Christie Aroulanda

Philippe Lesot
email: philippe.lesot@universite-paris-saclay.fr

Prof Aharon Loewenstein

Molecular Enantiodiscrimination by NMR Spectroscopy in Chiral Oriented Systems: Concept, Tools and Applications

Keywords: Chirality, Conformational distribution, Dynamics, Enantiotopicity, NMR spectroscopy, Oriented solvents, PBLG, Stereochemistry

published or not. The documents may come

Molecular enantiodiscrimination by NMR spectroscopy in chiral oriented systems: Concept, tools, and applications

Christie Aroulanda, Philippe

I. INTRODUCTION

Nuclear Magnetic Resonance (NMR) spectroscopy nowadays plays a central place in various analytical domains of the modern organic chemistry involving chirality, from the enantiopurity determination to the threedimensional (3D) structure elucidation through the intra-molecular distribution of isotopes [START_REF] Emsley | Milestones in the first fifty years of NMR[END_REF].

From the point of view of stereochemistry and the enantiomorphism concept, the study of molecular chirality/ isotopic chirality / prochirality (see Figure 1) is an ongoing challenge for NMR spectroscopists who have developed many useful approaches to solve many chemist's analytical problems [START_REF] Eliel | Stereochemistry of organic compounds[END_REF].

Among them, the anisotropic NMR spectroscopy using orientantionally oriented chiral environments has provided a set of new original tools by making accessible all order-dependent intramolecular, residual anisotropic NMR interactions, such as residual chemical shift anisotropy (RCSA), residual dipolar coupling (RDC), and residual quadrupolar coupling (RQC) for spin I > ½, while preserving high spectral resolution [START_REF] Emsley | NMR in anisotropic systems, theory[END_REF]. The origin of the term "residual" is associated with the idea that any solute interacting with a mesomorphic oriented media is both aligned (inside the magnetic field of spectrometer, B 0 ) and mobile. This combination leads to anisotropic interactions whose the magnitude is reduced by a factor ranging from 10 -5 to 10 -1 (depending of type of liquid crystals used), with respect to the values that would be observed in a crystalline solid by solid-state NMR (SS NMR) (see Figure 1a) ».

Figure 1 (a) Schematic description of the molecular orientational order in an isotropic liquid (no order), in a liquid crystal (order) and in a crystalline solid (order and position). (b) Example of an (arbitrary) reference axis system (a, b and c axes)

attached to a molecule oriented in a helical-polymer based chiral nematic system. The molecular order parameters and atomic coordinates will be described in accordance with this axis system. Figure partially adapted from Refs. [4] with permission.

The original analytical potential of NMR in enantiopure (non-racemic) oriented solvents lies on its ability to orient differently in average on the NMR timescale both enantiomers of chiral molecules and enantiotopic elements of prochiral solutes, thus leading to distinct NMR spectra or signals to be detected [4,[START_REF] Lesot | Enantiotopic discrimination in the NMR spectrum of prochiral solutes in chiral liquid crystals[END_REF][START_REF] Sarfati | Theoretical and experimental aspects of enantiomeric differentiation using natural abundance multinuclear NMR spectroscopy in polypeptide liquid crystals[END_REF].

In the frame of this special issue, "Chirality in France", we overview in this compendium (mainly written for chemists working with molecular (pro)chiral objects) various key aspects of NMR in weakly aligning chiral solvents based on organic lyotropic liquid crystals (LLCs), and in particular the methodologies developed at Orsay (University of Paris-Sud/Paris-Saclay) to study chiral and prochiral compounds in relation with the needs of chemists. Various analytical aspects based on the enantiorecognition phenomena in relation to molecular symmetry as the enantiopurity determination of mixtures, the stereochemistry analysis, the determination of the natural isotopic fractionation in prochiral molecules, as well as the determination of the molecular configuration and conformation of analytes will be examined. Herein, we will discuss the most representative applications illustrating the original potential of this NMR tool, in particular when using polypeptide-based chiral LLCs as enantiodiscriminating anisotropic systems. A special attention will be also paid to the problem of the conformational analysis in weakly orienting environments.

| Enantiomorphism concept and NMR spectroscopy

Nature is highly enantioselective by producing paramount families of natural chiral molecules (as carbohydrates or amino-acids) with a given D or L stereochemistry. In contrast, chemists involved in asymmetric syntheses are able to make new compounds while controlling the stereochemistry of their molecular target. Enantiomorphism covers both the concept of chirality and prochirality, associated respectively with enantiomers of chiral compounds or enantiotopic elements in prochiral molecules [START_REF] Eliel | Stereochemistry of organic compounds[END_REF].

Enantiomers of a chiral molecule are mirror images objects of each other by a plane of symmetry, but are not superimposable. Majority of chiral molecules possess a stereogenic center (as an asymmetric tetrahedral carbon atom, for instance), but it is not a prerequisite as observed in cases of axial chirality or atropoisomerism. Prochiral compounds are simply defined as any molecule that can be transformed into a chiral one by a single isotopic substitution. Chirality can therefore originate from isotopic substitution around a stereogenic center or not, and called isotopic chirality (see Figure 2) [START_REF] Eliel | Stereochemistry of organic compounds[END_REF][START_REF] Lesot | Enantiotopic discrimination in the NMR spectrum of prochiral solutes in chiral liquid crystals[END_REF]. ("classical" chirality and "(D/H) isotopic" chirality) with a single stereogenic center (tetrahedral carbon atom) deriving from their deuterated or hydrogenated prochiral precursors and characterized by methylene enantiotopic directions, pro-R/pro-S.

Figure 2 Examples of R/S-enantiomeric pairs of a model C1-symmetry chiral molecule

Enantiotopic elements in prochiral compounds are atoms, groups of atoms or internuclear directions exclusively exchangeable by a plane of symmetry (C s symmetry), or an improper symmetry axis (S 4 symmetry).

Often, prochiral molecules possess a prostereogenic center, as in case of a CH 2 group in C 2v -symmetry molecules [START_REF] Eliel | Stereochemistry of organic compounds[END_REF][START_REF] Lesot | Enantiotopic discrimination in the NMR spectrum of prochiral solutes in chiral liquid crystals[END_REF].

R 1 R 2 H H pro-S pro-R R 1 R 2 D H (S) σ R 2 R 1 D H (R) σ 1.

| Limits of classical enantiodiscriminating NMR approaches routinely used

NMR tools in isotropic liquids have been initially developed for the determination of the enantiomeric excess (ee in %) of mixtures of chiral compounds. The approaches employed in routine require the use of an enantiopure chiral partner reacting or interacting with the chiral analyte. Basically, three classes of enantiopure agents can be used: i) the chiral derivatizing agents (CDAs) able to form covalent bonds with enantiomers and convert them into stable diastereoisomers; ii) the chiral lanthanide-induced shift reagents (CLSRs) such as chiral lanthanide complexes (based on europium or ytterbium central cation), able to coordinate differently both enantiomers and create diastereoisomeric complexes; and iii) the chiral solvating agents (CSAs) including chiral cyclodextrines [START_REF] Parker | NMR determination of enantiomeric purity[END_REF][START_REF]To find information, an entire issue of Chemical Review was devoted to cyclodextrins[END_REF][START_REF] Wenzel | Chiral reagents for the determination of enantiomeric excess and absolute configuration using NMR spectroscopy[END_REF][START_REF] Wenzel | Discrimination of chiral compounds using NMR spectroscopy[END_REF].

While CDAs prequisite a preliminary chemical transformation prior to the analysis, CLSRs and CSAs can be used in situ in the NMR sample. If tools involving CLSRs and CSA are rather easy to setup, they require however an enantiopure agent that must be carefully selected according to the (specific) chemical functionalities (hydroxyl group, carboxylic acid, amine, …) of the investigated chiral compound. With the exception of cyclodextrines (chiral macrocycles), these agents are often rarely suitable for apolar chiral compounds [9, 10].

| Molecular order, chirality and NMR consequences

First, introduced by Saupe and Englert in 1963 [11], NMR spectroscopy in liquid crystals is a fascinating analytical tool combining the advantages of the SS NMR (detection of anisotropic NMR interactions) and NMR performed in conventional liquids (acquisition of high-resolution spectra) [START_REF] Emsley | NMR spectroscopy using liquid crystal solvents[END_REF][START_REF] Veracini | Nuclear magnetic resonance of liquid crystals[END_REF][START_REF] Dong | Nuclear magnetic resonance spectroscopy of liquid crystals[END_REF]. The detection of intramolecular anisotropic NMR interactions originates from the partial alignment (orientational ordering) of molecules in LC (analyte and solvent) with the static magnetic field, B 0 of the spectrometer, while averaging out of most intermolecular anisotropic interactions.

The orientational behavior of an oriented molecule (solute or solvent) can be described by its orientationalconformational probability function [START_REF] Emsley | NMR spectroscopy using liquid crystal solvents[END_REF][START_REF] Veracini | Nuclear magnetic resonance of liquid crystals[END_REF][START_REF] Dong | Nuclear magnetic resonance spectroscopy of liquid crystals[END_REF][START_REF] Zannoni | Quantitative description of orientational order: rigid molecules, nuclear magnetic resonance of liquid crystals[END_REF]. In practice, for a rigid molecular structure or a given conformation in a flexible compound, the dependence of NMR interactions on the molecular orientation, which affects the NMR spectra, can be described to a good approximation by a second-rank tensor, denoted also Saupe order matrix, which is much easier to work with. At the maximum, in a nematic uniaxial solvent, the molecular orientation can be described by five non-zero independent order parameters, S aa , S bb -S cc , S ab , S ac and S bc , where a, b and c refer to the axes of a molecular-fixed reference frame (see Figure 1b). Actually, this number can change depending on two key factors, accounting for the symmetry of the solute interacting within its solvent: i) the point group symmetry of the analyte, ii) the nature of the mesophase used, namely achiral or chiral. According to these two criteria, Table 1 lists the number of independent order parameters needed to suitably describe the orientational behaviour of an oriented rigid molecule (chiral, prochiral, achiral), or a flexible one in a given molecular conformation [START_REF] Merlet | The relationship between molecular symmetry and second-rank orientational order parameters for molecules in chiral liquid crystalline solvents[END_REF]. A special attention will be paid to molecules of symmetry C s , C 2v , S 4 and D 2d as discussed in Section V.

From the NMR point of view, the existence of a molecular alignment leads to the detection of all intramolecular anisotropic NMR interactions (RCSA, RDC, RQC) which are no longer reduced to zero due to fast and random (Brownian) molecular motions. These order-dependent interactions will be described briefly below.

In this sense, NMR spectra of a given nucleus recorded in anisotropic media are much more analytically informative than those recorded in isotropic ones.

Contrarily to solid phase, and even if analytes are uniformely oriented with respect to the magnetic field axis, they still continue to rotate and diffuse as in liquids, thus generally leading to high-resolution NMR spectra with full width at half-maximum (FWHM) of lines equal to 1 to 5 Hz.

| Advantages of NMR in chiral oriented environments

Isotropic and anisotropic NMR spectroscopies alone are basically blind to the chirality concept. In both cases, revealing the molecular enantiomorphism is only possible if a chiral (enantiopure) additional component or a non-racemic environment (seen as an enantio-stimulus) is used, and can create diastereomorphous interactions at the origin of the enantiodiscrimination mechanisms. This second approach was first explored by Snyder et al, in 1968, using a strongly orienting cholesteric-type mesophase and 1 H NMR [START_REF] Sackmann | Nuclear magnetic resonance spectra of enantiomers in optically active liquid crystals[END_REF].

