domain: chim sde sdv

text:
Alexandre Poulhazan

Alexandre A Arnold

Dror E Warschawski

Isabelle Marcotte

Solid-state NMR study of microalgal membranes and cell walls

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X.1 Overview

Microalgae are microscopic algal cells also known as phytoplankton. The cell wall and plasma membrane protect the organelles in the cytoplasm, such as the nucleus, the chloroplast, the reticulum, as well as the sugar and lipid reserves (Figure X.1). About 30,000 species of microalgae are found in fresh-and saltwater, where they are at the basis of the aquatic food chain [START_REF] Richmond | Handbook of microalgal culture: applied phycology and biotechnology[END_REF]. As such, global warming or pollution effects on this resource can have an impact on all higher organisms.

Microalgae bear an important socio-economical potential. They are already used in human and animal nutrition, and have been exploited for their therapeutic activity related to antimicrobial [START_REF] Falaise | Antimicrobial compounds from eukaryotic microalgae against human pathogens and diseases in aquaculture[END_REF]), antiviral (de Jesus Raposo et al., 2015), antioxidant, antiinflammatory [START_REF] Rasool | Anti-inflammatory effect of Spirulina fusiformis on adjuvant-induced arthritis in mice[END_REF] or anti-cancer [START_REF] Gutiérrez | Malyngolide dimer, a bioactive symmetric cyclodepside from the panamanian marine cyanobacterium Lyngbya majuscula[END_REF] properties. Another interesting characteristic of microalgae is their ability to sequestrate and bioaccumulate heavy metals [START_REF] Posadas | Microalgae cultivation in wastewater[END_REF][START_REF] Piccini | A synergistic use of microalgae and macroalgae for heavy metal bioremediation and bioenergy production through hydrothermal liquefaction[END_REF]. They thus stand as an elegant avenue for contaminated water treatment. Microalgae are also used in the development of third-generation biofuels, which partially or completely come from solar instead of fossil energy [START_REF] Abo | Microalgae to biofuels production: a review on cultivation, application and renewable energy[END_REF].

On a more fundamental level, microalgae are great tools for the study of photosynthesis. For example, carbon metabolism was deciphered in Chlorella by Melvin Calvin, Nobel Prize in 1961. The dynamics of photosynthesis has been investigated in detail in Chlamydomonas [START_REF] Eberhard | The dynamics of photosynthesis[END_REF]. New transgene strategies are available, mainly studies of Chlamydomonas reinhardtii [START_REF] Doron | Transgene expression in microalgae -from tools to applications[END_REF][START_REF] Baier | Intron-containing algal transgenes mediate efficient recombinant gene expression in the green microalga Chlamydomonas reinhardtii[END_REF], which has been called the 'photosynthetic Escherichia coli' because of its optional heterotrophic metabolism, allowing the use of non-photosynthetic mutants.

The microalgal cell wall, membranes and lipids are key players for both biochemical processes and biotechnological applications. Their selective permeability and mechanical or osmotic shock resistance make the cell interfaces and membranes a source of inspiration for biomaterial development. Furthermore, lipids in general are valuable nutritional ingredients with long-chain and ω-3 poly-unsaturated fatty acids (FAs), but also as precursors for biodiesel production from triacylglycerol (TAG). Understanding the composition, dynamics and architectures of these assemblies is essential to bioengineer new microalgal strains with efficient photosynthesis and easy lipid extraction from the biomass, or to design novel materials based on natural products with interesting biological/physical properties. Finally, microalgal membrane structure and composition (see Figure X.1) can vary depending on temperature, pH or interaction with exogenous molecules, making them excellent global warming reporters. Solid-state nuclear magnetic resonance (SS-NMR) is an invaluable tool providing nanoscopic information on the structure, dynamics and interactions of large biological molecules with virtually no limit in molecular size. This is why SS-NMR has proven its capabilities to investigate whole cell constituents, as well as their organization and membrane interactions. However, the cell complexity leads to crowded NMR spectra in which many signals potentially overlap. Therefore, SS-NMR approaches are being developed to tackle such challenging systems [START_REF] Warnet | In-cell solid-state NMR: an emerging technique for the study of biological membranes[END_REF][START_REF] Werby | Spectral comparisons of mammalian cells and intact organelles by solid-state NMR[END_REF][START_REF] Narasimhan | When small becomes too big: expanding the use of in-cell solid-state NMR spectroscopy[END_REF]. Considering the environmental and economical importance of microalgae, the objective of this chapter is to present how SS-NMR can provide information on the molecular composition and organization of microalgal membranes and the cell wall. Before presenting different NMR methodologies to address these cell compartments, the composition of microalgal membranes will first be summarized.

X.2 Microalgal cell membrane composition and architecture

Microalgal membranes are composed of lipids, proteins, carbohydrates, phytosterols as well as pigments. Although lipids ensure membrane integrity and efficient cell impermeability, the detailed lipidome of microalgae has been described for only a handful of species such as C. reinhardtii and Nannochloropsis oceanica [START_REF] Zhao | The permeability of puerarin loaded poly(butylcyanoacrylate) nanoparticles coated with polysorbate 80 on the blood-brain barrier and its protective effect against cerebral ischemia/reperfusion injury[END_REF][START_REF] Nguyen | The green microalga Chlamydomonas reinhardtii has a single omega-3 fatty acid desaturase that localizes to the chloroplast and impacts both plastidic and extraplastidic membrane lipids[END_REF]. Membranes are also rich in proteins that maintain the cell structure and metabolism, and in carbohydrates bound to both lipids and proteins, allowing key processes such as cell recognition or protein activity. Pigments are found in chloroplast membranes (Figure X.1) where they are involved in light energy capture and dissipation. Indeed, a lipid membrane also protects organelles in the cytoplasm, such as the reticulum and the nucleus. Moreover, lipid droplets are also present inside the cells. [START_REF] Harwood | Membrane Lipids in Algae[END_REF]. Lipid droplets incorporate TAGs and ergosterol, as a dense energy storage, surrounded by a single phospholipid monolayer. Energy can also be stored as complex carbohydrates, namely starch or chrysolaminarin.

X.2.1 Microalgal cell wall and plasma membrane

The cell wall primarily acts as a shell and thus makes biomolecules such as TAGs harder to harvest and exploit. It also protects the cell from mechanical constraints and external stress, and is involved in cell-cell and cell-substrate adhesion, protection from bacteria as well as sexual reproduction [START_REF] Domozych | The cell walls of green algae: a journey through evolution and diversity[END_REF]. Although the plasma membrane is not the largest membrane in the cell, it must be studied since it faces a wide range of stresses such as temperature changes and osmotic shocks in its environment. Moreover, multiple specific transporters are found within this membrane which is the last barrier crossed by molecules before accessing the cytosol. It is also where cell signalling is initiated. One of the first studies of the lipid composition of microalgal plasma membranes focused on the green algae Hydrodictyon africanum. Phospholipids, such as phosphatidylcholine (PC), phosphatidylserine (PS) and cardiolipin (CL), were mainly identified [START_REF] Bailey | Phospholipid composition of the plasma membrane of the green alga, Hydrodictyon africanum[END_REF]. The microalgal cell membrane is mostly composed of polar lipids [START_REF] Harwood | Membrane Lipids in Algae[END_REF] and its composition can adapt to external stress or cell state [START_REF] Mimouni | Lipids from microalgae[END_REF]. [START_REF] Vieler | The lipid composition of the unicellular green alga Chlamydomonas reinhardtii and the diatom Cyclotella meneghiniana investigated by MALDI-TOF MS and TLC[END_REF][START_REF] Nguyen | The green microalga Chlamydomonas reinhardtii has a single omega-3 fatty acid desaturase that localizes to the chloroplast and impacts both plastidic and extraplastidic membrane lipids[END_REF].

The relationship between the plasma membrane and the microalgal cell wall is so intimate that considering the membrane without the cell wall would make no sense in terms of dynamics and shape. Indeed, the cell wall synthesis and its remodelling proteins are associated to the membrane [START_REF] Popper | Plant and algal cell walls: diversity and functionality[END_REF] and some fibrillar assemblies anchored in the cell wall traverse the plasma membrane [START_REF] Domozych | Algal cell walls[END_REF], making membrane and cell wall boundaries hard to define precisely (Figure X.1). For microalgae such as Chlorophyceae, the cell wall is composed of very organized carbohydrates and glycoproteins [START_REF] Rashidi | Detailed biochemical and morphologic characteristics of the green microalga Neochloris oleoabundans cell wall[END_REF], so that glycan patterns and structures are important to determine. While electron microscopy (EM) has provided micrometer resolution images important for taxonomic determination, SS-NMR can potentially provide additional structural and molecular-level dynamic information of the cell membrane on different timescales.

X.2.2 Photosynthetic organelles (thylakoids and chloroplast)

There are several organelles in the algal cell and a membrane borders them all. The chloroplast has, by far, the largest membrane surface, with three layers of densely packed lipid membranes (Figure X.1). This explains why the lipid profile of other organelles is difficult to determine without getting any chloroplast membrane or pigment contamination during purification. The lipid and FA chain profiles of microalgae are usually given for the whole organism (Figure X.2). The membrane surface of each organelle has not been measured or estimated yet, even if a 3D reconstitution made by in situ cryo-electron tomography of C. reinhardtii has been reported [START_REF] Engel | Native architecture of the Chlamydomonas chloroplast revealed by in situ cryoelectron tomography[END_REF].

The lipid composition of chloroplasts membranes is very different from that of plasma membranes as they mostly contain galactolipids and few phospholipids. It varies according to the species but monogalactosyldiacylglycerol (MGDG, 40 to 55%), digalactosyldiacylglycerol (DGDG, 15-35%), sulfoquinovosyldiacylglycerol (SQDG, 10-20%) and phosphatidylglycerol (PG, 10-20%) are the main components [START_REF] Harwood | Membrane Lipids in Algae[END_REF]. In Chlamydomonas, the chloroplast envelopes and thylakoid membranes have slightly different lipid compositions [START_REF] Mendiola-Morgenthaler | Isolation of chloroplast envelopes from Chlamydomonas. Lipid and polypeptide composition[END_REF]. On the other hand, MGDG/DGDG and SQDG/PG ratios seem to be regulated in chloroplast membranes of various species and could be critical for proper thylakoid development [START_REF] Boudière | Glycerolipids in photosynthesis: Composition, synthesis and trafficking[END_REF]. Betaine lipids have been detected in a few brown algae, such as diacylglycreyl-O-(N,N,N-trimethyl) homoserine (DGTS) and diacylglyceryl-O-(hydroxymethyl)-(N,N,N-trimethyl)β-alanine (DGTA) [START_REF] Murakami | Betaine lipid is crucial for adapting to low temperature and phosphate deficiency in Nannochloropsis[END_REF]. Finally, recent work mentions glycolipids as microalgal membrane stabilizers because, during water or cold stress, the chloroplast membrane can be enriched in oligogalactolipids, produced by MGDG conversion [START_REF] Hölzl | Chloroplast lipids and their biosynthesis[END_REF].

