIG. 7 :

 7 FIG. 7: Log-log plots of pre-multiplied streamwise energy spectra at selected wall distances for Re θ = 8100 above and Re θ = 20600 below. Vertical lines show y 2π k x = (y/2π)(2π/δ) = y δ

FIG. 8 :

 8 FIG.8: Same as figure7in lin-log plots.

FIG. 9 :FIG. 12 :FIG. 13 :

 91213 FIG. 9: Lin-lin plots of a 2 versus λ/δ (left) and streamwise energy spectra plotted at wall distances y + = 41, 64, 88 and 125 (from top to bottom) at Re θ = 8100.

  (b).

11 •FIG. 14 :

 1114 FIG.14:From top to bottom: (a) Lin-log plots of pre-multiplied streamwise energy spectra at selected wall distances obtained with HWA in the Re θ = 19100 turbulent boundary layer of Tutkun et al.[START_REF] Tutkun | Two-point correlations in high reynolds number flat plate turbulent boundary layers[END_REF]. Least-square fits in a range bounded from above byx δ/(2π) = 2 but extended to wavenumbers as small as k x δ/(2π) close to 10 -1 . (b) Lin-log plots of pre-multiplied streamwise energy spectra at selected wall distances from the Re τ = 5200 turbulent channel flow DNS of Lee & Moser[START_REF] Lee | Direct numerical simulation of turbulent channel flow up to Re τ 5200[END_REF]. Least-square fits in the range 0.25 ≤ k x /(2π) ≤ 2.

FIG. 15 : 1 x

 151 FIG. 15: An example of a detected wall-attached flow structure for Re θ = 20600 and the u(x) signal through this structure at three different y + positions. The red line in the repeated binary image indicates the y + position where the signal u(x) is recorded (y + = 195, 450, 1110). The black/red line in the u(x) versus x/δ plots is the raw (un-filtered) PIV fluctuating streamwise fluctuating velocity outside/inside the detected flow structures. The dashed blue line is our model signal, equal to 0 outside and to the average value of u inside the detected structures. 27

2 FIG. 16 :

 216 FIG. 16: PDFs of streamwise lengths λ wall-attached structures at y + = 195 for Re θ = 20600 (top) and y + = 125 for Re θ = 8100 (bottom) over a set of thresholds

  can

2 FIG. 17 :

 217 FIG. 17: PDFs of streamwise lengths λ of wall-attached structures of positive streamwise fluctuating velocity with u th = u 300 + at selected wall distances for Re θ = 20600 (top) and Re θ = 8100 (bottom). The fits shown here are for y + = 195 at Re θ = 20600 and y + = 198 at Re θ = 8100.

FIG. 18 :

 18 FIG.18: From top to bottom: (a) Exponents p obtained from the best power-law fit of a 2 ∼ (λ/δ) p . (b) p + q versus y + . These fits are obtained over the range of scales investigated for the high-speed regions and the resulting exponents are plotted with the 95% confidence intervals for these fits. The y + positions and the two Reynolds numbers Re θ are those in figures 9 to 12.

  much smaller than 1 because N M is much smaller than N o and that (8) is valid in the range 1/λ o k x 1/λ i where λ o scales with but is much larger than δ. For a good correspondence with the scalings of the Townsend-Perry attached eddy model one needs to take λ i ∼ y and

TABLE I :

 I Values of the constants C 1 and C 2 in the form -C 1 + C 2 (λ/δ) -2 of the PDF of λ/δ. In the Re θ = 20600 case, the fit is over a range of λ/δ bounded from above by 3.8 and

	from below by 0.49 (y + = 90), 0.55 (y + = 195), 0.54 (y + = 305), 0.58 (y + = 450) and 0.78
	(y + = 630). In the Re θ = 8100 case, the fit is over a range of λ/δ bounded from above by
	3.4 and from below by 0.54 (y + = 52), 0.53 (y + = 88), 0.53 (y + = 125), 0.56 (y + = 198)
	and 0.65 (y + = 306).

power law exponent -2, i.e. D = 1, in all cases. Given the form of N (λ) hypothesised in sections II and III, we fit the PDF of λ/δ with a functional form -C 1 + C 2 (λ/δ) -2 (where

C 1 /N M = C 2 /N o ).

The fit is shown in figures 5 and 6 and is effectively the same for both Reynolds numbers and all values of y + in the mean flow's approximate log region. The constants C 1 and C 2 are reported in table I. They are indeed fairly constant over the range of wall distances and for both Reynolds numbers. Identical results are obtained for wallattached structures with positive streamwise fluctuating velocity except that C 1 ≈ 0.02 for both Reynolds numbers and C 2 ≈ 0.29 for Re θ = 8100 (see Table

III

in Appendix). It is worth noting that the lower bound of the range where the PDF of λ/δ is well approximated by -C 1 + C 2 (λ/δ) -2 seems to increase slightly with increasing y + .