Phenomenologically, the distinct interactions between the R-and S-isomers and the enantiopure (nonracemic) molecules of the chiral liquid crystal (CLC) results in different average R-and S-orientations with respect to the B 0 (Z axis). These molecular orientations can be described using two Saupe's order matrices, {𝑆 !" ! } and {𝑆 !" ! }, whose elements can be expressed as [START_REF] Lesot | Enantiotopic discrimination in the NMR spectrum of prochiral solutes in chiral liquid crystals[END_REF][START_REF] Emsley | NMR spectroscopy using liquid crystal solvents[END_REF][START_REF] Veracini | Nuclear magnetic resonance of liquid crystals[END_REF][START_REF] Zannoni | Quantitative description of orientational order: rigid molecules, nuclear magnetic resonance of liquid crystals[END_REF]:

𝑆 !" ! !" ! = ! ! 3 cos 𝜃 ! ! ! !" ! cos 𝜃 ! ! ! !" ! -𝛿 !" ( 1
)
where 𝛿 is the Kronecker function (𝛿 !" =1 if 𝛼 = 𝛽 and 𝛿 !" =0 for 𝛼 ≠ 𝛽). From the matrix elements of 𝑆 !"

! !" ! , we can derive any local order parameters, 𝑆 !" ! !" ! , associated with any internuclear direction ij (vector r ij ) expressed in the molecular reference axis system (RAS) initially defined, (a, b, c) S or R , on the basis of the so-called director cosines following Eq. 2 [START_REF] Lesot | Enantiotopic discrimination in the NMR spectrum of prochiral solutes in chiral liquid crystals[END_REF]:

𝑆 !" ! !" ! = cos 𝜃 ! !" ! !" ! cos 𝜃 ! !" ! !" ! 𝑆 !" ! !" ! !"!!,!,! (2)
All anisotropic NMR interactions (tensorial properties) are related to these order parameters. Note finally that 𝑆 !" ! !" ! can be also described as a function of the angle between the internuclear vector, r ij, and the axis of the magnetic field B 0 , 𝜃 !" ! ! , for each enantiomer, namely [4,[START_REF] Lesot | Enantiotopic discrimination in the NMR spectrum of prochiral solutes in chiral liquid crystals[END_REF][START_REF] Sarfati | Theoretical and experimental aspects of enantiomeric differentiation using natural abundance multinuclear NMR spectroscopy in polypeptide liquid crystals[END_REF]: S aa // n-fold axis a Compounds with these point groups have a lower effective symmetry and a change in the location of PAS of the order matrix.

𝑆 !" ! !" ! = ! !"# ! (! !" ! ! ) ! !" ! !! ! , (3)
In addition to the possibility to spectrally discriminate enantiomers, NMR in CLCs offers two other analytical advantages. The first one is to provide new tools to discriminate diastereoisomers, but also diastereotopic directions, not always differentiated in isotropic liquids. The second, and much more important, is its ability to spectrally differentiate enantiotopic elements (as atoms, groups of atoms, or internuclear directions) in prochiral molecules, whereas there are very few isotropic NMR approaches able to do so. In this review, the concept of spectral enantiodiscrimination covers both the differentiation of enantiotopic elements within a single molecular (prochiral) object or that of enantiomers, which can be formally considered as the discrimination of enantiomeric elements but involving two molecular (chiral) objects.

II. NMR TOOLS VERSUS MAGNETICALLY ACTIVE NUCLEUS

Any anisotropic NMR interaction and its associated observable noted, 𝑂𝑏𝑠 ! !" ! , can be simply described as the product of a constant, K NMR , associated to a given NMR interaction, and a local or molecular of parameter, that we will note 𝑆 !"

! !" ! . This description applies to the case of enantiomers (A, B = R or S stereodescriptors) or enantiotopic elements (A, B = pro-R or pro-S stereodescriptors), so we can write:

𝑂𝑏𝑠 ! !" ! = 𝐾 !"# × 𝑆 !" ! !" ! , (4)
assuming safely that 𝐾 !"# is identical for both enantiomers or enantiotopic elements.

As seen, the magnitude of 𝑂𝑏𝑠 ! !" ! both depends on the magnitude of 𝐾 !"# and 𝑆 𝐴 or 𝐵 , and subsequently on: i) the degree of alignment of the CLC used, strong (S = 10 -2 to 10 -3 ) or weak (S = 10 -3 to 10 -6 ), ii)

the magnitude of the factor 𝐾 !"# in regards of the spectral features of nucleus considered, e.g. its gyromagnetic ratio, γ, and its spin number I. Among the detectable nuclei in almost all organic molecules (CHNO), a particular attention can be paid to 1 H (99 % abundant, I = ½), the 13 C nuclei (1 % abundant, I = ½), and the 2 H nuclei (0.0155 % abundant, I = 1, a small nuclear quadrupolar moment, QM). Quadrupolar nuclei as 17 O or 14 N can be theoretically used but are analytically less interesting in anisotropic NMR due to their inherent quadrupolar properties (spin I = 5/2 and 1, respectively, and large QM values) reducing drastically their T 1 , T 2 relaxation times

[18].
In terms of spectral enantiodiscrimination efficiency, the difference of between R and S NMR observables, Δ𝑂𝑏𝑠 , depends only on the magnitude of 𝐾 !"# . Considering only 1 H, 13 C, and 2 H nuclei, and based on numerous empirical observations, three important observables useful in practice can be retained, and can be roughly classified in terms of both enantio-efficiency and experimental feasibility/ease as follows: 2 H-RQCs >> 13 C-RCSAs > ( but their analytical potentialities depend also to further factors such as, the amount of the analyte available, its MW, its chemical nature and its structural complexity [START_REF] Sarfati | Theoretical and experimental aspects of enantiomeric differentiation using natural abundance multinuclear NMR spectroscopy in polypeptide liquid crystals[END_REF][START_REF]Brevard C in Handbook of high-resolution multinuclear NMR[END_REF][START_REF] Lesot | Visualization of enantiomers in a polypeptide liquid-crystal solvent through carbon-13 NMR spectroscopy[END_REF][START_REF] Meddour | Proton-decoupled carbon-13 NMR spectroscopy in a lyotropic chiral nematic solvent as an analytical tool for the measurement of the enantiomeric excess[END_REF].

| Recording " routine" nuclei

Hydrogen NMR. Hydrogen nuclei can be exploited for discriminating enantiomeric or enantiotopic direction using differences in ( 1 H-1 H)-RDCs of enantiomers. In this case, each pair of interacting nuclei, 1 H i and 1 H j , produces distinct D HH defined in Hz as [4,[START_REF] Lesot | Enantiotopic discrimination in the NMR spectrum of prochiral solutes in chiral liquid crystals[END_REF]:

𝐷 ! ! ! ! ! ! ! !" ! = -𝑘 !! × ! !! ! !" ! ! !" ! , (5)
with:

𝑘 !! = 𝜇 ! 4π h𝛾 ! γ ! 4π ! = 120.09 kHz. A !! .
In this equation, r ij is the internuclear distance between two interacting nuclei (i and j), γ H is the gyromagnetic ratio of 1 H and µ 0 is the magnetic permeability of free space.

As a consequence of Eq. 5, the D HH values can be positive, null or negative. This new term adds to the scalar coupling, J HH , leading to total spin-spin coupling, T HH , with: T HH = J HH + 2D HH , for a classical AX spin system or T HH = 3D HH for a spin system constituted by two and three equivalent (homotopic) nuclei, A 2 or A 3 . An example of 1 H-1 H spectral differentiation (AX spin system) is shown in Figure 3b. D HH can be positive, null (magic angle)

or negative, and consequently T HH can be positive or negative as well as null when J HH = -D HH /2. Carbon-13 NMR. Naturally abundant at 1.07 % relative to Hydrogen, 13 C nuclei can be also efficiency used to discriminate enantiomers on the basis of a difference of anisotropic contribution to the electronic shielding,

𝜎 ! !"#$% ! !" !
that depends of Sαβ parameter. Thus, for an enantiomeric/enantiotopic pair, their respective resonance frequencies (in Hz), 𝜈 C i

13

!

and 𝜈 C i

13

! , can be written as [START_REF] Lesot | Visualization of enantiomers in a polypeptide liquid-crystal solvent through carbon-13 NMR spectroscopy[END_REF][START_REF] Meddour | Proton-decoupled carbon-13 NMR spectroscopy in a lyotropic chiral nematic solvent as an analytical tool for the measurement of the enantiomeric excess[END_REF]:

𝜈 ! ! !" ! !" ! = !! ! !" 1 -𝜎 ! !"# -𝜎 ! ! !"#$% ! !" ! , (6)
where 𝜎 C i

13

!"# is the isotropic contribution to the electronic shielding.

As a dilute spin (1 % with respect to 1 H) removing any 13 C-13 C interactions, this type of discrimination is simply observed when hydrogen nuclei are decoupled ( 13 C-{ 1 H}) leading to simple NMR spectra consisting of sum of independent 13 C resonances. Largest discriminations are genrally obtained with sp and sp 2 carbon atoms.

An example of spectral 13 C differentiation (A spin system) in a CLC is shown in Figure 3a.

When no 1 H decoupling is applied, enantiodiscrimination is possible on the basis of a difference of ( 13 C-1 H)-RDCs according to Eq. 7 [START_REF] Lesot | Visualization of enantiomers in a polypeptide liquid-crystal solvent through carbon-13 NMR spectroscopy[END_REF][START_REF] Meddour | Proton-decoupled carbon-13 NMR spectroscopy in a lyotropic chiral nematic solvent as an analytical tool for the measurement of the enantiomeric excess[END_REF]:

𝐷 ! !" ! ! ! ! ! !" ! = -𝑘 !" ! !" ! !" ! ! !" ! , (7)
with 𝑘 !" = 30.09 kHz. 𝐴 ! . As for 1 H, D CH adds to J CH (T CH = J CH + 2D CH ). Note here that direct scalar coupling, Deuterium NMR. Anisotropic 2 H spectra are dominated by the quadrupolar interaction resulting to from the interaction between the electric quadrupole moment of 2 H (Q) and the electric field gradient (EFG) along the C-D bond, with a good approximation. In practice, and as for the two other anisotropic interactions described above, we have access to the 2 H residual quadrupolar couplings, 2 H-RQCs, of molecules (isotopically enriched or normal), that corresponds to the distance (or quadrupolar splitting) between both components of a 2 H quadrupolar doublet ( 2 H-QD). As second isotope of 1 H, deuterium nuclei are naturally present in any hydrogenated molecules, with a very a low relative abundance of 1.55 x 10 -2 % (respect to 1 H) namely about 100 less than for 13 C). This drawback is compensated by three analytical advantages [START_REF] Sarfati | Theoretical and experimental aspects of enantiomeric differentiation using natural abundance multinuclear NMR spectroscopy in polypeptide liquid crystals[END_REF][START_REF] Lesot | Natural abundance deuterium NMR spectroscopy in polypeptide liquid crystals as a new and incisive means for enantiodifferentiation of chiral hydrocarbons[END_REF][START_REF] Lesot | Natural abundance deuterium NMR spectroscopy: developments and analytical applications in liquids, liquid crystals and solid phase[END_REF][START_REF] Lesot | Natural abundance 2 H NMR spectroscopy[END_REF]: i) a small Q D value leading to highresolution 2 H-{ 1 H} NMR spectra, ii) a distribution of anisotropic 2 H signals on only two components (2I peaks with I = 1), named a deuterium quadrupolar doublet ( 2 H-QD), leading to acceptable signal-to-noise ratio (SNR),

iii) a very high sensitivity of the 2 H quadrupolar interaction relative to a difference of local ordering.