X.2.3 Lipid droplets and free fatty acids

The lipid droplet (LD) is an important organelle for biofuel production due to the TAGs that they contain. Little is known about microalgae LDs but interest is growing due to their high potential as an alternative to fossil fuels. TAGs produce twice as much energy than starch and protein per unit weight. Moreover, microalgae are of great interest for nutrition purposes because of the essential long-chain and unsaturated FAs that are contained in the LDs [START_REF] Spolaore | Commercial applications of microalgae[END_REF].

The number, size and composition of LDs vary greatly among microalgal cell types but also according to the life cycle and metabolic stress. For example, nitrogen, iron, sulfur and phosphorus starvation are commonly used strategies to increase lipid storage in microalgae such as C. reinhardtii [START_REF] Hu | Microalgal triacylglycerols as feedstocks for biofuel production: perspectives and advances[END_REF][START_REF] Wang | Algal lipid bodies: stress induction, purification, and biochemical characterization in wild-type and starchless Chlamydomonas reinhardtii[END_REF][START_REF] Goold | Microalgal lipid droplets: Composition, diversity, biogenesis and functions[END_REF]. The size of LDs can range from nm to µm and they are very dynamic in terms of molecular motion, anabolism and catabolism. LDs are easily observed in the cytosol using lipophilic dyes such as Nile red or Bodipy, thus facilitating their quantification by microscopy or flow cytometry, as well as transmission electron microscopy which has been used to further describe their structure [START_REF] Goodson | Structural correlates of cytoplasmic and chloroplast lipid body synthesis in Chlamydomonas reinhardtii and stimulation of lipid body production with acetate boost[END_REF]. Lipid droplets share the same overall structure with a hydrophobic oil core of storage lipids that mainly comprise TAGs (80-90% by weight) and sterol esters, surrounded by a phospholipid monolayer that also contains proteins (only 5% by weight) (Figure X.1). Biochemical and microscopy studies have shown that the endoplasmic reticulum and plastid membranes are involved in LD production [START_REF] Fan | A chloroplast pathway for the de novo biosynthesis of triacylglycerol in Chlamydomonas reinhardtii[END_REF][START_REF] Goodson | Structural correlates of cytoplasmic and chloroplast lipid body synthesis in Chlamydomonas reinhardtii and stimulation of lipid body production with acetate boost[END_REF]. LDs are also associated with lipid homeostasis and several cellular signalling processes [START_REF] Farese | Lipid droplets finally get a little R-E-S-P-E-C-T[END_REF].

The FA composition of TAGs is similar to that of whole cells [START_REF] Goold | Microalgal lipid droplets: Composition, diversity, biogenesis and functions[END_REF]. However, the profile of the LD enveloping monolayer is very specific. For example, in C. reinhardtii [START_REF] Wang | Algal lipid bodies: stress induction, purification, and biochemical characterization in wild-type and starchless Chlamydomonas reinhardtii[END_REF] and Haematococcus pluvialis [START_REF] Peled | Isolation of a novel oil globule protein from the green alga Haematococcus pluvialis (Chlorophyceae)[END_REF], DGTS, sulfoquinovosyldiacylglycerol (SQDG) and traces of DGDG are its major component. The lipids around the LD core are thought to be intermediates for acyl chain production in TAGs. Specific proteins are associated with LDs [START_REF] Yoneda | Identification of a major lipid droplet protein in a marine diatom Phaeodactylum tricornutum[END_REF][START_REF] Wang | A lipid droplet-associated protein involved in lipid droplet biogenesis and triacylglycerol accumulation in the oleaginous microalga Phaeodactylum tricornutum[END_REF][START_REF] Wang | Proteomics analysis of lipid droplets from the oleaginous alga Chromochloris zofingiensis reveals novel proteins for lipid metabolism[END_REF]. In C. reinhardtii, the predominant protein is the very hydrophobic Major Lipid Droplet Protein (MLDP) [START_REF] Goold | Microalgal lipid droplets: Composition, diversity, biogenesis and functions[END_REF], but other LD-associated proteins have been identified [START_REF] Vieler | A lipid droplet protein of Nannochloropsis with functions partially analogous to plant oleosins[END_REF], and complete proteomics describe more than 200 proteins linked to LDs [START_REF] Nguyen | Proteomic profiling of oil bodies isolated from the unicellular green microalga Chlamydomonas reinhardtii: with focus on proteins involved in lipid metabolism[END_REF].

X.3 Technical considerations

All the microalgal cell compartments presented in the previous section have a large diversity in molecular composition and structure. However, sophisticated techniques and appropriate stable isotope labelling allow the determination of molecular structures, dynamics and interactions in complex systems such as microalgae by SS-NMR.

X.3.1 Observable nuclei for microalgal membranes studies

Since protons ( 1 H) are the most sensitive nuclei for biological applications in NMR, 1 Hbased techniques have been used to study microalgae, as will be illustrated below. The large abundance of protons in biological samples is an advantage in terms of sensitivity, but a drawback in terms of resolution. Therefore, other nuclei with higher specificity are frequently exploited. Phosphorous-31 ( 31 P), for example, has 100% natural abundance and high gyromagnetic ratio. Phosphorus is mainly found in ATP, DNA, phosphorylated proteins but also in phospholipids and, therefore, 31 P NMR is an excellent probe to determine the membrane composition and dynamics [START_REF] Overall | In Situ Monitoring of Bacteria under Antimicrobial Stress Using 31P Solid-State NMR[END_REF]. Carbon is present in all organic molecules of the cell but only the 13 C isotope, which has a low (1%) natural abundance, generates an NMR signal. Since its gyromagnetic ratio is one quarter that of 1 H, 13 C has a lower sensitivity and isotopic enrichment is required for experiments to be time efficient. Finally, 15 N NMR can also be used to study microalgae, but is even less sensitive than 13 C and with very low natural abundance (0.4%), so 15 N NMR can only be performed with isotopic labelling.

Uniform 13 C-and/or 15 N-labelling of microalgae is easily carried out with affordable carbon and nitrogen sources, i.e., sodium bicarbonate (NaH 13 CO3) and ammonium chloride ( 15 NH4Cl), respectively. The cost of growing uniformly labelled C. reinhardtii, for example, can be estimated to less than 100$/L of medium with an optimal cell concentration of 3 to 4×10 6 cells/mL. Depending on metabolism or the targeted molecule, the medium might also require more expensive 13 C-labelled glucose, sodium acetate [START_REF] Venters | Uniform 13C isotope labeling of proteins with sodium acetate for NMR studies: application to human carbonic anhydrase II[END_REF] or CO2 [START_REF] Akhter | Identification of aquatically available carbon from algae through solution-state NMR of whole 13C-labelled cells[END_REF].

Targeted labelling of specific molecules of interest, such as lipids, amino acids, carbohydrates or pigments, within the whole organism is an interesting alternative approach. This strategy can be applied to 13 C or 15 N but also to other reporter nuclei, such as 2 H [START_REF] Pius | 2H solid-state nuclear magnetic resonance investigation of whole Escherichia coli interacting with antimicrobial peptide MSI-78[END_REF][START_REF] Tardy-Laporte | A 2H solid-state NMR study of the effect of antimicrobial agents on intact Escherichia coli without mutating[END_REF][START_REF] Warnet | A 2H magic-angle spinning solid-state NMR characterisation of lipid membranes in intact bacteria[END_REF][START_REF] Molugu | Concepts and methods of solid-state NMR spectroscopy applied to biomembranes[END_REF][START_REF] Salnikov | Structure, topology, and dynamics of membrane-inserted polypeptides and lipids by solid-state NMR spectroscopy: investigations of the transmembrane domains of the DQ beta-1 subunit of the MHC II receptor and of the COP I protein p24[END_REF] (see Chapter X Booth) or 19 F [START_REF] Matsumori | Orientation of fluorinated cholesterol in lipid bilayers analyzed by 19F tensor calculation and solidstate NMR[END_REF][START_REF] Shi | In situ 19F NMR studies of an E. coli membrane protein[END_REF][START_REF] Afonin | 19F NMR screening of unrelated antimicrobial peptides shows that membrane interactions are largely governed by lipids[END_REF][START_REF] Kozorog | 19F NMR studies provide insights into lipid membrane interactions of listeriolysin O, a pore forming toxin from Listeria monocytogenes[END_REF], often chosen to replace protons in key locations. These specific labelling approaches have rarely been applied to microalgae as they require a good knowledge of algal metabolism, but they hold great promise in particular for in vivo SS-NMR studies, as performed with intact bacteria [START_REF] Booth | Recent progress on the application of 2H solid-state NMR to probe the interaction of antimicrobial peptides with intact bacteria[END_REF].

X.3.2 Solution and solid-state NMR

The NMR line width of a given nucleus depends on its interactions with neighbouring nuclei or electrons, including dipolar and quadrupolar interactions, chemical shift anisotropy but also anisotropic magnetic susceptibility effects. Most of these interactions are orientationdependent but averaged out under fast and isotropic tumbling, such as for small molecules in solution, resulting in very narrow NMR lines.

In solution NMR, molecules can be discriminated according to their diffusion correlation time. Such spectral filtering hence simplifies the spectra and has been applied to study whole microalgal cells [START_REF] Courtier-Murias | Comprehensive multiphase NMR spectroscopy: Basic experimental approaches to differentiate phases in heterogeneous samples[END_REF][START_REF] Akhter | Identification of aquatically available carbon from algae through solution-state NMR of whole 13C-labelled cells[END_REF]. Among the available "diffusion filters", diffusion editing (DE) removes the signal of mobile molecules, while inverse diffusion editing (IDE) only shows the spectrum of fast diffusing molecules. Single pulse experiments will obviously show both types of signals.

Macromolecules and complex assemblies such as cell extracts, or even whole cells such as microalgae, are considered as "solid" on the NMR time scale. The orientation dependence of the interactions presented above leads to broad lines that often overlap with neighbouring signals, resulting in the loss of spectral resolution. Such systems are thus studied using SS-NMR techniques. High resolution can be restored by using strategies that replace fast and isotropic tumbling by coherent averaging. One such strategy is magic-angle spinning (MAS), where the sample is spun at high frequency at an angle of 54.7° with respect to the magnetic field direction. Depending on the available spinning frequency, interactions can be completely averaged out, resulting in narrower lines. The required spinning frequency depends on the observed nuclei and the interactions that they experience. One may wonder if high spinning frequencies would be deleterious to soft eukaryotic cells. Microalgae are very resistant organisms, owing to their protective and rigid cell wall. It has been shown that C. reinhardtii can spend 24 h in a SS-NMR rotor spinning at 10 kHz, at room temperature and without any nutriments, with around 80% survival and cell morphology preserved [START_REF] Poulhazan | Unambiguous ex situ and in cell 2D (13)C solid-state NMR characterization of starch and its constituents[END_REF].

In the case of protons in rigid solids, homonuclear dipolar interactions are very strong and often cannot be efficiently averaged out using standard MAS. Therefore, most 1 H MAS SS-NMR studies are restricted to very dynamic small molecules where the remaining dipolar couplings and anisotropic magnetic susceptibility can be cancelled by MAS at moderate frequencies (typically below 10 kHz). This is the field of high-resolution magic-angle spinning NMR (HRMAS). In 1 H HRMAS, as in solution NMR, the lipid droplet signals dominate those of membrane lipids whereas higher spinning frequencies would be required to observe intense membrane phospholipid, carbohydrate or protein signals. HRMAS allows for direct measurements of non-liquid soft tissues and cells, generally by 1 H NMR, and is typically used to characterize metabolites, even in intact cells, as is summarized in the in-cell section.