TABLE II :

 II Number of structures detected over a set of three negative thresholds for Re θ = 20600 and Re θ = 8100 complete picture of streamwise energy spectra will also need to integrate the spectral model of Vassilicos et al. [31].

TABLE III :

 III Values of the constants C 1 and C 2 in the form -C 1 + C 2 (λ/δ) -2 of the PDF of λ/δ.

Acknowledgements

The work was carried out within the framework of the CNRS Research Foundation on Ground Transport and Mobility, in articulation with the ELSAT2020 project supported by the European Community, the French Ministry of Higher Education and Research, the Hauts de France Regional Council. The authors gratefully acknowledge the support of these institutions. JCV also acknowledges the support of ERC Advanced Grant 320560.

(or very weak power law) dependence on y exists for the rms streamwise turbulent velocity in the outer part of the inertial range of wall-distances. If one assumes the lower bound to scale as 1/δ and therefore an energy spectrum E 11 (k x ) of the form (i) E 11 (k x ) ∼ U 2 τ δ for 0 ≤ k x ≤ B 1 /δ, (ii) E 11 (k x ) ∼ U 2 τ δ(k x δ) -1-p for B 1 /δ ≤ k x ≤ B 2 /y (where B 1 and B 2 are dimensionless constants and p may be a function of y as in figure 13(a) and (iii) comparatively negligible energy at wavenumbers k x > B 2 /y, then we should have

This expression for (u + ) 2 tends to

as p → 0 which is the Townsend logarithmic dependence on y corresponding to p ≡ 0 (see Townsend [START_REF] Townsend | The structure of turbulent shear flow[END_REF], Perry & Abell [START_REF] Perry | Asymptotic similarity of turbulence structures in smooth-and rough-walled pipes[END_REF] and Perry et al [START_REF] Perry | A theoretical and experimental study of wall turbulence[END_REF]) . This logarithmic dependence [START_REF] Smits | High-reynolds number wall turbulence[END_REF] results from the assumption that the upper bound of the range of wavenumbers where ( 10)-( 11) may hold with p ≡ 0 scales as 1/y. Slightly non-zero values of p give slight deviations from this logarithmic dependence, of the form [START_REF] Vallikivi | Spectral scaling in boundary layers and pipes at very high Reynolds numbers[END_REF].

Using the values of p obtained in this work and plotted versus y + in figure 13 for our two values of Re θ , it is not possible to fit [START_REF] Vallikivi | Spectral scaling in boundary layers and pipes at very high Reynolds numbers[END_REF] to the data in the lower plot of figure 1 from y + = 41 to 306 in the Re θ = 8100 case and from y + = 90 to 742 in the Re θ = 20600 case.

These are the y + ranges where ( 10)- [START_REF] Mckeon | Asymptotic scaling in turbulent pipe flow[END_REF] has been established for our data and they should therefore also be the ranges where [START_REF] Vallikivi | Spectral scaling in boundary layers and pipes at very high Reynolds numbers[END_REF] holds if the spectral model of the previous paragraph is good enough. However, in spite of the three adjustable dimensionless constants (B 1 , B 2 and an overall constant of proportionality), ( 12) cannot fit the entire y + range for which this model has been designed, that is a y + range which includes both the p < 0 and the p > 0 regions.

A most suspect part of the spectral model used to derive ( 12) is its low wavenumber part.

Appendix A: Effects of threshold levels and sign

Our results have no significant dependence on threshold u th in the range -0.2u 300 + to -0.6u 300 + . An example of this lack of threshold dependence can be seen in the PDFs of λ/δ which we plot in figure 16. We also report in table II the number of structures educed by the algorithm described in subsection V B for the three negative threshold values -0.2u 300 + , -0.4u 300 + and -0.6u 300 + . Figures 9 to 13 have been obtained for u th = -0.4u 300 + but we checked that they remain very similar without deviations from our conclusions if the threshold u th is chosen in the range -0.2u 300 + to -0.6u 300 + .

As mentioned in subsection V B, this paper's analysis can be repeated equally well on structures of positive streamwise fluctuating velocity. We provide examples of results obtained with u th = 0.4u 300 + in figure 17 and table III. There are indeed no significant differences in the results for the low and high speed attached flow regions, except for a lower but extreme Reynolds numbers," Phys. Rev. Lett. 108, 094501 (2012).