For an enantiomeric/enantiotopic pair (R/S or pro-R/pro-S), the splittings (in Hz) between each component of a given 2 H-QD, ∆𝜈 ! ! ! ! ! !" ! , for given deuterium site, i, can be written as [START_REF] Lesot | Natural abundance deuterium NMR spectroscopy in polypeptide liquid crystals as a new and incisive means for enantiodifferentiation of chiral hydrocarbons[END_REF][START_REF] Lesot | Natural abundance deuterium NMR spectroscopy: developments and analytical applications in liquids, liquid crystals and solid phase[END_REF][START_REF] Lesot | Natural abundance 2 H NMR spectroscopy[END_REF]:

∆𝜈 ! ! ! ! ! !" ! = ! ! 𝑄𝐶𝐶 !" × 𝑆 !" ! !" ! (8)
with

𝑄𝐶𝐶 !" = (𝑒 ! 𝑄 D i 𝑞 C-D i )/ℎ and 𝑆 !" ! !" ! = ! ! 3 cos ! 𝜃 !" ! !" ! -1 .
A priori, the 2 H quadrupolar coupling constant, denoted 𝑄𝐶𝐶 !" , is assumed to be identical for two enantiomers or enantiotopic sites. The magnitude of this term depends on the 13 C hybridization state (QCC C-Di ≈ 170 ± 5 kHz, 185 ± 5 kHz and 210 ± 5 kHz, for sp 3 , sp 2 and sp carbons, respectively) and 𝑆 !"

! !" ! is the order parameter of the C-D i axis (actually the electric field gradient along C-D) relative to B 0. A schematic example of spectral 2 H differentiation (a single deuteron) in a CLC is shown in Figure 3c. Here again, the ∆𝜈 ! ! !" ! value can be positive, negative or null (rare). As we will discuss below, this interaction is particularly well adapted to visualize monodeuterated enantio-isotopomers associated with enantiotopic directions of CH 2 groups in prochiral molecules (see Figure 1).

| Playing with "exotic" nuclei

In practice, any magnetically active, naturally abundant nuclei present in a molecule, such as 19

| Spectral analysis by 2D NMR tools

The wealth of information content of anisotropic interactions combined to the doubling of signals in CLCs has a "cost" in terms of spectral complexity that can be overcome by using 2D NMR experiments. Adapted to specificities of oriented NMR as in case of spin-1 nuclei, various types of homo or heteronuclear NMR experiments have been designed to accelerate the analysis of ½-and 1-spin nuclei spectra, and even both. For the former, most interesting experiments are based on a "J-resolved" type scheme in which only the homo-or heteronuclear coupling interaction evolves during the t 1 dimension, combined with a selective excitation (concept of SERF NMR) to reduce the amount of spectral information or to select one of both enantiomers. In the second case, the several 2D experiments referred to under the generic name of "QUOSY" (QUadrupole Ordered SpectroscopY) have been proposed.

2D experiments involving spins-½ nuclei

Due to long-range ( 1 H-1 H)-RDCs or ( 1 H-13 C)-RDCs, 1 H 1D spectra as well as the proton-coupled 13 C 1D spectra can be complex to analyse mainly due to their weakly resolved spectral patterns at each inequivalent site [START_REF] Giraud | Analyses of enantiomeric mixtures using chiral liquid crystals[END_REF]. In practice, 1 H-COSY-type 2D experiments can provide help for simplifying the analysis of chiral molecules [30], including or not selective conditions (excitation of a given 1 H site) (see Figure 4) [START_REF] Prabhu | Band selective small flip angle COSY: A simple experiment for the analyses of 1 H NMR spectra of small chiral molecules[END_REF]. Better is the possibility to separate the homo-or heteronuclear spin-spin total couplings and the chemical shifts on both spectral dimensions using 2D experiments based on 1 H or 13 C "J-resolved"-type schemes or other schemes as 1 H J-HSQC-BIRD correlation sequence (see Figure 4) for which only the coupling interaction evolves during the t 1 dimension [START_REF] Ziani | Visualisation of enantiomers via insertion of a BIRD module in X-H correlation experiments in chiral liquid crystal solvent[END_REF]. Associated with a selective excitation of nuclear site of interest, further spectral simplifications (family of homo or heteronuclear G-SERF experiments) are obtained (see Figure 4) [START_REF] Farjon | Heteronuclear selective refocusing 2D NMR experiments for the spectral analysis of enantiomers in chiral oriented solvents[END_REF][START_REF] Nath | Enantiodiscrimination and extraction of short and long-range homo-and heteronuclear residual dipolar couplings by a spin selective correlation experiment[END_REF][START_REF] Nath | Selective detection of single-enantiomer spectrum of chiral molecules aligned in the polypeptide liquid crystalline solvent: transition selective one-dimensional 1 H-1 H COSY[END_REF][START_REF] Giraud | Nuclear magnetic resonance using a spatial frequency encoding: application to J-edited spectroscopy along the sample[END_REF][START_REF] Farjon | SERF-filtered experiments: New enantio-selective tools for deciphering complex spectra of racemic mixtures dissolved in chiral oriented media[END_REF].

2D experiments involving spin-1 nuclei. 2 H quadrupolar interaction has been initially and extensively exploited with deuterated chiral analytes. However, as the isotopic enrichment is not always easy to perform (generally considered as time-consuming by chemists), and does not guarantee a spectral discrimination on the enriched site(s), either, it becomes preferable to detect 2 H nuclei at the natural abundance level, despite its low abundance level [START_REF] Lesot | Natural abundance deuterium NMR spectroscopy in polypeptide liquid crystals as a new and incisive means for enantiodifferentiation of chiral hydrocarbons[END_REF][START_REF] Lesot | Natural abundance deuterium NMR spectroscopy: developments and analytical applications in liquids, liquid crystals and solid phase[END_REF].

As any dilute spin (like 13 C spins), the detection of 2 H-2 H couplings (J and D) at natural abundance is not possible yet, and so the proton-decoupled, anisotropic natural abundance deuterium (ANAD-{ 1 H}) spectra correspond to the sum of independent 2 H-QDs associated with all monodeuterated diastereo-isotopomers and/or enantio-isotopomers of the mixture. Consequently, the ANAD 1D spectra in weakly orienting CLC can become overcrowded even for rather structurally-simple chiral molecules, because: i) the small range of 2 H-RQCs (generally below 1000 Hz) compared to the range of δ aniso ( 2 H); ii) the detection of 2n 2 H-QDs (for a molecule of n inequivalent hydrogen site) if all sites are differentiated in CLC. To facilitate the analysis of overcrowded ANAD spectra, several types of 2D/3D-NMR experiments (homoand heteronuclear QUOSY) have been designed [START_REF] Merlet | Two-dimensional deuterium NMR spectroscopy of chiral molecules oriented in a polypeptide liquid crystal: application for the enantiomeric analysis through natural abundance deuterium NMR[END_REF][START_REF] Merlet | Description of natural abundance deuterium 2D NMR experiments in weakly ordered liquid crystalline solvents using a tailored cartesian spin-operator formalism[END_REF][START_REF] Lesot | 2D-NMR strategy dedicated to the analysis of weakly ordered, fully deuterated enantiomers in chiral liquid crystals[END_REF][START_REF] Lafon | Modified z-gradient filtering as a new mean to obtain phased deuterium autocorrelation 2D NMR spectra in oriented solvents[END_REF][START_REF] Lafon | Deuterium three-dimensional NMR experiments for analysing weakly aligned, isotopically enriched molecules[END_REF][START_REF] Lesot | Enantiomeric analysis using natural abundance deuterium 3D NMR spectroscopy in polypeptide chiral oriented media[END_REF][START_REF] Lafon | Homo-and heteronuclear 2D NMR approaches to analyse a mixture of deuterated unlike/like compounds using weakly ordering chiral liquid crystals[END_REF][START_REF] Lesot | Experimental detection of achiral and chiral naturally abundant 13 C-2 H isotopomers by 2D-NMR in liquids and chiral oriented solvents[END_REF] as schematically depicted in Figure 5. The common aim of all QUOSY's is to correlate the two components of each 2 H-QD and assign them on the basis of their anisotropic chemical shift, considering that δ( 2 H) aniso ≈ δ( 2 H) iso ≈ δ( 1 H) iso . In this class of (autocorrelation) experiments, δ( 2 H) aniso 's and Δν Q ( 2 H)'s can evolve in both t 1 and t 2 dimensions (Q-COSY-type) or be refocused during t 1 (Q-resolved or δ-resolved experiments). Excitation of double quantum coherence deuterium (spin I = 1) is also possible (Q-DQ type), but needs an optimization of a delay, not required for other experiment). The most useful experiments are the Q-COSY 2D and Q-resolved 2D experiments (and their phased variants) for which a tilt procedure (post acquisition procedure) can be also applied on 2D spectra removes the 2 H-QDs in F 2 dimension, and so the F 2 projection formally resembles the isotropic NAD NMR spectra [START_REF] Merlet | Two-dimensional deuterium NMR spectroscopy of chiral molecules oriented in a polypeptide liquid crystal: application for the enantiomeric analysis through natural abundance deuterium NMR[END_REF][START_REF] Lesot | 2D-NMR strategy dedicated to the analysis of weakly ordered, fully deuterated enantiomers in chiral liquid crystals[END_REF]. All QUOSY 2D experiments using the "Fz" extension possess a « Z-filter » block (specifically designed for spins I = 1) within the pulse sequence to obtain phased 2 H 2D spectra. Homonuclear 3D experiments (as 3D Q-DQ) [START_REF] Lafon | Modified z-gradient filtering as a new mean to obtain phased deuterium autocorrelation 2D NMR spectra in oriented solvents[END_REF][START_REF] Lafon | Deuterium three-dimensional NMR experiments for analysing weakly aligned, isotopically enriched molecules[END_REF] were also designed, as well as heteronuclear versions of QUOSYs experiments. These latter correlate 13 C and To experimentally fight the inherent low sensibility of ANAD 2D NMR, the use of higher magnetic fields (> 9.4 T) and selective 2 H probes (standard probe or better, a cryogenic probe [START_REF] Kovacs | Cryogenically cooled probes, a leap in NMR technology[END_REF] is a simple way to increase the sensitivity of NAD 1D/2D experiments. Besides to reduce the experimental time (T exp ) of ANAD QUOSY nD experiments that required a large number of scans for each t 1 transient, a new type of data acquisition in the direction dimension t 1 as the non unform sampling (NUS) approaches combined with an adapted reconstruction such as the covariance (Cov) method or compressed sensing (CS) processing can be applied [START_REF] Lafon | Fast and high-resolution stereochemical analysis by non-uniform sampling and covariance processing of anisotropic natural abundance 2D 2 H-NMR datasets[END_REF][START_REF] Kazmierczuk | Criteria for sensitivity enhancement by compressed sensing: practical application to anisotropic NAD 2D-NMR spectroscopy[END_REF][START_REF] Lesot | Fast acquisition of multidimensional NMR spectra of solids and mesophases using alternative sampling methods[END_REF].