High-resolution SS-NMR developments over the last 30 years have provided robust methodologies to characterize the structure and dynamics of biological membrane components (lipids and proteins) by observing 13 C signals under MAS, sometimes in combination with 15 N since these nuclei are found in all peptide bonds. Since the internuclear interactions involved are smaller than for 1 H, standard MAS is often enough to generate high-resolution one-(1D) and multidimensional spectra. In some cases, line narrowing by MAS results in membrane components spectra that are similar to those obtained by solution NMR.

While solution NMR and HRMAS are restricted to molecules with "fast" dynamic properties, SS-NMR can reveal and exploit the dynamic heterogeneity of biological membranes. To do so, adequate "dynamic filters" that select only rigid or dynamic components within the sample are utilized. This results not only in spectral simplification but also provides information regarding the molecular environment. The most common filters take advantage of proton-to-carbon polarization transfer schemes. For example, INEPT (nuclei enhancement by polarization transfer) will only be efficient for very dynamic molecules (with tumbling times from 10 -15 to 10 -9 s) and only detect carbons directly bound to protons. On the other hand, cross polarization (CP) will only show carbons from more rigid molecules (with correlation times from 10 -9 to 1 s). Single pulse 13 C experiments will show both types of signals. Other types of filters, that will not be detailed here, have also been developed [START_REF] Reckel | In-cell solid-state NMR as a tool to study proteins in large complexes[END_REF][START_REF] Renault | Cellular solid-state nuclear magnetic resonance spectroscopy[END_REF][START_REF] Warnet | In-cell solid-state NMR: an emerging technique for the study of biological membranes[END_REF][START_REF] Matlahov | Hidden motions and motion-induced invisibility: Dynamics-based spectral editing in solid-state NMR[END_REF]. Since cells are not pure solids or pure liquids, the research group of Simpson has developed tools to study such heterogeneous samples with an approach called comprehensive multiphase (CMP) NMR [START_REF] Simpson | Environmental comprehensive multi-phase NMR[END_REF][START_REF] Liaghati Mobarhan | In vivo comprehensive multiphase NMR[END_REF]. Indeed, the dynamical heterogeneity of intact cells requires the use of both solution and SS-NMR approaches to obtain a complete picture of their composition and function. CMP-NMR probes thus combine lock, gradients and improved shimming capabilities, typical of solution NMR probes, with access to moderate magic-angle spinning frequencies (vide infra) of a few kHz and high-power pulses typical of SS-NMR probes. With this type of equipment, monitoring different types of molecules is possible: from dissolved metabolites up to very rigid storage molecules or proteins.

X.4. NMR of lipid extracts and model membranes

Considering the complexity of microalgal cells, simple and straightforward approaches are used to garner information on membrane lipids by NMR. First, lipids can be extracted, dissolved in organic solvents, and studied by simple 1 H, 13 C, or 31 P solution NMR. This approach has been applied, for example, on whole cell extracts of Pavlova spp. [START_REF] Kobayashi | Accumulation of cyclitols functioning as compatible solutes in the haptophyte alga Pavlova sp[END_REF] or Neochloris oleoabundans [START_REF] Beal | Lipid analysis of Neochloris oleoabundans by liquid state NMR[END_REF] for lipid assignment and quantification in combination with other techniques [START_REF] Challagulla | Advances in techniques for assessment of microalgal lipids[END_REF]. The use of lipid extracts obviously neglects important information such as the lipid environment, collective properties and membrane dynamics.

Another strategy to study microalgal membrane properties is to employ model systems with lipid composition similar to the mimicked membranes. Although part of the living organism complexity is sacrificed, model membranes provide valuable information on specific biophysical properties and interactions with the lipids. Mixtures of galactolipids such as MGDG and DGDG have been studied by 2 H and 31 P static SS-NMR as early as 1985 to investigate phase transitions in microalgae and chloroplast membranes [START_REF] Brentel | Phase equilibria of mixtures of plant galactolipids. The formation of a bicontinuous cubic phase[END_REF][START_REF] Lindblom | Phase equilibria of membrane lipids from Acholeplasma laidlawii: importance of a single lipid forming nonlamellar phases[END_REF][START_REF] Borovyagin | Polymorphism in galactolipid/phosphatidylglycerol model membranes initiated by chlorophylls: 31P-NMR and electron-microscopy studies[END_REF]. Galactolipid structures and dynamics were later probed by 13 C SS-NMR using MAS [START_REF] Adebodun | Spectroscopic studies of lipids and biological membranes: carbon-13 and proton magic-angle samplespinning nuclear magnetic resonance study of glycolipid-water systems[END_REF][START_REF] Gross | Multidimensional NMR in lipid systems. Coherence transfer through J couplings under MAS[END_REF][START_REF] Castro | NMR studies of membranes composed of glycolipids and phospholipids[END_REF] or in fast-tumbling bicelles by solution NMR [START_REF] Howard | Membrane and solution conformations of monogalactosyldiacylglycerol using NMR/molecular modeling methods[END_REF]Prestegard, 1995, Ye et al., 2013). Figure X.3 shows a 1 H-13 C proton-detected local field (PDLF) study that allowed measurement of the order parameters at various carbon positions in model unlabelled DMPC/MGDG membranes. This revealed that MGDG reduced the order parameters along DMPC molecules [START_REF] Castro | NMR studies of membranes composed of glycolipids and phospholipids[END_REF]. Simplified thylakoid model membranes made of palmitoyl-oleoyl phosphocholine (POPC) and MGDG were used to determine the interactions of Hcf106 -a peptide found in the thylakoid of microalgae -with its host membrane lipids [START_REF] Zhang | Solid-state NMR investigations of peptide-lipid interactions of the transmembrane domain of a plantderived protein, Hcf106[END_REF][START_REF] Zhang | Investigating the interaction between peptides of the amphipathic helix of Hcf106 and the phospholipid bilayer by solid-state NMR spectroscopy[END_REF]. Finally, the light harvesting complex II (LHCII) of C. reinhardtii was also studied by 13 C, 15 N-NMR reconstituted in a thylakoid-like membrane environment made up of MGDG, DGDG, SQDG and PG (Azadi-Chegeni et al., 2018b). [START_REF] Castro | NMR studies of membranes composed of glycolipids and phospholipids[END_REF] with permission.

X.5. In vivo, in-cell and in situ NMR

In-cell SS-NMR tackles complex samples that are very hard to characterize using other atomic-resolution techniques such as EM, X-ray crystallography or solution NMR. While in vivo NMR addresses the challenge of studying living cells, in situ NMR examines cell extracts -the cell is no longer intact but molecules are still in their native local environment. In vivo, in-cell and in situ SS-NMR have only recently been applied to microalgae but a wide range of applications are possible, as will be presented in the next sections.

X.5.1 1 H-NMR based approaches

As previously explained, HRMAS provides insights into cellular metabolism by producing spectra with resolution sometimes comparable to that of solution NMR; and also allows spectral editing according to the dynamics of the constituents. HRMAS is applied to a variety of biological samples including human or mice cell lines, or tissues for clinical studies, allowing diagnosis and follow up of organism responses to treatments [START_REF] Kaebisch | Applications of high-resolution magic angle spinning MRS in biomedical studies I-cell line and animal models[END_REF]. It is a common tool in metabolomics, especially for detecting metabolites associated with lipids such as inositol, phosphoryl choline, FAs and cholesterol. 1 H HRMAS is also used in lipid profiling and characterization of the saturated/unsaturated FA ratio [START_REF] Griffin | Assignment of 1H nuclear magnetic resonance visible polyunsaturated fatty acids in BT4C gliomas undergoing ganciclovir-thymidine kinase gene therapy-induced programmed cell death[END_REF][START_REF] Kabli | In vivo metabolite profile of adult zebrafish brain obtained by high-resolution localized magnetic resonance spectroscopy[END_REF][START_REF] Berry | Highresolution magic angle spinning nuclear magnetic resonance of intact zebrafish embryos detects metabolic changes following exposure to teratogenic polymethoxyalkenes from algae[END_REF]. Terpenes found in chloroplast membranes are interesting molecules also detected by HRMAS [START_REF] Gaysinski | Extraction, purification, and NMR analysis of terpenes from brown algae[END_REF]. Finally, because lipids are an important part of cell metabolism, several studies have focused on alterations of lipid metabolism using HRMAS to detect lipid droplet perturbation, for example in cancer cell lines [START_REF] Pan | The size of cytoplasmic lipid droplets varies between tumour cell lines of the nervous system: a 1H NMR spectroscopy study[END_REF].

Considering the capabilities of HRMAS, this approach has been used to study microalgae, including their membranes and the in vivo intracellular C/N ratio [START_REF] Bondu | The use of HRMAS NMR spectroscopy to study the in vivo intra-cellular carbon/nitrogen ratio of Solieria chordalis (Rhodophyta)[END_REF][START_REF] Hashim | HR-MAS NMR spectroscopy: a practical guide for natural samples[END_REF]. Native membranes are difficult to study because they are rigid structures on the NMR time scale, thus leading to broad and overlapping lines, with notable exceptions [START_REF] Simon | HRMAS NMR analysis of algae and identification of molecules of interest via conventional 1D and 2D NMR: sample preparation and optimization of experimental conditions[END_REF]. Merkley and Syvitsk (Merkley and Syvitski, 2011) successfully assigned lipids and carbohydrates in whole Nannochloropsis granulata and monitored the unsaturated/saturated FA ratio as a function of cell harvest time using multidimensional 1 H-13 C HRMAS. Also, Chauton et al. published several studies on different microalgae species, such as C. mülleri [START_REF] Webb | HR MAS NMR spectroscopy of marine microalgae, Part 2: 13C and 13C HR MAS NMR analysis used to study fatty acid composition and polysaccharide structure[END_REF], Dunaliella sp. [START_REF] Chauton | HR MAS NMR spectroscopy of marine microalgae, Part 1: Classification and metabolite composition from HR MAS 1H NMR spectra and multivariate analysis[END_REF], Chaetoceros muelleri [START_REF] Chauton | High-resolution magic angle spinning NMR analysis of whole cells of Chaetoceros muelleri (Bacillariophyceae) and comparison with C-13-NMR and distortionless enhancement by polarization transfer C-13-NMR analysis of lipophilic extracts[END_REF] and Thalassiosira pseudonana [START_REF] Chauton | High-resolution magic angle spinning 1H NMR analysis of whole cells of Thalassiosira pseudonana (Bacillariophyceae): Broad range analysis of metabolic composition and nutritional value[END_REF]. In these works, detailed assignments of lipids were reported, allowing quantification in response to growth conditions. HRMAS was also used to classify microalgae using statistical data treatments such as Principal Component Analysis (PCA). NMR studies performed on other cells and organisms will certainly inspire future investigations of microalgal lipids. For example, HRMAS has been used to characterize erythrocyte membranes and perform permeability studies in Daphnia magna [START_REF] Bruno | Water exchange across the erythrocyte plasma membrane studied by HR-MAS NMR spectroscopy[END_REF][START_REF] Kovacevic | Evaluation of Daphnia magna metabolic responses to organic contaminant exposure with and without dissolved organic matter using 1H nuclear magnetic resonance (NMR)-based metabolomics[END_REF]. Recent work on living earthworms (Eisenia fetida) and freshwater shrimps (Hyalella azteca) have improved the lipid signal treatment based on relaxation and diffusion spectral filtering [START_REF] Mobarhan | Comprehensive multiphase NMR applied to a living organism[END_REF][START_REF] Hassan | Improvements in lipid suppression for 1H NMR-based metabolomics: Applications to solution-state and HR-MAS NMR in natural and in vivo samples[END_REF]).