III. ENANTIODISCRIMINATING LYOTROPIC MESOPHASES

In addition to the early historical liquid-crystalline systems known as "thermotropics" which strongly align any analytes [START_REF] Saupe | High-resolution nuclear magnetic resonance spectra of orientated molecules[END_REF][START_REF] Emsley | NMR spectroscopy using liquid crystal solvents[END_REF][START_REF] Veracini | Nuclear magnetic resonance of liquid crystals[END_REF][START_REF] Dong | Nuclear magnetic resonance spectroscopy of liquid crystals[END_REF], it exists another type of organosoluble (or hydrosoluble) weakly orienting systems named "lyotropic liquid crystals" (LLC). These latter phases are made up of molecular components that do not possess intrinsic mesogenic properties as in case of thermotropics. Achiral or chiral LLCs are often formed by long molecules (rod-like shaped objects) such as polymers, that mixed with suitable (organic) solvent under appropriate conditions (concentration, temperature and pressure) lead a uniform/homogeneous oriented medium in the magnetic field of the NMR spectrometer [50, 51].

| Polypeptide-based organosoluble polymers

Despite, the paradigm shift associated with NMR in CLCs as innovating approach for enantiodiscrimination purposes proposed from 1967 [START_REF] Sackmann | Nuclear magnetic resonance spectra of enantiomers in optically active liquid crystals[END_REF], the analytical complexity of 1 H spectra recorded in thermotropic (cholesteric) systems (strongly orienting solvents) has greatly reduced its attractivity of this tool, and only a few reports exist [START_REF] Lafontaine | Visualization of enantiomers in cholesteric solvents through deuterium NMR[END_REF][START_REF] Canet | Visualization of enantiomers through deuterium NMR in cholesterics optimization of the chiral liquid-crystal solvent[END_REF]. In fact, a revolutionary development came with the use of weakly-ordering chiral ordering agents (lyotropic system) consisting of polypeptide polymers dissolved in conventional organic solvents [START_REF] Bayle | Enantiomeric analysis in a polypeptide lyotropic liquid-crystal through proton decoupled deuterium NMR[END_REF],

concomitantly with the development water-compatible dilute liquid crystalline media developed for the structure elucidation of biomolecules [START_REF] Tjandra | Direct measurement of distances and angles in biomolecules by NMR in a dilute liquid crystalline medium[END_REF].

Neat polypeptide. Among organo-compatible, weakly aligning enantiodiscriminating mesophases, a special attention has been paid to rod-like systems made of α-helicoidally chiral polymers. In this family of LLC, where Length poly >> Diameter poly , the stereogenic center can be present in the central backbone (as polypeptidic polymers with achiral side chains), on the flexible side chain (as polyacetylenic or poly(arylisocyanide) polymers) or simultaneously on both these structural elements (polypeptides bearing a chiral side chain) (see Figure 6).

So far, the most investigated homopolypeptide polymer as enantiodiscriminating CLCs and implied in many applications (in particular at Orsay) is made of poly-γ-benzyl-L-glutamate (PBLG) whose AC is of L-type, and a vast literature exists on its physical properties [START_REF] Panar | Magnetic ordering of poly(γ-benzyl-L-glutamate) solutions[END_REF][START_REF] Guha | Polypeptide liquid crystals: magnetic susceptibily, twist elastic constant, rotational viscosity coefficient, and PBLG sidechain conformation[END_REF][START_REF] Czarniecka | Polypeptide liquid crystals: a deuterium NMR study[END_REF][START_REF] Poliks | Poly-γ-benzyl-L-glutamate: order parameter, oriented gel, and novel derivatives[END_REF]]. However, other polypeptide polymers, such as poly-ε-carbobenzyloxy-L-lysine (PCBLL) [START_REF] Aroulanda | Investigation of enantioselectivity of three polypeptide liquidcrystalline solvents using NMR spectroscopy[END_REF][START_REF] Thiele | Simultaneous assignment of all diastereotopic protons in strychnine using RDCs: PELG as alignment medium for organic molecules[END_REF], poly-γ-ethyl-L-glutamate (PELG) [START_REF] Aroulanda | Investigation of enantioselectivity of three polypeptide liquidcrystalline solvents using NMR spectroscopy[END_REF][START_REF] Thiele | Simultaneous assignment of all diastereotopic protons in strychnine using RDCs: PELG as alignment medium for organic molecules[END_REF][START_REF] Hansmann | Enantiodifferentiating properties of the alignment media PELG and PBLG -a comparison[END_REF], can provide robust enantiodiscriminating lyotropic systems. Obviously, the enantiomers of these polymers (PBDG, PCBDL, ...) are also enantiodiscriminating systems, but with inverted results in terms or enantio-orientation.

Over the last decade, new and original variants of enantiodiscriminating polypeptide-based polymers have been reported such as helicoidally chiral systems made of: i) poly-β-phenethyl-L-aspartate (PPLA) [START_REF] Schwab | Polyaspartates as thermoresponsive enantiodifferentiating helically chiral alignment media for anisotropic NMR spectroscopy[END_REF], ii) polyβ-benzyl-L-aspartate (PBLA) [START_REF] Hirschmann | Molecular weights: the key of lyotropic liquid crystalline of poly-βbenzyl-L-aspartate[END_REF], iii) poly-γ-p-biphenylmethyl-L-glutamate (PBPMLG) [START_REF] Jeziorowski | Poly-γ-p-biphenylmethyl-glutamate as enantiodifferentiating alignment medium for NMR spectroscopy with temperature-tunable properties[END_REF], iv) poly-γ-(S)-2methylbutyl-L-glutamate (PSMBLG) [START_REF] Hansmann | Synthesis of poly-γ-S-2-methylbutyl-L-glutamate and poly-γ-S-2methylbutyl-D-glutamate and their use as enantiodiscriminating alignment media in NMR spectroscopy[END_REF]. The idea related to the use of PSMBLG is to reinforce the enantiorecognition efficiency and increase spectral enantio-separations by adding a second asymmetric center in the side chain [START_REF] Hansmann | Synthesis of poly-γ-S-2-methylbutyl-L-glutamate and poly-γ-S-2methylbutyl-D-glutamate and their use as enantiodiscriminating alignment media in NMR spectroscopy[END_REF]. Finally, we can also mention the design of a polypeptide co-polymer (a polymer bearing two types of sidechain) made phenethyl and benzylpolyaspartate [START_REF] Schwab | Thermoresponsive alignment media in NMR spectroscopy: helix reversal of a co-polyaspartate at ambient temperatures[END_REF]. Mixture of polypeptides with opposite absolute configuration. Contrarily to organic solutions of a single polypeptide, mixtures prepared by mixing equal amounts by weight of polypeptides (with close degree of polymerization (DP)) of same nature but with opposite absolute configuration (e.g PBLG and its enantiomer PBDG) lead to achiral liquid crystals (ALCs) where enantiodiscrimination phenomenon is inactive. In these achiral mesophases, noted « PBG », both enantiomers exchange rapidly, between the L-and D-helical vicinities of each polypeptide on the NMR time scale. This result in identical average magnetic interactions for enantiomers, and no difference on their NMR spectra is therefore expected [START_REF] Canlet | Deuterium NMR stereochemical analysis of threo-erythro isomers bearing remote chiral centres in racemic and non-racemic liquid crystalline solvents[END_REF][START_REF] Courtieu | Evolution of the Saupe order parameters of enantiomers from a non-chiral to a chiral liquid crystal solvent: an original light on the absolute configuration problem[END_REF].

From analytical viewpoint, the comparison of NMR spectra recorded in the chiral mesophases (PBLG or PBDG) and the achiral mesophase (PBG) obtained by compensation is interesting for several reasons: i) the simplification of spectra by reduction of the number of resonances, ii) an help for the identification of 2 H signals, iii) the checking whether the doubling of spectra (signals) originates from an enantiorecognition effect in the CLC, or is related to another (dynamic) phenomena (exchange), iv) the determination of relative sign of 2 H-RQC for a given site. For instance, recording 2 H spectra in PBG can be very useful for assigning the 2 H-QD associated with diastereotopic positions in methylene groups of chiral molecules, both in isotopically enriched solutes or at natural abundance level (see below). (of same family) with different types of side chains, such as PBLG and PCBLL (same AC), may lead to uniform a new monophasic oriented system, allowing the discrimination of both enantiomers and enantiotopic directions in prochiral molecules. Interestingly in such mixed systems, the magnitude of anisotropic observables and enantiodiscriminations can be adjusted by varying the relative proportion of the two polypeptides in the mixture [START_REF] Lesot | 2 H NMR studies of two-homopolypeptide lyotropic mesophases: toward the quantification of solute-fiber interactions[END_REF][START_REF] Serhan | Investigation of solute-fiber affinity and orientational ordering of norbornadiene interacting with two-polypeptide chiral liquid crystalline solvents by NAD NMR[END_REF]. These enantiodiscriminating media can therefore be proposed as an alternative to single-polypeptide mesophases for stereochemical applications. Note finally that, whatever their polymeric ratio used, the mixture of two polypeptides of chemically different compensated by the same mixture of their respective enantiomers leads to an achiral orientationally ordered environment [71].

| Polymeric-based LLCs optimisation

The enantiorecognition efficiency and analytical potential of polypeptide systems, and notably PBLG, have been intensively explored and studied for various practical reasons: i) PBLG is soluble in a large range of organic cosolvents, ii) preparation of samples is a rather simple and alignment of polymeric mesophase with the B 0 field is rapid (< 60 s), iii) the possibility to play with the concentrations of phase components (analyte, polymer, cosolvent), iv) and last but not least, their commercial availability. Playing with all these advantages, PBLG-based mesophases revealed to be able to discriminate enantiomers of almost all classes of organic chiral molecules, but also various aspects of enantiomorphism, such as chirality by virtue of the isotopic H/D substitution, or the enantiotopy in prochiral molecules (see below).

Concentration and temperature. Compared to thermotropic liquid crystals, organic LLCs are much more "flexible" systems because they can accommodate a large amount of many types of organosoluble solute, while maintaining their liquid-crystalline properties on rather large range of temperatures (depending of the co-solvent).

Also, the degree of alignment of LLC, and in turn the magnitude of anisotropic interactions, can be controlled by adjusting the amount of polymer relative to the co-solvent in the phase or modifying the degree of polymerisation (DP) of polypeptide [START_REF] Texier-Bonniot | Analytical contribution of deutrium 2D-NMR in oriented solvents to 2 H/ 1 H isotopic characterization: the case of vanillin[END_REF].

Polarity of organic co-solvent and additives. One of the most remarkable interest of PBLG-based oriented matrices is their ability to form LLCs with various organic co-solvents, ranging from polar (DMF, TMU, NMP, pyridine, ...) to weakly polar (toluene, benzene, 1,1,2,2-tetrachloroethane, CH 2 Cl 2 , CHCl 3 , …). This large variety of co-solvents allows an optimal optimisation of the phase toward the nature of the analyte studied (polar, apolar, neutral, charged), but also the possibility to optimize the chiral environment in terms of enantiodiscriminations [START_REF] Texier-Bonniot | Analytical contribution of deutrium 2D-NMR in oriented solvents to 2 H/ 1 H isotopic characterization: the case of vanillin[END_REF]. Thus, the replacement of low polar solvent by higher polar one can significantly enhance the enantiodiscrimination properties of the phase, as shown in case of chiral amines [START_REF] Solgadi | NMR in chiral polypeptide liquid crystals: the problem of amines[END_REF] or fatty acids [START_REF] Lesot | Recent advances in the analysis of the site-specific isotopic fractionation of metabolites such as fatty acids using anisotropic natural abundance 2 H NMR spectroscopy: application on conjugated linolenic methyl esters[END_REF][START_REF] Serhan | Analysis of NAD 2D-NMR spectra of saturated fatty acids in polypeptide aligning media by experimental and modeling approaches[END_REF]. In practice, the multi-parameters choice of co-solvent is never trivial whilst a wrong choice can affect the quality of discrimination [START_REF] Texier-Bonniot | Analytical contribution of deutrium 2D-NMR in oriented solvents to 2 H/ 1 H isotopic characterization: the case of vanillin[END_REF][START_REF] Lesot | Recent advances in the analysis of the site-specific isotopic fractionation of metabolites such as fatty acids using anisotropic natural abundance 2 H NMR spectroscopy: application on conjugated linolenic methyl esters[END_REF].