An alternative to HRMAS studies for 1 H detection is the emerging use of ultrahigh frequency MAS. Although membrane lipids and proteins can be studied at intermediate frequencies (15 kHz [START_REF] Gopinath | Probing the conformationally excited states of membrane proteins via 1H-detected MAS solid-State NMR spectroscopy[END_REF]), ultra-fast MAS (> 50 kHz) allows detection of protons in rigid molecules with increased sensitivity and a resolution comparable to that of solution NMR. Combined with 13 C and 15 N labelling of samples, sequential assignment of membrane proteins or other complex molecules is possible, even in their native environment, e.g. proteins involved in photosynthesis [START_REF] Beal | Lipid analysis of Neochloris oleoabundans by liquid state NMR[END_REF][START_REF] Medeiros-Silva | 1H-detected solid-state NMR studies of water-inaccessible proteins in vitro and in situ[END_REF]. New information on biomembrane structure is expected to emerge from this approach, in particular for the study of membrane proteins in microalgae.

X.5.2 1D 13 C solid-state NMR methods

As stated above, SS-NMR can be exploited to obtain valuable information on cell samples. Depending on the spectral editing used, both rigid and dynamic molecules can be identified and their dynamics in situ revealed -information that is lost using solution NMR of cell extracts, for example. Whole uniformly 13 C-labelled C. reinhardtii were studied by our group in 2015 by 1D 13 C SS-NMR with MAS, along with Pavlova lutheri and Nannochloropsis oculata [START_REF] Arnold | Identification of lipid and saccharide constituents of whole microalgal cells by 13C solid-state NMR[END_REF]. This work allowed the assignment of different lipids, carbohydrates including glycans, as well as rigid cell-wall proteins using different dynamic filters. As shown in Figure X.4, the more mobile molecules of C. reinhardtii are revealed by 13 C SS-NMR using the INEPT filter, the most intense peaks being ascribed to methyl groups from the lipid acyl chains around 30 ppm. This spectrum is comparable to a 13 C solution NMR spectrum using the IDE filter, with a slightly lower resolution. [START_REF] Arnold | Whole cell solid-state NMR study of Chlamydomonas reinhardtii microalgae[END_REF]). Solution NMR experiments were performed at 500 MHz at room temperature (adapted from Akhter et al. [START_REF] Akhter | Identification of aquatically available carbon from algae through solution-state NMR of whole 13C-labelled cells[END_REF]).

On the exact same sample, a CP filter revealed the more rigid molecules such as starch, with C1 carbon at 101 ppm. Starch is invisible in solution NMR, even with a DE filter. These experiments highlight that many lipids are very mobile, while proteins in the cell wall are much more rigid assemblies. In 2018, Azadi-Chegeni et al. published similar work on C. reinhardtii whole cells (Azadi-Chegeni et al., 2018a, Azadi-Chegeni et al., 2018b). For example, they assigned several component resonances in whole cell samples and quantified the dynamics in terms of correlation times and order parameters of these molecules at different temperatures (Azadi-Chegeni et al., 2018a). They also investigated LHCII in native membranes and showed that they favored protein conformational stability. These works represent a step toward the understanding of chloroplast complexity in terms of molecular composition and dynamics. Another interesting study reports the consumption of Chlorella vulgaris lipids by the daphnia Hyalella azteca using comprehensive multiphase NMR and 13 C-labelled microalgae. As shown in Figure X.5, good resolution was obtained and lipid consumption could be monitored with this new probe. Multidimensional solid-state experiments were also recorded with 1 H-13 C correlations from 13 C-labelled algae and 2 H-13 C correlations from 2 H-13 C uniformly labelled algae.

X.5.3 2D-13 C and 15 N-NMR experiments

With isotopically-labelled samples sensitivity is greatly increased, enabling the use of multidimensional SS-NMR which also significantly enhances spectral resolution. 2D SS-NMR NMR has already proven successful in studying biological membranes in situ or in whole cells, mainly bacteria [START_REF] Andronesi | Determination of Membrane Protein Structure and Dynamics by Magic-Angle-Spinning Solid-State NMR Spectroscopy[END_REF][START_REF] Schanda | Atomic model of a cell-wall cross-linking enzyme in complex with an intact bacterial peptidoglycan[END_REF][START_REF] Romaniuk | Bacterial cell wall composition and the influence of antibiotics by cell-wall and whole-cell NMR[END_REF][START_REF] Booth | Recent progress on the application of 2H solid-state NMR to probe the interaction of antimicrobial peptides with intact bacteria[END_REF][START_REF] Bouhlel | Labelling strategy and membrane characterization of marine bacteria Vibrio splendidus by in vivo 2H NMR[END_REF] but also bovine retinas [START_REF] Verhoeven | Ultra-high-field MAS NMR assay of a multispin labeled ligand bound to its G-protein receptor target in the natural membrane environment: electronic structure of the retinylidene chromophore in rhodopsin[END_REF]. More closely related to algal membranes, the work on bacterial chlorosomal antenna by Van Rossum et al. is the first to mention and assign galactolipid signals in a native thylakoid membrane protein study [START_REF] Van Rossum | A refined model of the chlorosomal antennae of the green bacterium Chlorobium tepidum from proton chemical shift constraints obtained with high-field 2-D and 3-D MAS NMR dipolar correlation spectroscopy[END_REF]. As proteins are part of the membrane architecture, it is important to present some of the SS-NMR studies carried out on these proteins in microalgal membranes. Protein conformations, dynamics and interactions with different chloroplast partners in photosynthesis have been described in several model organisms in a membrane environment using multidimensional SS-NMR, for example by the De Groot group, for green and purple bacteria light harvesting complexes [START_REF] Huang | Partial site-specific assignment of a uniformly (13)C, (15)N enriched membrane protein, light-harvesting complex 1 (LH1), by solid state NMR[END_REF]McDermott, 2008, Alia et al., 2009) but also bacteriorhodopsin [START_REF] Varga | Advances towards resonance assignments for uniformly--13C, 15N enriched bacteriorhodopsin at 18.8 T in purple membranes[END_REF]. They have also shown the interaction between photosynthesis-associated proteins and their chromophore, using a single amino acid labelling approach [START_REF] Alia | Ultrahigh Field MAS NMR Dipolar Correlation Spectroscopy of the Histidine Residues in Light-Harvesting Complex II from Photosynthetic Bacteria Reveals Partial Internal Charge Transfer in the B850/His Complex[END_REF]. To better understand the photosynthesis machinery in an environment close to native, the LHCII protein complex was purified from C. reinhardtii with some residual lipids and examined by multidimensional SS-NMR [START_REF] Pandit | First solid-state NMR analysis of uniformly 13C-enriched major light-harvesting complexes from Chlamydomonas reinhardtii and identification of protein and cofactor spin clusters[END_REF], Pandit et al., 2013[START_REF] Sunku | Insights into the photoprotective switch of the major light-harvesting complex II (LHCII): a preserved core of arginine-glutamate interlocked helices complemented by adjustable loops[END_REF]. LHCII switching its conformation from light harvesting to photoprotective mode through nonphotochemical quenching was studied under MAS (<15 kHz) NMR and at temperatures below -30°C.

The only multidimensional SS-NMR study of a light harvesting system in native microalga membranes, without any purification or reconstitution steps, was performed on isolated C. reinhardtii thylakoid membranes [START_REF] Pandit | An NMR comparison of the light-harvesting complex II (LHCII) in active and photoprotective states reveals subtle changes in the chlorophyll a ground-state electronic structure[END_REF]. The resolution was sufficient to report LHCII conformational dynamics -a phenomenon impossible to observe using detergents or protein crystals. The results are thus specific to the dynamics in membranes of a photosynthetic organism, which prevents photo-damage. By using in vivo 2D solid-state NMR and dynamic spectral edition on 13 C-labelled microalgae, we improved our previous 1D NMR results for C. reinhardtii, showing that precise assignments of carbohydrates, lipids and proteins are possible [START_REF] Arnold | Whole cell solid-state NMR study of Chlamydomonas reinhardtii microalgae[END_REF]. Figure X.7 compares rigid and mobile molecules detected by 13 C-13 C SS-NMR using either INEPT-or CP-based filters. A large range of microalgal membrane lipid classes could thus be identified in situ, with those from thylakoid membranes dominating due to their abundance. In addition to lipids and rigid proteins, assignment and characterization of carbohydrates in situ is a necessary step towards a better understanding of carbohydrates architecture in the cell walls of microalgae, and their ability to face osmotic shocks, for example.

X.6. Conclusion and future prospects

In this chapter, we have reviewed the NMR techniques that can provide molecular-level information on the structure, as well as dynamics, of microalgal membranes constituents. So far, little is known about these membranes reflecting their overall complexity but they are a key component to understanding the photosynthetic machinery, and to bioengineer new strains for biofuel or food supplements.

We have shown the versatility of NMR, which can detect nuclei, such as 1 H, 31 P, 13 C, 15 N, 2 H and 19 F, in either single or multidimensional experiments. It is applicable to small or large, dynamic or rigid molecules, to metabolites, model membranes, cell extracts, whole cells and even living cells. For NMR, isotopic labelling is generally required. While uniform labelling is routinely performed, as demonstrated here with examples of fully 13 C-labelled microalgae, new approaches can certainly be developed to specifically label molecules of interest within whole microalgae, providing that the biosynthetic pathways are known. This strategy has notably been applied to bacteria and to 2 H in vivo SS-NMR study of their membranes [START_REF] Booth | Recent progress on the application of 2H solid-state NMR to probe the interaction of antimicrobial peptides with intact bacteria[END_REF].

Depending on the sample and cellular constituent studied, solution or SS-NMR approaches are available. In the solid state, they include static or MAS samples, spinning at moderate or high frequencies. New techniques continue to emerge, including the use of dynamic filters and comprehensive multiphase NMR, which are useful to discriminate membrane lipids from lipid droplets, for example.

Beside broad molecular identification, NMR focuses on high-resolution structure determination of biomolecules. Microalgal membrane proteins and photosynthetic systems are important targets since they are involved in metabolism and transport. Their structures will need to be determined to understand these mechanisms [START_REF] Ladizhansky | Applications of solid-state NMR to membrane proteins[END_REF]. Revealing the structure of microalgae cell walls (including their connexion to the membrane) is also an area where SS-NMR can contribute, either within whole cells or extracts, as has been done with higher plants to obtain atomic resolution of cellulose nanostructures (Dick-Pérez et al., 2011, Wang and[START_REF] Wang | Solid-state NMR investigations of cellulose structure and interactions with matrix polysaccharides in plant primary cell walls[END_REF]. Such information is essential for a fundamental understanding of microalgal species and also for commercial exploitation of this aquatic resource.