The main limitation of PBLG and more generally of all homopolypeptide-based CLCs is their difficulty to dissolve highly polar analytes, and their inability to dissolved water-compatible molecules. This drawback can be overcome by adding a small proportion (1% w/w) of polar additives, such as DMSO, in order to (slightly) increase the amount of solute in solution in the mesophase [START_REF] Marx | Enhancing the orienting properties of poly(γ-benzyl-L-glutamate) by means of additives[END_REF]. Another more drastic option is to record their NMR spectra in β-peptide-based helical systems in the presence of water [START_REF] Thiele | Lyotropic liquid crystalline phases from helical βpeptides as alignment media[END_REF] or using non-peptide other watercompatible phases (see Section 3.4).

| Alternatives to polymeric chiral systems

From 2010, various new chiral mesophases made of helically chiral polymers have been synthesised and described as enantiodiscriminating oriented media (see Figure 6b). Without being exhaustive, we can mention the oriented chiral systems some of them, made of: i) poly(guanidine), poly(arylisocyanide), poly(arylacetylene) containing chiral side chains, poly(isocyanopeptide) [START_REF] Arnold | Polyguanidines as chiral orienting media for organic compounds[END_REF][START_REF] Dama | Polyisocyanides as a new alignment medium to measure residual dipolar couplings for small organic molecules[END_REF][START_REF] Meyer | Polyacetylenes as alignment enantiodifferentiating alignment media[END_REF][START_REF] Krupp | Phenylalanine-based polyarylacetylenes as enantiomer-differentiating alignment media[END_REF][START_REF] Dama | Polyacetylenes as a new alignment medium to measure residual dipolar couplings for chiral organic molecules[END_REF][START_REF] Li | Helical polyisocyanopeptides as lyotropic liquid crystals for measuring residual dipolar couplings[END_REF]. Another interesting source of weakly oriented media can be found with compressed or stretched polymer gels. The analytical potential of both these equivalent concepts for introducing anisotropy in the sample have been demonstrated from the mid-2000's by various groups [START_REF] Kobza | Stretched gelatin gels as chiral alignment media for the discrimination of enantiomers by NMR spectroscopy[END_REF][START_REF] Naumann | Tunable-alignment chiral system based on gelatin for NMR spectroscopy[END_REF][START_REF] Gayathri | Residual dipolar couplings (RDCs) analysis of small molecules made easy: fast and tuneable alignment by reversible compression/relaxation of reusable PMMA gels[END_REF].

| Hydrosoluble chiral environments

Amphiphilic chiral systems. Historically, the first descriptions of spectral enantiodiscriminations of small chiral molecules (L/D-alanine) in water-based chiral LLCs was reported by Tracey et al. who used enantiopure sodium decyl-2-sulfate as an interesting model membrane system in 1975 [START_REF] Tracey | The interaction of D and L-alanine with an optically active model membrane system[END_REF], and then the potassium N-dodecanoyl-Lalaninate based system in 1984 [START_REF] Tracey | Effects of composition on cholesteric behavior in the lyotropic mesophase system of potassium N-dodecanoyl-l-alaninate[END_REF][START_REF] Solgadi | Enantiomeric discrimination of water-soluble compounds using deuterium NMR in a glucopon/buffered water/n-hexanol chiral lyotropic liquid crystal[END_REF]. More recently enantiodiscriminating phases made of glucopon (a mixture mainly composed of alkyl (octyl and decyl) α-and β-mono-and diglucosides)/ hexanol/ buffered water [89], as well as alanine-based surfactant solutions [90] were described. The main drawback of these aligning media is the complexity of their preparation and/or their ability to orient uniformly in the magnetic field, and as a result, the number of efficient and very robust systems proposed remains rather limited.

When ignoring any enantiodiscriminations issues, a large variety of water-based mesophases have been mainly explored and used to elucidate the 3D structure of large water-soluble biomolecules, such as proteins, based on the analysis of anisotropic interactions such as RDCs, in particular following the first work of Bax's group using hydrosoluble dilute liquid-crystalline media [START_REF] Tjandra | Magnetic field dependence of nitrogen-proton J splittings in 15 N-enriched human ubiquitin resulting from relaxation interference and residual dipolar coupling[END_REF]. In this field, we can mention the use of bicelles, micelles, bacteriophages, non-ionic dilute liquid crystalline media, polyacrylamide-based gel, collagen-based gels, DNA LLCs [55, [START_REF] Clore | Measurement of residual dipolar couplings of macromolecules aligned in the nematic phase of a colloidal suspension of rod-shaped viruses[END_REF][START_REF] Ottiger | Characterization of magnetically oriented phospholipid micelles for measurement of dipolar couplings in macromolecules[END_REF][START_REF] De Alba | NMR dipolar couplings for the structure determination of biopolymers in solution[END_REF][START_REF] Prestegard | Residual dipolar couplings in structure determination of biomolecules[END_REF][START_REF] Lorieau | Liquid cystalline phase of G-tetrad DNA for NMR study of detergent-solubilized proteins[END_REF]. Interestingly, the review by Prestegard et al.

| Amount of analyte and limits of detection

The sensitivity of NMR experiments depends on many parameters. We can mainly mention: i) the strength of the magnetic field (B 0 ), the type of probe (cryogenic probe or not, direct or inverse probe, selective or broadband), ii) the spectral characteristics of the detected nuclei (spin number, gyromagnetic ratio and relaxation times), iii) and the spin density present in the sample. This last criterion is directly related to the abundance level of nucleus as well as the mass (m) of analyte available in the sample (5-mm o.d. tube), the molecular weight (MW) and the symmetry of solute and the type of chiral mixtures analyzed, (enantiopure, scalemic or racemic). In anisotropic NMR, it exists another important parameter related to the ability of the (chiral or not) liquid-crystalline mesophases to dissolve or not a large amount of a given analyte.

Generally, thermotropic phases, oriented gel based media but also DNA-based system (and more generally amphilic systems) are not ideal solvents to dissolve a large amount of analyte (5-20 mg), limiting the use to 1 H and 13 C NMR as well as 2 H NMR for isotopically-enriched molecules. In this last case, few mg of solute are generally enough for a detection. LLCs made of chiral polymers (polypeptide or polyacetylenic) provides more flexibility, accepting larger amounts of solute (up to 100 -120 mg) depending on the choice of the organic cosolvent used. Using anlytes with MW < 150 -300 g/mol and sufficient long T 2 , the detection of low abundant nuclei as 13 C is easy while the 2 H signals at natural abudance level is still possible with reasonable SNR (10 -50) and experimental time (15 -20 h). For smaller MWs, the analyte amount can be safely divided by a factor of two, at least.

Obviously, for a given m and MW, the combination of high magnetic fields with cryogenic selective probe boosts considerably the sensitivity of anisotropic NMR, reducing the experimental times while improving the (low) detection limit. However in all cases, the amount of analyte has to choose considering all parameters mentionned above and the type of applications required (dynamic experiments, measurement of anisotropic data, enantiomeric determination, isotopic studies, …).

IV. ILLUSTRATIVE EXAMPLES OF ENANTIODISCRIMINATIONS AND ENANTIOPURITY CONTROL

| Proton NMR

Due to their high isotopic abundance (99.985%), their γ and spin (I = 1/2), Hydrogen atoms are basic nuclear spies for revealing spectral enantiodiscriminations based on a difference of ( 1 H-1 H)-RDCs (see Eq. 4). However, even for small-size molecules, a dense homonuclear network made of (long and short range) 1 H-1 H residual dipolar interactions is obtained. This results in complex spectra, often poorly resolved, much more difficult to analyze than isotropic 1 H spectra as illustrated in Figure 7 in the case of (±)-1,2-epoxypropane in PBLG phase Experimentally, there is no doubt that proton NMR is the most sensitive and fastest tool for routine use.

However due to numerous dipolar 1 H-1 H couplings, anisotropic 1 H spectra usually became generally inextricable (even for small molecules), and hence very difficult to analyze by non-experts. Although they are more time consuming than 1 H NMR experiments, the analysis of 1 H coupled or decoupled 13 C/X NMR spectra remains more simple for two reasons: i) the 13 C/X-RCSA values, even larger than 1 H-RCSA, do not change significantly the overall chemical shifts profile on the analyzed spectra, and ii) the magnitudes of (X-1 H)-RDC are usually lower than their scalar couplings counterpart, keeping generally first-order type spectral patterns to analyze. Under these conditions, only slight perturbations compared with the conventional isotropic spectra are expected and generally observed, thus facilitating their interpretation.

Several homonuclear 2D NMR approaches (as the GSERF family) have been explored to improve the resolution of 1 H spectra of enantiomeric mixtures in CLCs, and to extract the structural information related to the numerous anisotropic interactions. Most of these methods rely on the ability of the NMR pulse sequences to separate the evolution of chemical shifts and 1 H-1 H spin-spin coupling interactions for each enantiomer of a mixture as shown in Figure 7b [START_REF] Giraud | Analyses of enantiomeric mixtures using chiral liquid crystals[END_REF] (see Section 2.3 above).

To be complete, we can mention the latest methodological developments associated with the pure shift 1 H NMR experiments, first developed by Zanger [100], that lead to 1 H decoupled 1 H NMR spectra where each inequivalent proton atom gives rise to a single line both in isotropic and anisotropic solvents [START_REF] Giraud | Analyses of enantiomeric mixtures using chiral liquid crystals[END_REF][START_REF] Prabhu | Application of z-COSY experiment and its variant for accurate chiral discrimination by 1 H NMR[END_REF]. Although powerful, this technique is, however, not yet simple to be implemented as a routine NMR tool because: i) the optimisation of some specific experimental parameters remains challenging, ii) NMR sequence and adapted spectral reconstruction algorithms are not always compatible with NMR hardware and software, iii) this approach is not optimal with second order 1 H spin systems, as often observed in polypeptide-based liquid-crystalline solvents. As an alternative to this challenge, we can limit the effect of second-order on 1 H NMR spectral patterns using for instance LLCs with lower aligning power as done in [101]. By using low molecular weight PBLG, it has been shown that smaller RDCs were obtained and a better resolution on 1 H signals observed on the F 2dimension of the 1 H-13 C HSQC experiments. Limitations of this approach are the commercial availability of such low-DP polypeptides, as well as the weaker stability of these mesophases.

| Carbon-13 NMR

13 C-{ 1 H} 1D NMR. The complexity of 1 H spectra in CLC mainly due to a dense network of 1 H-1 H scalar and dipolar couplings disappear when detecting "dilute" nuclei, namely weaky abundant as 13 C (1.1%) or even 2 H (0.015%) for which no coupling is simply detected. In this context, 13 C-{ 1 H} 1D NMR in CLCs rapidly emerged as an efficient alternative for chirality purposes, without any isotopic enrichment, and a wide collection of enantiomers were explored to be discriminated on the basis of 13 C-RCSA difference. Recording 13 C spectra at highest magnetic fields (see Eq. 5) becomes a fantastic advantage in terms of enantiodiscrimination magnitudes

(∆𝜈 ! ! !"
! !" ! ∝ strength of B 0 ) as well as sensitivity, leading robust determinations of ee's calculating by peak deconvolution or even by peak integration. Depending on the magnitude of discriminations, it is easy to determine ee( 13 C)'s within an error range of 2 -5%.