Finally, Dynamic Nuclear Polarization (DNP) coupled with SS-NMR has been shown to be very powerful for studies of biological assemblies, such as bacterial cell walls [START_REF] Wang | Cellulose structural polymorphism in plant primary cell walls investigated by high-field 2D solid-state NMR spectroscopy and density functional theory calculations[END_REF], or membrane proteins [START_REF] Becker-Baldus | Enlightening the photoactive site of channelrhodopsin-2 by DNP-enhanced solid-state NMR spectroscopy[END_REF], with unprecedented sensitivity. Using this approach on microalgae should improve our knowledge of the complex entanglement of proteins, lipids and carbohydrates in cellular external matrices. It is exciting to see that, while in-cell SS-NMR is still the source of many new developments, it has now reached the maturity to be applied to whole microalgae to give better understanding of living organisms at an atomic level.

Figure X. 1 .

 1 Figure X.1. Microalgal cell with organelles and membrane structures. The cell wall is generally composed of glycoproteins, cellulosic material, carbonate or silica and the plasma membrane is rich in phospholipids. Chloroplast membranes have different galactolipid-tophospholipid ratios, going from 19:81 in the chloroplast outer layer to 2:98 in thylakoid membranes[START_REF] Harwood | Membrane Lipids in Algae[END_REF]. Lipid droplets incorporate TAGs and ergosterol, as a dense energy storage, surrounded by a single phospholipid monolayer. Energy can also be stored as complex carbohydrates, namely starch or chrysolaminarin.

Figure X. 2 .

 2 Figure X.2. Main lipids and fatty acids found in microalgae. Galactolipids and betaine lipids are the most abundant in microalgae, and mainly located in photosynthetic membranes. Phospholipids are found in other membranes. Triacylglycerols (TAGs) are stored in lipid droplets. The lipid profile and fatty acid chain composition are species-and mediumdependent. As a reference, values for C. reinhardtii are indicated[START_REF] Vieler | The lipid composition of the unicellular green alga Chlamydomonas reinhardtii and the diatom Cyclotella meneghiniana investigated by MALDI-TOF MS and TLC[END_REF][START_REF] Nguyen | The green microalga Chlamydomonas reinhardtii has a single omega-3 fatty acid desaturase that localizes to the chloroplast and impacts both plastidic and extraplastidic membrane lipids[END_REF].

Figure

  Figure X.3. 1 H-13 C MAS (5 kHz) NMR PDLF spectra of a model membrane made of unlabelled DMPC and MGDG, recorded at 42°C. The 1D 13 C NMR spectrum is displayed on the left. Reprinted from (Castro et al., 2007) with permission.

Figure X. 4 .

 4 Figure X.4. Comparison of 13 C labelled C. reinhardtii whole cell spectra by solution and solidstate NMR using either single pulse or filtered experiments. Solid-state experiments were performed at 600 MHz at 25°C with a 10 kHz MAS frequency (adapted from Arnold et al.[START_REF] Arnold | Whole cell solid-state NMR study of Chlamydomonas reinhardtii microalgae[END_REF]). Solution NMR experiments were performed at 500 MHz at room temperature (adapted from Akhter et al.[START_REF] Akhter | Identification of aquatically available carbon from algae through solution-state NMR of whole 13C-labelled cells[END_REF]).

Figure

  Figure X.5. 13 C CMP NMR (15°C and 1.1 kHz spinning frequency) spectra of uniformly 13 Clabelled Chlorella vulgaris microalgae with and without H. azteca daphnia. The pink region highlights the glucose peaks that increase when H. azteca is in contact with the algae. Reprinted from (Liaghati Mobarhan et al., 2019) with permission.

Figure X. 6 .

 6 Figure X.6. 13 C-13 C solid-state NMR spectra of LHCII from C. reinhardtii in: (A) its native thylakoid environment (red contours), reconstituted in β-n-dodecyl-D-maltoside detergent buffer (light blue contours); and (B) pure extracted protein (black contours) and in a reconstituted thylakoid membrane (red contours). (C) The LHCII crystal structure, and (D) chlorophyll molecular structure are also given. A is adapted from (Pandit et al., 2013), and B-D are adapted from (Azadi-Chegeni et al., 2018b), with permission.

  Figure X.6 compares 13 C-13 C spectra of LHCII in its native thylakoid environment and reconstituted in detergent. This can be compared also to 13 C-13 C spectra of the same protein extracted from its membrane or reconstituted in a model thylakoid-like membrane, as done by Azadi-Chegeni et al. (Azadi-Chegeni et al., 2018b).

Figure X. 7 .

 7 Figure X.7. 13 C-13 C in vivo solid-state NMR spectra of uniformly 13 C-labelled C. reinhardtii recorded at a MAS frequency of 26.7 kHz using either INEPT-(in red) or CP-based (in blue) filters. From (Arnold et al., 2018) with permission.

X.7 Acknowledgements

This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada (grant RGPIN-2018-06200 to I.M.) and the Centre National de la Recherche Scientifique (UMR 7203 to D.E.W.). A.P. would like to thank the PROTEO network, the Faculty of Science of the Université du Québec à Montréal (UQAM) and the Fonds Québécois de la Recherche sur la Nature et les Technologies for the award of scholarships. I.M. is a member of Ressources Aquatiques Québec (RAQ) and PROTEO research networks.


Erna K Wieduwilt

Giovanni Macetti

Alessandro Genoni

Climbing Jacob's Ladder of Structural Refinement: Introduction of a Localized Molecular Orbital-Based Embedding for Accurate X-ray Determinations of Hydrogen Atoms Positions

Positions of hydrogen atoms in molecules are fundamental in many aspects of chemistry.

Nevertheless, most of molecular structures are obtained from refinements of X-ray data exploiting the independent atom model (IAM), which uses spherical atomic densities and provides bond lengths involving hydrogen atoms that are too short compared to the neutron reference values. To overcome the IAM shortcomings, the wave function-based Hirshfeld atom refinement (HAR) method has been recently proposed, emerging as a promising strategy able to give element-hydrogen bond distances in excellent agreement with the neutron ones in terms of accuracy and precision. In this Letter, we propose a significant improvement of HAR based on the idea of describing the crystal environment explicitly in the underlying wave function calculation through a quantum mechanical embedding strategy that exploits extremely localized molecular orbitals. Test-bed refinements on a crystal structure characterized by strong intermolecular interactions are also discussed.

KEYWORDS: X-ray structural refinement, Quantum Crystallography, Hirshfeld Atom Refinement, Extremely Localized Molecular Orbitals, QM/ELMO embedding technique, hydrogen atoms positions Nowadays experimental crystal structures are undoubtedly important points of departure to start evaluating molecular properties and to perform molecular computations or simulations.

Of course, the higher the accuracy of the crystal structure, the better the starting point. In this framework, a crucial role is actually played by positions and thermal parameters of hydrogen atoms, which are fundamental in many aspects and fields of chemistry. [START_REF] Halliwell | Free Radicals in Biology and Medicine[END_REF][START_REF] Hynes | Hydrogen Transfer Reactions[END_REF][START_REF] Jena | Materials for hydrogen storage: Past, present, and future[END_REF][START_REF] Desiraju | Crystal Engineering -The Design of Organic Solids[END_REF][START_REF] Desiraju | The Weak Hydrogen Bond in Structural Chemistry and Biology[END_REF][START_REF] Hoser | Dynamic quantum crystallography: lattice dynamical models refined against diffraction data. II. Applications to L-alanine, naphthalene and xylitol[END_REF] To obtain reliable hydrogen structural parameters, one should resort to neutron diffraction, which really enables to determine hydrogen atoms positions accurately and precisely because neutrons are strongly diffracted by atomic nuclei. Nevertheless, the use of this experimental technique is intrinsically and strongly limited by the fact that it requires nuclear reactors or spallation sources. For this reason, X-ray diffraction remains the method of choice, although it is often neglected that practically almost all the X-ray-based structural determinations rely on the very basic independent atom model (IAM), [START_REF] Compton | The Distribution of Electrons in Atoms[END_REF] an approximation that considers the electron density of the molecule under exam as sum of tabulated, spherical atomic electron distributions previously obtained through quantum mechanical calculations on isolated atoms. Therefore, IAM completely neglects the aspherical deformation of the atomic electron densities due to chemical bonds. Consequently, since X-rays are diffracted by electrons and hydrogen atoms have only one electron, IAM unavoidably leads to element-hydrogen (E-H) bond distances that are significantly shorter than those observed from neutron diffraction measurements.

To partially overcome this drawback the E-H bond lengths can be adjusted to tabulated and averaged neutron diffraction-based values. [START_REF] Allen | International Tables for Crystallography[END_REF][START_REF] Allen | Bond lengths in organic and metal-organic compounds revisited: X-H bond lengths from neutron diffraction data[END_REF] However, this approach lacks flexibility and cannot be applied to hydrogen atoms in particular bonding situations. Therefore, refinement strategies that enable to obtain accurate and precise E-H bond lengths specifically for systems/crystals under examination were proposed in the framework of quantum crystallography [10][START_REF] Genoni | Quantum Crystallography: Current Developments and Future Perspectives[END_REF][START_REF] Massa | Quantum Crystallography: A perspective[END_REF][START_REF] Genoni | Quantum Crystallography in the Last Decade: Developments and Outlooks[END_REF][START_REF] Grabowsky | The Advent of Quantum Crystallography: Form and Structure Factors from Quantum Mechanics for Advanced Structure Refinement and Wavefunction Fitting[END_REF] over the years. Most of them tried to account for the asphericity of the atomic electron distributions in chemical bonding [START_REF] Stewart | Coherent X-Ray Scattering for the Hydrogen Atom in the Hydrogen Molecule[END_REF] through the multipole models of the electron density, [START_REF] Stewart | Electron Population Analysis with Rigid Pseudoatoms[END_REF][START_REF] Hansen | Testing Aspherical Atom Refinements on Small-Molecule Data Sets[END_REF] exploiting pseudoatoms databanks [START_REF] Dittrich | The invariom model and its application: refinement of D,L-serine at different temperatures and resolution[END_REF][START_REF] Dittrich | How to easily replace the independent atom model -the example of bergenin, a potential anti-HIV agent of traditional Asian medicine[END_REF][START_REF] Bendeif | The experimental library multipolar atom model refinement of L-aspartic acid[END_REF][START_REF] Bąk | Verification of Structural and Electrostatic Properties Obtained by the Use of Different Pseudoatom Databases[END_REF][START_REF] Dadda | Charge-density analysis and electrostatic properties of 2-carboxy-4-methylanilinium chloride monohydrate obtained using a multipolar and a spherical-charges model[END_REF][START_REF] Jha | TAAM: A reliable and user friendly tool for hydrogen atom location using routine X-ray diffraction data[END_REF] or ad hoc multipole model parameters for the hydrogen atom corresponding to its electron density in the H2 molecule [START_REF] Destro | Bond lengths, and beyond[END_REF] .