Illustrative examples. This tool has been implied in the analysis of enantioselectivity of many and various asymmetric synthesis. Thus, we can mention: i) the double diastereoselection in [2+3] cycloaddition reactions of chiral oxazoline N-oxides and their application to the kinetic resolution of a racemic α,β-unsaturated δ-lactone Among other important results reported, this approach has been also successively applied to discriminate enantiomers planar chirality of chiral as manganese/chromium tricarbonyl complexes ((η 6 -arene)X(CO) 3 with X: Mn or Cr) obtained with new asymmetric reactions [107, 108] (see Figure 9a). Finally, the possibility to differentiate enantiomers of chiral molecules having heteronuclear asymmetric atoms as in case of chiral sulfoxides show the analytical versatility of this tool (see Figure 9b). 4). However, the presence of long-range ( 13 C-1 H)-RDCs combined with doubling of 13 C-1 H spectral patterns when the discrimination occurs can considerably complicate the spectral analysis at a given 13 C site. Nevertheless, for chiral analytes exhibiting isolated methyl groups, it may be advisable to examine the proton-coupled carbon-13 signals of these CH 3 sites.

To illustrate this idea, we present in Figure 10a an expansion of the proton-coupled 13 C spectrum of methyl group of (±)-2-bromopropionic acid in PBLG. The expected (ideal) spectral pattern might be is the sum of two quadruplets dedoubled due to the direct 𝑇 !" coupling (with

𝑇 !! ! ! ! ≠ 𝑇 !! ! ! !
) and the long-range coupling with the proton of stereogenic site (with

𝑇 !"!!! ! ! ! ≠ 𝑇 !"!!! ! ! !
). At the temperature selected here, one enantiomer

shows no long-range coupling ( 𝑇 !"!!! ! ! ! = 0).
Here the large separations between R-and S-resonances would allow a simple determination of ee in case of a scalemic mixture (0 < ee(%) < 100). Figure 10b reports another interesting example of 13 C enantiodiscrimination of an isolated methyl group bonded now on an asymmetry sulphur atom. In the case of this (±)-S-methyl-S-p-tolyl-N-tosylsulfoximine, the broadening of 13 C resonances observed [START_REF] Sarfati | Theoretical and experimental aspects of enantiomeric differentiation using natural abundance multinuclear NMR spectroscopy in polypeptide liquid crystals[END_REF][START_REF] Parker | NMR determination of enantiomeric purity[END_REF][START_REF]To find information, an entire issue of Chemical Review was devoted to cyclodextrins[END_REF][START_REF] Wenzel | Chiral reagents for the determination of enantiomeric excess and absolute configuration using NMR spectroscopy[END_REF][START_REF] Wenzel | Discrimination of chiral compounds using NMR spectroscopy[END_REF] originates mainly from the long-range D CH with aromatic protons of the analyte.

Extraction of ( 13 C-1 H)-RDCs for each enantiomer can be simplified using heteronuclear correlation 2D experiments of 13 C-detected "resolved" type or 1 H-detected "HSQC" type (see Figure 4) [START_REF] Ziani | Visualisation of enantiomers via insertion of a BIRD module in X-H correlation experiments in chiral liquid crystal solvent[END_REF]. In heteronuclear resolved-type 2D experiments for which only the J CH 's and D CH 's evolve during the t 1 -dimension (noted Tresolved or J/D-resolved experiments) lead to a full separation of δ( 13 C)'s and T CH couplings on F 2 and F 1 dimensions, respectively, and so a better visualisation of enantiodiscriminations (see Figure 11a).

Figure partially adapted from Refs. [21] and [26] with permission.

Elegant and efficient modification of this T-resolved experiment is to replace the 180° hard pulse by a bandselective 180° pulse in order to simplify the 13 C spectral pattern on a given site when enantiodifferentiation is difficult to identify and therfore quantify due to too many lines overlaps [START_REF] Farjon | Heteronuclear selective refocusing 2D NMR experiments for the spectral analysis of enantiomers in chiral oriented solvents[END_REF]. This experiment, called HetSERF, requires the calibration and optimisation of a shaped band-selective pulse, originally a REBURP-type pulse, easily implementable on routine NMR spectrometers through a dedicated "plug-and-play" module nowadays available on recent NMR softwares [110]. Those J-resolved-type experiments were, over the years, well integrated in the strategy of elucidation of molecules in order to univocally establish their 3D structure [111].

Figure 10 Two typical examples of dedoublet quadruplets associated with the 1 H coupled 13 C signals (100.3 MHz) of a methyl group bonded to a stereogenic C or S atom, respectively: (a) of (±)-2-bromopropionic acid, (b) (±)-S-methyl-S-P-tolyl-Ntosylsulfoximine. Figure partially adapted from Refs. [20] and [26] with permission.

Alternative to these 13 C experiments is the used of 1 H-detected J/D-HSQC experiments where 1 H signals are now detected, thus offering a significant gain in sensitivity compared to the 13 C-detected sequences. Figure 11b shows an example of chiral analysis by 1 H-J-HSQC-BIRD experiments where the 13 C BIRD NMR module (inserted in the middle of the t 1 evolution period of 13 Finally, in the world of heteronuclear 13 C-1 H experiments, the 13 C-detected "HETCOR" sequence without any 13 This double information can be exploited with chiral and prochiral molecules as shown in Figure 14 in the case of (±) -[1-2 H]-1-octyn-3-ol and [ 2 H 11 ]-dibenzylmethanol, respectively. natural monodeuterated isotopomers of a molecule. Achievement of high-field NMR spectrometers, the high quality of modern electronic or even NMR tubes as well as the possible use of cryogenic probe (not a prerequisite) made anisotropic natural abundance deuterium (ANAD) NMR an emerging tool for chiral analysis purposes, but also for other important applications (as isotope fractionation analysis or structure/configuration determination) as we will describe below.

First ANAD-{ 1 H} 1D NMR spectra of a chiral analyte recorded in PBLG phase (ternary mixture) were reported from 1998 using again (±)-PE as model analyte [123] before the development of arsenal of QUOSY 2D sequences designed to simply overcrowded ANAD spectra (see above) [START_REF] Merlet | Two-dimensional deuterium NMR spectroscopy of chiral molecules oriented in a polypeptide liquid crystal: application for the enantiomeric analysis through natural abundance deuterium NMR[END_REF]. Figure 15a shows the NAD-{ 1 H} 1D-NMR spectrum of (±)-hept-3-yn-2-ol dissolved in PBLG/CHCl 3 and recorded at 14.1 T with a selective 2 H cryogenic probe, within 30 minutes [START_REF] Lafon | Deuterium three-dimensional NMR experiments for analysing weakly aligned, isotopically enriched molecules[END_REF][START_REF] Kovacs | Cryogenically cooled probes, a leap in NMR technology[END_REF]. As seen the various position of two components of each QD as well as the overlapping peaks on the 1D spectrum makes the data analysis difficult. In contrast, the analysis Qresolved-Fz 2D spectrum (after a tilting process) is direct, visualizing enantiodiscriminations (in F 1 dimension) for each inequivalent 2 H site (F 2 dimension) with a simple assignment of 2 H-QDs on the basis of their δ( 2 H). Here the presence of four 2 H-QDs for the methylene group 6, indicating the discrimination of pair of enantioisotopomers associated to both diastereotopic hydrogen sites of CH 2 . The identification of 2 H-QDs for the diastero-isotopomeric pair can be performed by recording the ANAD spectrum in the PBG achiral mesophase, thus eliminating the doubling of signal due to the enantiodiscrimination (see above). and can play an important role because: i) chiral precursors have rather low MW (100-150 g/mol), ii) large amounts of analytes are available during the initial stages of a total synthesis, allowing preparation of anisotropic samples with rather large quantities, that can be even analysed using NMR spectrometers without cryogenic NMR probe.

A first illustrative example of the ANAD NMR in CLC can be found with the study of the enantioselectivity of the alkyne zipper reaction leading to hept-6-yn-2-ol from the hept-3-yn-2-ol, a possible chiral building block for preparing the dolatrienoic acid used to build the south fragment of dolastatine-14 (see Figure 16a From practical aspects, the detection of enantiodiscriminations on methyl groups (see also Section 4.2) is highly advantageous in terms of sensitivity and subsequently robustness of ee determination, because three equivalent monodeuterated isotopomers contribute to the ANAD signal. In case (±)-hept-6-yn-2-ol, the two methyl groups showed spectral differentiation, indicating that both sites are able to probe the chirality of the molecule despite the distance of the methyl group on the terminal alkyl chain. It can be noted, however, that this occurrence no longer exists for tridec-3-yn-2-ol recorded in PBLG/CHCl 3 since only the methyl group bonded to the stereogenic center shows a spectral separation. This situation illustrates the importance of position of probed 2 H site (in particular in linear flexible molecules) relative to the stereogenic center where the molecular asymmetry appears be seen as the strongest. Here the complex conformational dynamic of the methyl of the alkyl chain (8 consecutive cooperative rotors) with the averaged motion of three homotopic sites in each methyl) leads to a large molecular space explored with two effects: i) the reduction of 2 H-RQC value, ii) the absence of discrimination. This may be seen as a decorrelation effect between this methyl site and the asymmetric center, leading to a formal loss of the chirality memory at this specific site.

Apolar alkyl enantiodiscriminations.

Very few routine isotropic NMR approaches are able to discriminate enantiomers of unsaturated and saturated chiral hydrocarbons. Indeed, the absence of highly reactive or polar functions generally prevents to form stable diastereisomers (derivatizing agent) or diastereomeric adducts (lanthanide shift reagent or solvating agent [126]. Although exotic NMR tools have been successfully applied for discriminating some specific allenics, olefinics or aromatic chiral hydrocarbons, the enantiodifferentiations observed on the basis of isotropic shifts, δ aniso ( 1 H), does not guarantee a reliable enantiomeric purity determination. Besides enantiodiscriminations of highly flexible, apolar chiral alkyls, such as (±)-3methylehexane (MH) (the simplest chiral molecule), has remained an NMR important challenge over decades.

In 2000, ANAD 2D NMR in the PBLG phase has proved to be able to discriminate enantiomer of 3methylhexane (see Figure 17) and superior homologous, as 3-methylheptane or methyloctane, and a general tool 100%, I = 1/2), it becomes possible to detect 2 H- 19 F isotopomers (1.55 10 -2 % × 100 %), and then exploit the heteronuclear couplings, 2 H-X, in particular whether a 2 H cryogenic probe is used to boost the sensitivity. An example of analysis of coupling fine structures between 19 F and 2 H atoms at natural abundance level as well as A first example was reported in the case of isotopomer2 H- 13 C of a small prochiral molecule, the benzyl alcohol in PBLG. In Figure 18b is shown the differentiation of two 2 H-13 C enantio-isotopomers associated to the methylene group of benzyl alcohol. Finally, the comparison of the F 2 projection of anisotropic NASDAC 2D spectra and 1D 13 C-{ 1 H} 1D spectra allows to determine the 2 H to 13 C isotopic effect on δ( 13 C), without deuteration of the analyte.

| Probing "Atypical chirality"

Chiral discrimination in CLCs is not limited to one type of chirality, such as compounds bearing four distinct substituents around an asymmetric atomic centre as carbon, sulphur or phosphorus atoms (see above), including the case of chirality by virtue of the isotopic (H/D, H/T, D/T, H/D/T) substitution. Indeed, the concept of chirality is larger and covers many kinds of enantiomorphisms without stereogenic center. Thus, atropoisomerism as well as the axial, planar or helical chirality are other important form of chirality, that could be defined as "atypical" or "exotic", even if the condition of nonsuperposability of their enantiomers remains true Another interesting examples of "exotic" chirality can be found with the family of cone-shaped (rigid) chiral compounds (cyclotriveratrylene or CTV) that also exhibit an axial chirality but of C 3 symmetry. In case of enantiomers of (±)-nonamethoxy-CTV (see Figure 20a), discriminations were obtained by 2 H-{ 1 H} NMR both on the diastereotopic sites of three homotopic methylene groups (hexadeuterated solute) (see Figure 20b) as well as on the three homotopic methyl groups (1, 2, 3) using NAD 2D NMR in PBLG at 14.1 T (see Figure 20c).

Although nine deuterium sites contribute to the ANAD signal for each inequivalent methyl and each enantiomer, this last result is remarkable considering the MW of the analyte. The smallest 2 H-differentiation observed on the methyl groups compared to the methylene junctions, originates mainly from an averaging effect due to their double rotation of hydrogen (deuterium) site (C-O and O-CH 3 ) and their positions located at periphery of the molecular cone.

Finally, the enantiorecognition phenomenon in case of "exotic chirality" was not only observed by 2 H-{ 1 H} or NAD -{ 1 H} NMR on the basis of 2 H-RQC differences, but also revealed by 13 C-{ 1 H} NMR on the basis of 13 [START_REF] Meddour | Proton-decoupled carbon-13 NMR spectroscopy in a lyotropic chiral nematic solvent as an analytical tool for the measurement of the enantiomeric excess[END_REF], [START_REF] Lesot | Natural abundance deuterium NMR spectroscopy in polypeptide liquid crystals as a new and incisive means for enantiodifferentiation of chiral hydrocarbons[END_REF] and [START_REF] Berdagué | Multi-nuclear NMR of axially chiral biaryls in polypeptide orienting solvents: spectral discriminations and enantiorecognition mechanisms[END_REF] with permission.

| Cryptochirality

In the world of stereochemistry, the concept of "cryptochirality" (meaning "hidden chirality") was introduced to define molecules which are chiral but do not have measurable optical rotation in the ordinarily accessible UV/Visible spectral range (800 ± 200 nm), even in neat form. The idea of cryptochiral compounds, better called Associated to the key concept of chemical desymmetrization of a prochiral molecule, it could imagine to establish a parallel between the enantiodiscrimination phenomena occuring in a prochiral solutes interacting with polypeptide fibers of polymeric CLC and the chemical desymmetrization of those same prochiral molecules to produce chiral entities with the help of homochiral ligands. Indeed the role of the polypeptide and the chiral ligand is rather similar in regard to the face recognition phenomenon. The final aim would be to find an empirical correlation between both phenomena (ee versus magnitude of enantiotopic discrimination) in order to possibly predict, from NMR measurements, the ee that could be obtained by desymmetrizing prochiral substrates [141].

| Intriguing aspects of stereogenicity: the case of malononitrile

Malononitrile (MLN) is a typical example of rigid C 2v -symmetry compounds without any prostereogenic tetrahedral center (see Figure 26a). Strictly speaking, this particular molecular structure can be seen as nonprochiral, because it may be superimposed upon itselves by an overall rotation, thus producing a structure that is indistinguishable from the original. For chemists, this molecule could be defined as pro-prochiral, because a priori two steps would be need to convert MLN into a chiral structure. From NMR viewpoint, the two 1 H nuclei and two 13 C nuclei (in nitrile group) are isochronous.

From a stereochemical point of view, the one-bond 13 C-1 H directions in MLN are defined as homotopic and so cannot be differentiated even using a CLC. However, in this structure, the long-range 13 C .....1 H (nonbonded) internuclear directions are enantiotopic, and so expected to be spectrally distinguished in a CLC. Theoretically, an AXY spin system (A = 13 C of nitrile group and X/Y = 1 H) is expected, but experimentally, a typical second-order spectral pattern (AXX' spin system with X/X' = 1 H and δ(X) ≈ δ(X')) is observed for 13 C signals (C-2/C-3) of

VI. ( 2 H/ 1 H) ISOTOPIC FRACTIONATION IN PROCHIRAL COMPOUNDS

| Analysis of Fatty acids and derivatives

Investigation of ( 2 H/ 1 H) i isotopic fractionation by isotropic 2 H-{ 1 H} 1D-NMR, namely the measurment of the intramolecular non-statistical site-specific distribution of deuterium and its analytical applications have been extensively explored and developped by Martin et and co-wokers from 1981 [149]. This tool, best known as SNIF-NMR® tool (site-specific natural isotopic fractionation), has proved to be an invaluable NMR approach for investigating the natural isotope 2 H profile in biosynthesized compounds of interest such as ethanol, fatty acids, vanilline or limonene for instance [150].

In practice, the value of site-specific ( 2 H/ 1 H) i isotope ratios (expressed in ppm) measured by NMR under quantitative conditions (T R = 5 × T 1 ) is different from one hydrogenated site to another, and can be compared to the "Vienna Standard Means Ocean Water" (V-SMOW) value (1.55 10 -2 %). This molecular isotopic information provides crucial data to understand the enzymatic processes/pathway leading to a given biomolecule (kinetic isotopic effects (KIEs) associated, source of hydrogen, ...) [151, 152]. It can be also exploited to identify the geographical origin and/or botanical source of analyte, and hence to fight against counterfeiting and adulteration of substances sold as "natural" (authenticity/traceability investigations). Despite many practical applications, this approach has two drawbacks: i) it is not able to measure the ( 2 H/ 1 H) i isotopic ratios of enantiotopic sites in CH 2 groups of prochiral molecules due to the absence of external chiral stimuli, ii) its efficiency is basically limited by the small dispersion of δ( 2 H)'s expressed in Hz (γ( 2 H) = γ( 1 H)/6.515). In this context, quantitative ANAD NMR in CLC provides fruitful alternative enable to discriminate between enantio-or diastereo-isotopomers nuclei by exploiting the spectral distribution of 2 H-RQCs [153].

First successful applications of this approach were experimentally pioneered in 2004 with the analysis of 1,1'-bis(phenylthio)hexane (BPTH), a fragment of the methyl linoleate (ML), a prochiral fatty acid methyl ester (FAMEs) defined as an essential unsaturated fatty acid (UFA) (see Figure 27). Indeed, the analysis of ANAD-{ 1 H} 2D spectrum of BPTH (recorded at 61.4 MHz) revealed the difference of intramolecular isotope ratios of enantiotopic hydrogen sites in each CH 2 groups with a proportion of (R)-enantio-isotopomers were higher than those of the (S)-enantio-isotopomers [154, 155]. Taking advantage of NMR spectrometers operating at higher field (92.1 MHz for 2 H) and equipped with a selective 2 H cryogenic probe, the method was then easily extended to study more complex prochiral FAMEs with longer chains, such as C-18 mono-or poly-unsaturated FAMEs as well as conjugated, unsaturated FAMEs (CUFAs), such as methyl oleate (MO), methyl linolenate (MLn), methyl vernoleate (MV), methyl eleostearate (ME), methyl punicate (MP) (see Figure 27) [START_REF] Lesot | Recent advances in the analysis of the site-specific isotopic fractionation of metabolites such as fatty acids using anisotropic natural abundance 2 H NMR spectroscopy: application on conjugated linolenic methyl esters[END_REF]156,157] and finally it also been tested on saturated fatty acids (SFA) [START_REF] Serhan | Analysis of NAD 2D-NMR spectra of saturated fatty acids in polypeptide aligning media by experimental and modeling approaches[END_REF].

Figure 27 Successive biotransformations of methyl myristate (MM) into longer SFAs (palmitate (MP) and stereate (MS)), then leading to various UFAs, (9Z)-MO, (9Z, 12Z)-ML, (9Z, 12Z, 15Z)-MLn, (9Z, 12R, 13S)-MV or two CUFAs, (9Z, 11E, 13E)-ME and (9Z, 11E, 13Z)-MPu (X is a methyl group). Except MV, the other FAMEs are prochiral of molecules Cs-symmetry. Figure from ref. [154] with permission.

Actually, all the potential of the method was nicely illustrated in the case of ML oriented in the PBLG/pyridine phase in (see Figure 28a), because all inequivalent mono-deuterated enantio-isotopomers are spectrally detected in a single experiment, whereas isotropic 2 H 1D-NMR spectra resolve only 30% of inequivalent 2 H sites. As a new molecular isotopic information related to the discrimination of all (R/S)-enantioisotopomers, it is possible to determine the bio enantio-isotopomeric excess, (eie) (%) for each CH 2 group, according to the equation

[157]. 𝑒𝑖𝑒 % = ! ! / ! ! ! ! ! ! ! / ! ! ! ! ! ! / ! ! ! ! ! ! ! / ! ! ! ! ×100 (9)
From the eie values, it has been evidenced that at the odd-numbered CH 2 positions are larger than those measured at even-numbered CH 2 groups all along the chain (see Figure 28b). Interestingly, using chiral FAMES (as MV), the discrimination of diastereotopic positions in methylene groups allows the determination of the diastero-isotopomeric excesses, (DIEs) for each CH 2 group of the chiral molecule. Access to eie and die to key hydrogenated sites have enabled to answer two specific biochemical questions never solved previously: i) the stereoselectivity during the elongation (by FAS) and the desaturation

| The complex case of homogeneous triglycerides

In cells, FAs are actually stocked under the form of (homogenous or not) triglycerides (TGs) with various lengths and nature of flexible chain (C4 to C36), thus corresponding to the most complex class of lipids.

From a more stereochemical viewpoint, TGs defined as homogenous (three identical chains) are fantastic prochiral molecular structures containing both homotopic, enantiotopic and diastereotopic elements (atomic sites, groups or chains) [160]. In Figure 29a is detailed the various stereochemical relationships relationships of tributyrin (TB), a short-chain (4 carbon atoms) TG. In accordance with Altmann's definition [138], homogeneous TGs are flexible molecules of C s symmetry on average, containing: i) a plane of symmetry σ associated with two enantiotopic side chains for which hydrogen atoms in a given methylene group are diastereotopic, ii) a central chain (that is diastereotopic relative to the two side chains) but where hydrogen atoms in a given methylene group are enantiotopic. Interesting diastereo-and enantiodisciminations was therefore expected when using ANAD NMR CLC, in comparison with NAD 1D NMR in achiral isotropic medium that only exhibits seven resonances corresponding to all the diastereotopic sites of the molecule (50% of the molecule's inequivalent 2 H sites).

Amazingly, the analysis of all ANAD 1D subspectra extracted from the Q-COSY Fz map of TB recorded in PBLG/Py indicates that 95% of the unequivalent 2 H sites (i.e. 19 2 H-QDs out of 20) are spectrally discriminated (see Figure 29c). In other words, chiral ANAD NMR was able to discriminate between the two lateral enantiotopic chains, a and c, chains as well as the enantiotopic C-H directions in the central chain b, while the number of differentiated sites agrees with the theoretical number expected for a C s -symmetry molecule. In this example, the enantio/diastereorecognition mechanisms were able to separate isotopomers. Such an ideal situation was however not meet for trimyristate (TM), another homogenous TG with longer chain (C-14), since

VII. TOOLS FOR STRUCTURE DETERMINATION OF SMALL RIGID MOLECULES

The problem of meso/threo mixtures

The analysis of meso/threo mixtures is an interesting challenge. Indeed they can be difficult to study by isotropic NMR while the components are not always properly separated using a chiral stationary phase. Here again, NMR in oriented solvent, due to its ability to spectrally enantiodifferentiate groups of enantiotopic elements in meso isomers and couple of enantiomers of threo isomers, reveals to be a valuable tool for helping to describe quantitatively the mixture, allowing to determine diastreoisomeric and enantiomeric excesses (de and ee) and finally give unambiguous proof to evidence a reaction mechanism.