However, in this context, the most important step forward is represented by the more recent Hirshfeld atom refinement (HAR) method. [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][START_REF] Woińska | Hirshfeld atom refinement for modeling strong hydrogen bonds[END_REF][START_REF] Woińska | Hydrogen atoms can be located accurately and precisely by X-ray crystallography[END_REF][START_REF] Wall | Quantum crystallographic charge density of urea[END_REF][START_REF] Fugel | Probing the accuracy and precision of Hirshfeld atom refinement with HARt interfaced with Olex2[END_REF] At each iteration of the refinement, this technique requires a quantum chemical calculation that provides a molecular electron density that is afterwards subdivided into aspherical atomic contributions by exploiting the so-called Hirshfeld stockholder partitioning approach [START_REF] Hirshfeld | Spatial Partitioning of Charge Density[END_REF][START_REF] Hirshfeld | Bonded-Atom Fragments for Describing Molecular Charge Densities[END_REF] . Extensive test-bed refinements have clearly shown that HAR is generally able to provide E-H bond distances that are as accurate and precise as those obtained from neutron diffraction experiments, also with X-ray diffraction data at resolutions as low as 0.8 Å. [START_REF] Woińska | Hydrogen atoms can be located accurately and precisely by X-ray crystallography[END_REF] However, the real quality of ADPs resulting from HARs is still unclear and, in general, they have to be taken with caution. [START_REF] Köhler | Comparison of different strategies for modelling hydrogen atoms in charge density analyses[END_REF][START_REF] Malaspina | Hydrogen atoms in bridging positions from quantum crystallographic refinements: influence of hydrogen atom displacement parameters on geometry and electron density[END_REF] In fact, although physically reasonable, they do not seem as accurate as those resulting from the refinements of neutron data or as those estimated through SHADE [START_REF] Madsen | SHADE web server for estimation of hydrogen anisotropic displacement parameters[END_REF][START_REF] Madsen | SHADE3 server: a streamlined approach to estimate H-atom anisotropic displacement parameters using periodic ab initio calculations or experimental information[END_REF] (simple hydrogen anisotropic displacement estimator).

Initially based on Hartree-Fock (HF) and density functional theory (DFT) calculations, the HAR technique has been also recently extended to post-HF multi-determinant strategies. [START_REF] Wieduwilt | Post-Hartree-Fock methods for Hirshfeld atom refinement: are they necessary? Investigation of a strongly hydrogen-bonded molecular crystal[END_REF] Moreover, to limit its intrinsically higher computational cost and to extend it to the refinement of macromolecular crystal structures, HAR has been coupled with libraries of extremely localized molecular orbitals [START_REF] Meyer | Libraries of Extremely Localized Molecular Orbitals. 1. Model Molecules Approximation and Molecular Orbitals Transferability[END_REF][START_REF] Meyer | Libraries of Extremely Localized Molecular Orbitals. 2. Comparison with the Pseudoatoms Transferability[END_REF][START_REF] Meyer | Libraries of Extremely Localized Molecular Orbitals. 3. Construction and Preliminary Assessment of the New Databanks[END_REF] (ELMOs; [START_REF] Stoll | On the Use of Local Basis Sets for Localized Molecular Orbitals[END_REF][START_REF] Fornili | Determination of extremely localized molecular orbitals and their application to quantum mechanics/molecular mechanics methods and to the study of intramolecular hydrogen bonding[END_REF][START_REF] Genoni | A Novel Approach to Relax Extremely Localized Molecular Orbitals: the Extremely Localized Molecular Orbital-Valence Bond Method[END_REF][START_REF] Genoni | Optimal Virtual Orbitals to Relax Wave Functions Built Up with Transferred Extremely Localized Molecular Orbitals[END_REF][START_REF] Genoni | A novel extremely localized molecular orbitals based technique for the one-electron density matrix computation[END_REF][START_REF] Sironi | Extremely Localized Molecular Orbitals: Theory and Applications[END_REF][START_REF] Sironi | A new program to determine electron densities of polypeptides using extremely localized molecular orbitals[END_REF] see Supporting Information for more details about ELMOs and ELMO databanks). This gave rise to the HAR-ELMO approach 48 that enabled to successfully refine crystal structures of polypeptides, proteins and also organometallic compounds at a significantly reduced computational cost compared to traditional Hirshfeld atom refinements. In the same context, it is also worth mentioning the fragHAR method, [START_REF] Bergmann | fragHAR: towards ab initio quantum-crystallographic X-ray structure refinement for polypeptides and proteins[END_REF] which combines the traditional HAR technique with the molecular fractionation with conjugate caps (MFCC) strategy [START_REF] Zhang | Molecular fractionation with conjugate caps for full quantum mechanical calculation of protein-molecule interaction energy[END_REF] to speed up the calculations of QMbased structure factors for polypeptides and proteins.

Test calculations on HAR have also shown that taking into account the crystal environment is crucial to carry out better refinements, especially when strong intermolecular interactions (e.g., hydrogen bonds) are present. [START_REF] Fugel | Probing the accuracy and precision of Hirshfeld atom refinement with HARt interfaced with Olex2[END_REF] To accomplish this task, instead of performing expensive computations on large supermolecules that account for the existing interactions in the crystal, point charges obtained from the quantum mechanical calculation at the previous refinementstep are generally placed at symmetry-generated positions around the reference crystal unit within a given radius from any atom (default value set equal to 8 Å). [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][START_REF] Woińska | Hirshfeld atom refinement for modeling strong hydrogen bonds[END_REF] This simple strategy generally speeds up the convergence of the HAR procedure and, above all, it improves the agreement with neutron results.

In this Letter, we introduce a new and significant step forward for the embedding technique in HAR. Instead of simply using a cluster of point charges, in the refinement process we propose to mimic the crystal environment of the reference unit at a fully quantum mechanical level by exploiting a novel embedding approach: the quantum mechanics / extremely localized molecular orbital (QM/ELMO) method, [START_REF] Macetti | Quantum Mechanics/Extremely Localized Molecular Orbital Method: A Fully Quantum Mechanical Embedding Approach for Macromolecules[END_REF][START_REF] Macetti | Localized Molecular Orbital-Based Embedding Scheme for Correlated Methods[END_REF][START_REF] Macetti | Quantum Mechanics/Extremely Localized Molecular Orbital Embedding Strategy for Excited States: Coupling to Time-Dependent Density Functional Theory and Equation-of-Motion Coupled Cluster[END_REF] namely a technique that successfully enables to treat the most important (or chemically relevant) region of the system under examination at a higher level of theory (HF, DFT or post-HF levels), while the remaining part is described through frozen extremely localized molecular orbitals previously transferred from the recently constructed ELMO libraries [START_REF] Meyer | Libraries of Extremely Localized Molecular Orbitals. 3. Construction and Preliminary Assessment of the New Databanks[END_REF] or from proper tailor-made model molecules. Along this line, another possibility considered in this work is the coupling of HAR with the more recent QM/ELMO/MM technique, a three-layer approach where the outermost region is treated by means of classical Molecular Mechanics (MM). [START_REF] Macetti | Three-Layer Multiscale Approach Based on Extremely Localized Molecular Orbitals to Investigate Enzyme reactions[END_REF] As in a traditional Hirshfeld atom refinement, also for the new version of the method proposed in this work, it is necessary to define a reference crystal-unit, which must correspond at least to the asymmetric unit of the crystal structure that one wants to refine.

After this preliminary definition, the new embedded-HAR technique consists in the following self-consistent cycle:

1. Quantum chemical computation on the reference unit. In our case, instead of a simple QM calculation or a QM computation with an embedding cluster of point charges, we perform a QM/ELMO or QM/ELMO/MM calculation in which the chosen reference crystal unit is fully treated at QM level (HF, DFT or post-HF), while the crystal environment (i.e., the other symmetry-generated units within a given radius from any atom of the reference unit) is described through transferred extremely localized molecular orbitals and, in the QM/ELMO/MM case, also through a classical MM force field (see Figure 1). This allows a complete or partial quantum mechanical treatment of the embedding due to the crystal environment (see Supporting Information for more details about the QM/ELMO [START_REF] Macetti | Quantum Mechanics/Extremely Localized Molecular Orbital Method: A Fully Quantum Mechanical Embedding Approach for Macromolecules[END_REF][START_REF] Macetti | Localized Molecular Orbital-Based Embedding Scheme for Correlated Methods[END_REF][START_REF] Macetti | Quantum Mechanics/Extremely Localized Molecular Orbital Embedding Strategy for Excited States: Coupling to Time-Dependent Density Functional Theory and Equation-of-Motion Coupled Cluster[END_REF] and QM/ELMO/MM 54 methods).

Partitioning of the electron density into Hirshfeld atoms. The stockholder Hirshfeld

partitioning technique [START_REF] Hirshfeld | Spatial Partitioning of Charge Density[END_REF][START_REF] Hirshfeld | Bonded-Atom Fragments for Describing Molecular Charge Densities[END_REF] is used to subdivide the reference-unit electron density 𝜌(𝒓) (i.e., the electron density of the QM region) obtained at the previous step into aspherical atomic contributions {𝜌 ! (𝒓)} according to the following equation:

𝜌 ! (𝒓) = 𝑤 ! (𝒓) 𝜌(𝒓) (1)
,

where 𝑤 ! (𝒓) is the Hirshfeld partitioning function for the generic atom 𝐴. This is given by the spherically averaged atomic density 𝜌 ! " (𝒓) of atom 𝐴 divided by the sum of the spherically averaged densities associated with the atoms in the reference crystal-unit (namely, divided by the "promolecular" density of the reference crystal-unit): 4. Least-squares minimization of the statistical agreement 𝜒 . . The calculated and experimental structure factor amplitudes (here indicated as ./𝐹 𝒉 ,%-, /1 and ./𝐹 𝒉 /01 /1 , respectively) are compared to obtain the statistical agreement 𝜒 . :

𝑤 ! (𝒓) = 𝜌 ! " (𝒓) ∑ 𝜌 # " (𝒓) $%&"'( #)* (2)
𝜒 . = 1 𝑁 2 -𝑁 1 6 7𝜂/𝐹 𝒉 ,%-, / -/𝐹 𝒉 /01 /9 . 𝜎 𝒉 . 𝒉 (3)
where 𝑁 2 is the number of reflections taken into account, 𝑁 1 the number of parameters used in the model, and 𝜂 a scale-factor that sets the computed structure factor amplitudes on the same scale of the experimental ones. 𝜒 . is least-squared minimized with respect to 𝜂 and the atomic structural parameters (namely, atomic coordinates and ADPs) of the chosen reference unit. 5. Check of convergence. The convergence of the refinement is checked on the atomic structural parameters: if it is achieved, the procedure is halted and a refined crystal structure is obtained, otherwise the new geometry for the reference crystal unit is used to restart the cycle and, in particular, to carry out another QM/ELMO or QM/ELMO/MM computation (see point 1 above).

The new ELMO-based embedded-HAR technique has been implemented by interfacing the quantum crystallographic software Tonto 55 (Version No. 19.12.17 v. dd352d58), which enables to perform Hirshfeld atom refinements, with an in-house modified version of the quantum chemistry package Gaussian09, [START_REF] Frisch | Revision D.01[END_REF] where the QM/ELMO and QM/ELMO/MM approaches have been coded. Also Tonto (Version No. 19.12.17 v. dd352d58) was slightly modified to enable the reading of the QM/ELMO and QM/ELMO/MM formatted checkpoint files. In particular, our variants of Tonto and Gaussian09 were coupled through an in-house Bash script that follows the philosophy of the lamaGOET [START_REF] Malaspina | lamaGOET: an Interface for Quantum Crystallography[END_REF][START_REF] Malaspina | The advanced treatment of hydrogen bonding in quantum crystallography[END_REF] GUI (graphical user interface), according to which i) the geometry of the system under exam is passed from Tonto to a quantum chemical software in order to perform a QM calculation and ii) the resulting density matrix/wave function is afterwards read in Tonto for the subsequent structural refinement, repeating the procedure until convergence is reached. A similar strategy is also followed by the recently proposed NoSpherA2 system 59 that allows to perform HARs through the Olex2 software [START_REF] Dolomanov | 2: a complete structure solution, refinement and analysis program[END_REF][START_REF] Bourhis | The anatomy of a comprehensive constrained, restrained refinement program for the modern computing environment -Olex2 dissected[END_REF] by exploiting wavefunction calculations computed by means of traditional quantum chemistry packages.

To test the refinement method described above, we considered the crystal structure of xylitol, which is characterized by strong intermolecular interactions (particularly, hydrogen bonds).

The X-ray data used for the refinements were collected by Madsen and collaborators 62 at 122.4 K and up to a resolution of 1.22 Å -1 (no cutoffs were applied). Four different types of Hirshfeld atom refinements were performed: i) traditional HARs without embedding, ii) HARs with crystal environment given by a cluster of point charges (4 Å and 8 Å as radii to define the clusters), iii) HARs with crystal embedding treated at QM/ELMO level (4 Å and 8 Å as radii to define the ELMO regions), iv) HARs with crystal environment described at QM/ELMO/MM level with an ELMO region up to 4 Å and an MM region from 4 to 8 Å (see Figure 1 and also Table S1 in the Supporting Information). Refinements (i) and (ii) were performed by exploiting the lamaGOET GUI (https://github.com/lomalaspina/lamaGOET), while (iii) and (iv) were carried out using our in-house script. In all cases, the reference crystal-unit was described at DFT-B3LYP level of theory with basis-sets cc-pVDZ and cc-pVTZ. For all the QM/ELMO/MM computations, the MM subsystem was treated by interfacing our modified version of Gaussian09 with the Molecular Dynamics package AMBER 2016 63 and exploiting the general AMBER force field (GAFF) [START_REF] Wang | Development and Testing of a General Amber Force Field[END_REF] with charges generated using the AM1-BCC model [START_REF] Jakalian | Fast, efficient generation of high-quality atomic charges. AM1-BCC model: I[END_REF][START_REF] Jakalian | Fast, efficient generation of high-quality atomic charges. AM1-BCC model: II. Parameterization and validation[END_REF] . Finally, the results of the different HARs were compared to those obtained through the refinement of corresponding neutron data 67 measured at 122 K.

For the sake of completeness, figures of merits and descriptors that show the positive outcome of the performed refinements are shown in Table S2 (𝜒 . , 𝑅, maximum and minimum residual density values) and Figure S3 (plots of the Meindl-Henn fractal distributions [START_REF] Meindl | Foundations of residual-density analysis[END_REF] ). Here we will focus on the analysis of the obtained structural results.

To this regard, let us now consider the positions of the oxygen-bonded hydrogen atoms, which are also the atoms involved in strong hydrogen bond interactions with the other xylitol units in the crystal. The O-H bond distances resulting from the different variants of HAR are shown in Figure 2. Concerning the results obtained with the cc-pVDZ basis set, it is easy to observe that the Hirshfeld atom refinement without embedding and the HARs exploiting clusters of point charges significantly underestimate all the neutron reference values, while all the ELMO-embedded HARs provide bond lengths that optimally agree with the neutron results. The same trends were observed when basis-set cc-pVTZ was used for the underlying quantum chemical calculations. In fact, the O-H bond distances obtained through traditional HARs without embedding or with embeddings of point charges are systematically shorter than the considered reference values, while those resulting from refinements with the crystal environment described at ELMO or ELMO/MM level are in very good agreement with the benchmark results, except for the O3-H13 bond whose length is overestimated. Furthermore, we can interestingly observe that the results do not change when the size of the embedding region is varied (4 or 8 Å), but only when a different strategy to treat the crystal embedding is adopted.

The results for the C-H bond lengths are displayed in Figure S4 of the Supporting Information. In this case, for both the two considered basis-sets, we can see that the bond distances obtained through the different HARs are generally indistinguishable from the statistical point of view. This can be explained with the fact that most of C-H bonds in xylitol do not establish intermolecular interactions and, therefore, carrying out an embedded QM calculation is not really essential to improve the corresponding bond lengths. This is in agreement with a recent study conducted by Chodkiewicz et al., who also proposed extensions of HAR to other partitioning methods of the electron density. [START_REF] Chodkiewicz | Hirshfeld atom like refinement with alternative electron density partitions[END_REF] However, we can also observe that the two cases in which the distances obtained through ELMO-embedded HARs differ the most from those resulting from traditional HARs with cluster charges correspond to C-H bonds that are involved in hydrogen bond-like contacts with the surrounding xylitol units, namely bonds C4-H4 and C1-H1B (see also Figure S5 in the Supporting Information, reporting the network of inter-and intra-molecular contacts as obtained from PLATON analysis 70 ). For a more quantitative comparison, in Table 1 we reported the mean absolute deviations (MADs) of the determined bond distances from the corresponding reference neutron values.

Other than C-H and O-H bond lengths already discussed above, in this analysis we also specifically considered H⋯A (hydrogen-acceptor) distances and O-H⋯A (oxygen-hydrogenacceptor) angles related to the hydrogen bonds established by xylitol in the crystal.

Concerning the O-H distances, regardless of the used basis-set, for HARs based on QM/ELMO and QM/ELMO/MM calculations we have MADs that are significantly lower (even from a statistical point of view) than those resulting from HARs without embedding or with cluster charges. For example, considering the cc-pVTZ basis-set, the mean absolute deviation associated with the Hirshfeld atom refinement without embedding amounts to 0.029 (11) Å, it decreases to 0.024 (10) Å for the HARs with clusters of point charges, and it eventually drops to 0.006(6) Å when HAR is coupled with the QM/ELMO and QM/ELMO/MM strategies. Analogous results are observed for the H⋯A distances, although for the cc-pVTZ basis-set MADs differ by less than one standard deviation. Also for the O-H⋯A angles, the mean absolute deviations systematically decrease as the crystal environment is described more and more accurately. The same trends are observed for the C-H bond lengths, but, in agreement with the results already reported in Figure S4, the discrepancies associated with the ELMO-embedded refinements are not significantly lower than those corresponding to traditional HARs. Similar conclusions can be also drawn from the analysis of the mean ratios between HAR and neutron bond lengths, which are reported in the Supporting Information (see Table S3) along with the actual values of bond distances and angles associated with hydrogen bond-like contacts in the xylitol crystal structure (see Tables S4 andS5). Finally, before concluding this discussion on the statistical analyses of the obtained bond lengths, two further aspects have to be pointed out. First of all, it is again evident that it is the strategy through which the crystal environment is described during the refinement that impacts on the final results, and not the size of the embedding region (i.e., 4 and 8 Å radii always give identical results). Furthermore, it is also interesting to note that the standard deviations associated with the MADs for ELMO-embedded HARs are always lower than those for the other types of refinements, thus indicating that, on average, we have a lower variability of the discrepancies from the neutron results when the crystal environment is described more accurately. a) For each distance or angle, the first column reports the mean absolute discrepancies, while the second one provides the corresponding population standard deviations upon averaging.

As already mentioned above, thermal parameters of hydrogen atoms resulting from HARs do not generally seem as accurate as the obtained E-H bond lengths when they are compared to neutron reference values, although it is also important to bear in mind that differences between X-ray and neutron ADPs may be also caused by other effects, such as absorption, extinction, disorder, thermal diffuse scattering, multipole scattering and scan truncation. [START_REF] Blessing | On the Differences Between X-ray and Neutron Thermal Vibration Parameters[END_REF][START_REF] Larsen | Diffraction studies of crystals at low temperatures -crystallography below 77[END_REF] In this regard, we will now examine the anisotropic displacement parameters resulting from the Hirshfeld atom refinements carried out in this work with a dual goal: i) completing the analysis of the obtained structural data and, above all, ii) understanding whether the quantum mechanical treatment of the crystal environment can be also advantageous to improve the description of the (hydrogen) anisotropic displacement parameters.

In Figures S6 and S7 we showed the refined structures of xylitol with thermal ellipsoids corresponding to the obtained ADPs. From visual inspection, except for some differences at hydrogens H13, H14 and H15, all the performed HARs provided almost identical results, with ellipsoids for the hydrogen atoms that are sometimes more elongated and differently oriented compared to the neutron reference ones. To better visualize the discrepancies between the (Xray) HAR and neutron ADPs, we also reported PEANUT plots [START_REF] Hummel | Computer graphics program to represent atomic displacement parameters[END_REF] in Figures S8 andS9. The depicted differences for non-hydrogen atoms and hydrogen atoms bonded to carbons are visually identical for all the considered HAR strategies, while for hydrogen atoms bonded to oxygens the PEANUT isosurfaces corresponding to HARs with ELMO embedding are substantially different from those obtained through the other two variants of Hirshfeld atom refinement. For a more quantitative analysis, in Tables 2 and3 we reported the mean absolute deviations of the obtained diagonal and non-diagonal ADPs from the corresponding neutron values. For non-hydrogen atoms and hydrogens bonded to carbons, all the HARs practically gave identical results. On the contrary a trend is observed for the ADPs of hydrogen atoms bonded to oxygen, thus confirming what was qualitatively observed in the PEANUT plots. In fact, especially for the diagonal elements, the MADs resulting from HARs coupled with the QM/ELMO and QM/ELMO/MM calculations are much lower, although not statistically different, from those obtained through the other refinements. Analogous conclusions can be also drawn from the average ratios of the ADPs diagonal elements (see Table S6 in the Supporting Information). a) For each group of atoms, the first column reports the mean absolute discrepancies, while the second one provides the corresponding population standard deviations upon averaging. a) For each group of atoms, the first column reports the mean absolute discrepancies, while the second one provides the corresponding population standard deviations upon averaging.

To rationalize all the above-discussed results, we finally decided to compare the electron densities associated with the quantum mechanical calculations underlying the performed Hirshfeld atoms refinements. They were computed on the IAM structure of xylitol, which was the starting geometry for all the performed HARs. In this way, the obtained electron distributions were directly comparable. Furthermore, it is worth noting that the largest structural changes in HARs are usually observed at the first iteration, which is therefore the most crucial one and usually has the largest influence on the final outcome of the refinement.

The comparison of the electron densities computed on the starting IAM structure should thus allow us to explain the main reasons behind the different results obtained with the HAR variants considered in this study. To this aim, using as reference the electron distribution computed at B3LYP level without embedding, in Figure 3 we displayed all the electron density differences obtained with basis-set cc-pVTZ (see Figure S10 in the Supporting Information for the analogous plots obtained with basis-set cc-pVDZ). We can observe that, for all the refinements with embedding, the differences are mainly localized on all the hydroxy groups of xylitol. They are significantly more evident in the ELMO-embedded cases, for which we also notice some non-negligible variations in correspondence of the C4-H4 and C1-H1B bonds whose lengths were indeed those that changed the most among the C-H ones when ELMO-embedded HARs were performed. Concerning the density differences detected at the hydroxy groups for the ELMO-embedded calculations, we have charge depletions at the hydrogens and charge accumulations at the oxygens (particularly, in the directions pointing towards the surroundings hydrogens atoms), thus indicating charge density shifts ascribable to the hydrogen bond interactions. Therefore, it is clear that the ELMO embeddings allowed to significantly and successfully account for those electron density variations that are due to intermolecular interactions with the crystal environment, thus leading to O-H (and to some C-H) refined bond lengths that are in much better agreement with the neutron reference values. without the need of also including an outer Molecular Mechanics layer. Further tests will be performed to completely assess all capabilities and limitations of the new approach. However, borrowing the concept of Jacob's ladder used for the classification of exchange-correlation functionals in DFT, [START_REF] Perdew | s ladder of density functional approximations for the exchangecorrelation energy[END_REF] the strategy proposed in this work could also be seen as an attempt of introducing a new rung in the stairway that connects IAM to real crystal structures, thus paving the way to future couplings of HAR with other fully QM embedding methods of quantum chemistry [START_REF] Wesolowski | Frozen-Density Embedding Strategy for Multilevel Simulations of Electronic Structure[END_REF]76 order to carry out more and more accurate X-ray based structural refinements. Figures S8 and S9 depicting the PEANUT plots of the differences between (X-ray) HAR and neutron ADPs. Figure S10 showing electron density differences computed on the IAM structure with basis-set cc-pVDZ. Table S1 reporting the number of atoms and basis functions involved in the ELMO-embedded Hirshfeld atom refinements. Table S2 reporting the values of the figures of merit associated with the different HARs. Table S3 providing the mean ratios between HAR and neutron distances/angles. Tables S4 andS5 This material is available free of charge via the Internet at http://pubs.acs.org/.

Figure 1 .

 1 Figure 1. Schematic representations (with xylitol molecules in polyhedral rendering) of the crystal environment treatment in the ELMO-embedded Hirshfeld atom refinements: A) QM/ELMO and B) QM/ELMO/MM cases. Red: central QM unit; blue: ELMO subsystem; cyan: MM region.

3 .

 3 Computation of structure factors. The aspherical atomic density functions {𝜌 ! (𝒓)} resulting from point 2 are Fourier transformed to get the corresponding thermally averaged atomic form factors, which are then summed to obtain the global calculated structure factor amplitudes ./𝐹 𝒉 ,%-, /1 , with 𝒉 as the generic triad of Miller indices characterizing each reflection.

Figure 2 .

 2 Figure 2. O-H bond lengths resulting from the performed Hirshfeld atom refinements, along with the neutron crystal structure of xylitol with specification of the atomic labels.

Figure 3 .

 3 Figure 3. Electron density differences for xylitol obtained for the different embedding schemes compared to the calculation without embedding. All the electron densities were computed on the IAM structure exploiting the cc-pVTZ basis-set. Contour level: 0.003 e bohr -3 ; Colors: blue (positive) and red (negative). Orientation of the xylitol molecules as in Figure 2.

Figure

  Figure S5 depicting the network of inter-and intra-molecular contacts in the xylitol crystal

  reporting the actual values of bond distances and angles associated with inter-and intra-molecular contacts in the xylitol crystal structures. Table S6 giving the average ratios between HAR and neutron values of the ADPs diagonal elements. (PDF) Crystallographic information files (cif files) and files containing experimental and calculated structure factors (fco files) associated with all the performed Hirshfeld atom refinements of the xylitol crystal structure. (ZIP)

Table 1 .

 1 Mean absolute deviations of HAR distances and angles from the corresponding neutron ones, as resulting from the performed Hirshfeld atom refinements of the xylitol crystal structure.(a)

	Basis-set / Type of Embedding	C-H / Å	O -H / Å	H⋯A / Å	O -H⋯A / °
	cc-pVDZ
	no embedding	0.013	0.010 0.029 0.014 0.026 0.014	0.9	0.7
	4 Å cluster charges	0.012	0.009 0.023 0.012 0.020 0.012	0.7	0.7
	8 Å cluster charges	0.012	0.009 0.023 0.012 0.020 0.012	0.7	0.7
	4 Å ELMOs	0.009	0.007 0.004 0.002 0.005 0.002	0.4	0.4
	8 Å ELMOs	0.009	0.006 0.004 0.002 0.004 0.002	0.3	0.4
	4 Å ELMOs + MM (4 Å -8 Å)	0.009	0.006 0.004 0.002 0.005 0.002	0.3	0.4
	cc-pVTZ
	no embedding	0.011	0.008 0.029 0.011 0.025 0.011	0.8	0.7
	4 Å cluster charges	0.010	0.007 0.024 0.010 0.021 0.011	0.7	0.6
	8 Å cluster charges	0.010	0.006 0.024 0.010 0.021 0.011	0.7	0.6
	4 Å ELMOs	0.007	0.005 0.006 0.006 0.008 0.004	0.4	0.2
	8 Å ELMOs	0.007	0.005 0.006 0.006 0.007 0.004	0.5	0.2
	4 Å ELMOs + MM (4 Å -8 Å)	0.007	0.005 0.006 0.006 0.007 0.004	0.5	0.2

Table 2 .

 2 Mean absolute deviations (in Å 2 ) of the HAR diagonal elements of the ADPs from the

	corresponding neutron values ( 〈"U !"# $$	-U %&'()*+ $$	"〉 ) for different groups of atoms in xylitol. (a)
	Basis-set / Type of Embedding		Non-H	H bonded to C H bonded to O
	cc-pVDZ
	no embedding		0.0018 0.0007 0.0031 0.0026 0.0055 0.0054
	4 Å cluster charges		0.0018 0.0007 0.0030 0.0027 0.0043 0.0041
	8 Å cluster charges		0.0018 0.0007 0.0030 0.0027 0.0042 0.0041
	4 Å ELMOs		0.0018 0.0007 0.0035 0.0026 0.0028 0.0022
	8 Å ELMOs		0.0018 0.0007 0.0035 0.0027 0.0028 0.0022
	4 Å ELMOs + MM (4 Å -8 Å)	0.0018 0.0007 0.0035 0.0026 0.0028 0.0022
	cc-pVTZ
	no embedding		0.0018 0.0007 0.0049 0.0031 0.0097 0.0066
	4 Å cluster charges		0.0018 0.0007 0.0048 0.0029 0.0082 0.0055
	8 Å cluster charges		0.0018 0.0007 0.0048 0.0029 0.0082 0.0055
	4 Å ELMOs		0.0018 0.0007 0.0050 0.0028 0.0049 0.0036
	8 Å ELMOs		0.0018 0.0007 0.0050 0.0028 0.0048 0.0037
	4 Å ELMOs + MM (4 Å -8 Å)	0.0018 0.0007 0.0050 0.0028 0.0048 0.0037

Table 3 .

 3 Mean absolute deviations (in Å 2 ) of the HAR non-diagonal elements of the ADPs from the

	corresponding neutron values ( 〈"U !"# $,	-U %&'()*+ $,	"〉 , with 𝑖 ≠ 𝑗 ) for different groups of atoms in
	xylitol. (a)
	Basis-set / Type of Embedding	Non-H	H bonded to C H bonded to O
	cc-pVDZ
	no embedding	0.0002 0.0002 0.0035 0.0027 0.0042 0.0027
	4 Å cluster charges	0.0002 0.0002 0.0035 0.0027 0.0039 0.0025
	8 Å cluster charges	0.0002 0.0002 0.0035 0.0027 0.0039 0.0025
	4 Å ELMOs	0.0002 0.0002 0.0035 0.0028 0.0036 0.0021
	8 Å ELMOs	0.0002 0.0002 0.0035 0.0028 0.0036 0.0021
	4 Å ELMOs + MM (4 Å -8 Å)	0.0002 0.0002 0.0035 0.0028 0.0036 0.0021
	cc-pVTZ
	no embedding	0.0002 0.0002 0.0031 0.0019 0.0042 0.0038
	4 Å cluster charges	0.0002 0.0002 0.0030 0.0019 0.0037 0.0035
	8 Å cluster charges	0.0002 0.0002 0.0030 0.0019 0.0037 0.0035
	4 Å ELMOs	0.0002 0.0002 0.0032 0.0018 0.0032 0.0030
	8 Å ELMOs	0.0002 0.0002 0.0032 0.0018 0.0032 0.0030
	4 Å ELMOs + MM (4 Å -8 Å)	0.0002 0.0002 0.0032 0.0018 0.0032 0.0030

  To summarize, in this Letter we have proposed an improved version of the Hirshfeld atom refinement technique consisting in the coupling of the original HAR algorithm with the recently proposed embedding methods QM/ELMO and QM/ELMO/MM. Our preliminary validation tests on xylitol have shown that, through its more rigorous description of the crystal environment, the new version of Hirshfeld atom refinement is fundamental to obtain significantly more accurate positions of hydrogen atoms, especially if they are directly involved in intermolecular interaction networks. From the test calculations it also emerged that a 4 Å ELMO embedding is generally enough to obtain completely satisfactory results,

Lee, S. J. R.; Welborn, M.; Manby, F. R.; Miller, T. F., III Projection-Based Wavefunction-in-DFT Embedding. Acc. Chem. Res. 2019, 52, 1359-1368.

ACKNOWLEDGMENTS

The French Research Agency (ANR) is gratefully acknowledged for financial support of this work through the Young Investigator Project QuMacroRef (Grant No. ANR-17-CE29-0005-01). The High-Performance Computing Center EXPLOR of the University of Lorraine is thanked for providing computing time through the projects 2019CPMXX0966, 2019CPMXX0886 and 2019CPMXX1332. Fabien Pascale is also acknowledged for the setup and maintenance of our local cluster, which was used to perform most of the calculations reported in this paper. Hans-Beat Bürgi is thanked for providing us the PENAUT software and for useful discussions.

DATA AVAILABILITY

The xylitol structures obtained in the present study through the ELMO-embedded HAR technique with a 4 Å ELMO-layer (cc-pVDZ and cc-pVTZ basis-sets for the underlying quantum chemical calculations) have been deposited at the Cambridge Crystallographic Data Centre (CCDC) under the deposition numbers 2044846 and 2044847. This data can be obtained free of charge via www.ccdc.cam.ac.uk/data_request/cif. All models for all refinements plus the measured structure factors have been deposited with this journal in addition to the CCDC deposition to ensure reproducibility of the study.

AUTHORS INFORMATION

Corresponding Author

Alessandro Genoni (A.G.), Université de Lorraine & CNRS, Laboratoire de Physique et Chimie Théoriques (LPCT), 1 Boulevard Arago, F-57078 Metz, France; Phone: +33 (0)3 72 74 91 70; Fax: +33 (0)3 72 74 91 87. E-mail: Alessandro.Genoni@univ-lorraine.fr.

Notes

The authors declare no competing financial interests.

ASSOCIATED CONTENT Supporting Information. Details about theory, transfer and libraries of extremely localized molecular orbitals (with Figure S1 reporting some examples of plotted ELMOs and Figure S2 showing the schematic representation of reference frames and atomic triads required for the ELMOs rotation). Details about the QM/ELMO and QM/ELMO/MM methods. Figure S3 displaying the Meindl-Henn plots associated with the different Hirshfeld atom refinements.

Figure S4 showing the C-H bond lengths of xylitol resulting from the performed HARs.