As an example, we can mention the study of the 3-exo-trig cyclisation of anti-( E The principle of the analysis as schematically depicted in Figure 31a lies on the principle that the meso isomer in an ALC phase should to give a single 2 H-QD whose quadrupolar splitting is equal to the average of those two coupled 2 H-QDs expected in the CLC. Regarding the threo isomers, it exists a relationship between the diastereotopic deuterons and their 2 H-QDs, coupled to each other within each isomer [160]. The analytical strategy adopted was therefore to record the 2 H-COSY of deuterated mixture in ALC and CLC in order to exploit: i) the correlations between deuteron signals of the same isomer and ii) the relationship (in magnitude) existing between 2 H-QDs within the meso isomer on the one hand and the threo isomers on the other hand.

Finally, it was possible to fully individuate and assign the signals of each isomer. The deconvolution of the experimental 2 H spectrum for better accuracy in the determination of mixture composition leads to de = 35% ± 5% (d,l) and ee = 20% ± 5% (see Figure 31c). These results indicated a limited racemisation of the mixture during the chemical reaction, and therefore allowed to evidence that the opening of B into A with the reduction of B into C.

In the same framework, several homo-and heteronuclear 2D NMR approaches have been proposed to simplify the analysis of mixture of deuterated meso/threo compounds oriented in weakly ordering chiral liquid crystals [START_REF] Lafon | Homo-and heteronuclear 2D NMR approaches to analyse a mixture of deuterated unlike/like compounds using weakly ordering chiral liquid crystals[END_REF].

| Structure elucidation using the Saupe matrix: a computational approach

Determination of the cis/trans isomerism. Anisotropic NMR is a valuable tool in determining easily and rapidly the relative configuration within a given molecular group. Indeed, as RDC between two nuclei, i and j, are related to orientation of the associated i-j bond, it is possible to evidence if two internuclear directions, ij, are parallel or not, meaning that their relative configuration is either trans or cis, for instance.

In other word, it is possible to distinguish between stereoisomers by exploiting the univocal relationship between the geometry of a (rigid) its orientation into the aligned solvent and the observed NMR observables, namely here dipolar couplings as schematically depicted in Figure 32.

Figure 32 (a) Schematic description of the univocal relationship between a set of experimental anisotropic observables, the molecular geometry and the molecular order parameters (alignment tensor).

A first illustrative case was found with dihydropyridone (see Figure 33b) in 2003, whose cis and trans isomers were chromatographically separated into two different samples after their synthesis, without being possible to determine the configuration cis/trans by any standard anaytical method unambiguously [162]. Thus, in this example, it is expected that ( 13 C-1 H)-RDC couplings between internuclear pairs C6-H6 and C3-H3 are quite different in the cis isomer, unlike the trans isomer, due to a greater degree of collinearity of these C-H bonds in the trans isomer.

The sets of experimental 13 C-1 H and 1 H-1 H dipolar couplings observed for each sample were fitted towards the back-calculated dipolar from the Saupe order tensor and an optimized (semi-empirical method) for each stereoisomer. Figure 33c shows the correlation plot between both sets of data for each isomer. As seen, only experimental dipolar data extracted for a sample fit perfectly with the back-calculated dataset using the right geometry of the isomer present in the sample. In this study, an achiral aligned solvent could have been also used, but for a sake of facility within the preparation of the oriented samples, it was preferable to use an enantiopur polypeptide polymer, instead of a couple of enantiomers polypeptide leading to a racemic mixture .

Figure adapted from Ref. [162] with permission.

Exploiting again the univocal relationship between the geometrical data for a rigid solute given, and its NMR observables (here, the 2 H-RQCs), through the iterative process presented in Figure 34b, it was possible to define the most adapted molecular modelling method for these types of highly-stressed cycle. Indeed, the fitting procedure using the MM2-based geometry was not satisfactory at all, as the relative differences (RD) between experimental and back-calculated data were larger than 30% and the quality of the iteration in terms of root mean square (RMS) value was too poor to be acceptable (RMS > 20 Hz). On the contrary geometries resulting from semi-empirical methods or ab initio calculations are good enough to satisfactorily reproduce experimental data within 10% and RMS < 3 Hz).

Contrarily to previous example, NMR analysis of quadricyclane in a CLC is advantageous for two reasons: i) the possibility to spectrally differenciate the enantiotopic pairs that allows, in turn, to overdetermine the system during the iterative process (4 experimental data to fit in the CLC instead of 3 parameters to fit in an ALC), ii) the unambiguous determination of 2 H-RQCs values associated to enantiotopic directions by comparison with 2 H-RQCs measured in the ALC.

Case of chiral molecules: the Fenchone. As in the case of quadricyclane, it is valuable to exploit the analytical consistency between NMR observables and Saupe parameters for a given molecular geometry in order to solve stereochemical challenges in case of chiral solutes. Besides, it arises the opportunity to determine the Saupe matrix of each enantiomer (in racemic/scalemic mixture) to compare them and then evaluate the ability of a CLC to discriminate enantiomers. For this purpose, the fenchone (FCH), a typical chiral, rigid molecule, has been investigated using ANAD NMR [163]. For this bicyclic compound up to 20 2 H-QDs are expected to be observed for racemic/scalemic mixtures in the presence of a CLC if all 2 H enantio-isotopomers are discriminated (see Experimentally, a single oriented sample containing a scalemic mixture of FCH (ee of 30% with (2S,5R)-(+)-FCH as the enantiomer) has been prepared and analysed by proton decoupled NAD NMR. This strategy is advantageous for three reasons: i) the accuracy of the measurement of 2 H-RQC values (up to two orders of magnitude compared to ( 13 C-1 H)-RDCs), ii) the difference of amplitude of 2 H-QDs associated with the ee allowed easily to assign the signals belonging to the same stereoisomer, iii) the absence of bias in the determination of orientational order that can never be exluded when the analysis of two enantiomers is performed with two separated oriented samples containing a single isomer (as generally done when RDCs are used as anisotropic data).

Figure 35a).

In this example, 19 distinct 2 H RQCs (over 20) were extracted from the analysis of the NAD 2D-NMR spectrum ans exploted to determine the Saupe tensor for each enantiomer. As a control of relative signs of 2 H-QDs as well as to avoid ambiguities within the set of assigned lines for each enantiomer in the PBLG-based CLC, the analysis of FCH (with sample composition and temperature) in PBG-based ALC can be useful. The absolute signs of 2 H-QDs were obtained by comparison with those of ( 13 C-1 H)-RDCs associated with the corresponding C-H direction assuming that the ratio |Δν Q ( 2 H)/D CH | ≈ 11 -12. The set of experimental 2 H-RQCs for both enantiomers were tabulated as an input for the "MSpin-RQC" program (derived from the "MSpin-RDC" designed by , along with the Gaussian 09 file that contains the optimized geometry and computed EFG tensors at nuclear positions. The Saupe tensor parameters were iteratively computed (using a singular value decomposition (SVD) algorithm) for both enantiomers until the best agreement between experimental and calculated NMR observables was obtained. The schematic principle of the protocol is presented in Figure 35.

Herein, the so-called Cornilescu's Q-factor,

𝑄 = ! ! !"# ! !"#$!. !!"# ! !"#$. ! ! ! !"# ! !"#$%. ! , ( 10
)
is used as quality factor for evaluating the best fit: the smaller the value of Q, the better the agreement [165]. In Eq. 10, the term, 𝑂𝑏𝑠 ! !"#$%. , refers to the NMR anisotropic observable used and related to Saupe parameters experimentally measured without any distinction on their physical origin, i.e RCSA, RDC or RQC, and 𝑂𝑏𝑠 ! !"#$.

(namely also back-calculated data) are their properly calculated counterpart. Finally, the terms w n are normalized relative weighting factors allowing the use of three anisotropic observables simultaneously during the best-fit procedure to be combined. When the same experimental data, Obs ! !"#$%. , are used in the iterative process, then ) simply visualizes the quality of fit of experimental data.

w n = 1.

Figure from Refs. [2] with permission.

The 𝑄-factor represents a more general quality factor as entailing all possible experimental data available compared to RMS. In practice, if only one type of data are used, as in the case of FCH where only 2 H-RQCs were exploited, both parameters are equivalent from numerical point of view (w n =1). Excellent agreement was found between experimental and back-calculated 2 H-RQCs with 𝑄 factors equal to 0.007 and 0.013 for (S)-FCH and (R)-FCH in PBLG, respectively, and 0.008 in PBG. The excellent fit indicates a correct assignment of 2 H-QDs for both R and S isomers. As in the case of norbornene and quadricyclane, as an added value, it was possible to clearly confirm the spectral assignment performed in 1990 [166] and in 2014, [167] but was different from the one proposed in 1999, in particular for positions 15 and 16 (relative inverted position) [168]. Indeed, exchanging the RDC values for both positions in the bets-fit procedure make the quality factor increases up to 0.4 showing a less best-fit agreement.

Once performed, the MSpin-RQC computational programme displayed all pertinent data relative to the properties of the solute, such as the Saupe matrix elements, asymmetric parameter, as well as, the principal axis system (PAS) of the inertia tensor axis as seen on the graphical interface of the program (see Figure 36b). In addition, the Saupe tensor can be represented as a valued 3D surface. All these graphical utilities facilitates the visualization and the comparison the alignment properties reduced to simple vectors or surfaces, particularly in the case of two enantiomers. For illustration, Figure 36b shows an example of the graphical screen for enantiomers of FCH, in which the principal axis systems of the Saupe, S x' , S y' , S z' and inertia tensors I x' , I y' , I z' are simultaneously displayed. for chemists since single crystal X-ray diffraction spectroscopy is often not applicable in many cases, the analyte of interest failing to produce suitable crystals [170, 171]. Anisotropic NMR in weakly aligning media revealed an astonishing ability to solve structural problems of increasingly complex molecules (with a smaller and smaller amount of analyte), from orientational information inherent to anisotropic NMR observables. The reason is that anisotropic data encode valuabes 3D structural information including the molecular relative configuration.

During the last decade, a wide variety of natural substances have been investigated and their structure solved, confirmed or even revisited using ( 13 C-1 H)-or ( 1 H- Investigation of polystereogenic rigid molecules requires first the determination of all energeticallyreasonable (or possible) structures with a given relative configuration among the 2 n possible structures, for n asymmetric carbons. As in the case of small molecules (see above), it is possible to determine the relative configuration by comparing the experimental signed 2 H-RQC data and the back-calculated from order matrix with the help of the Cornilescu's Q-factor in the frame of an hyphenated computational protocol (using SVD algorithm). The selection of the most probable configuration corresponds to the smallest Q-value (with Q < 0.05), in combination with the best set of smaller RDs between 2 H-RQC Calc. and 2 H-RQC Exptl. values. In case of artemisinin, the smallest Q-factor was below 0.02, with a very small of RD average below 4 %.

This result has been obtained with the stereoisomer of absolute configuration 1S, 4R, 5S, 6R, 7S, 10R, 11R, namely the known absolute configuration. These results have therefore demonstrated that the 3D structure/relative configuration of complex bioactive molecules could be unambiguously determined using 2 H-RQCs, being in this case at 2 H natural abundance. Noteworty is that the Cornilescu's Q-factor does allow to consider any type of anisotropic observables related to Saupe matrix, both separately or in conjunction, One should be aware that not all anisotropic NMR measurables are equally sensitive to orientational orders as already stated in previous Section II, hence the possibility and necessity of weighting them during the best-fit procedure. Interestingly, recent work has shown that it is possible to successfully combine RDC and 13 C-RCSA data, extracted from the same molecule [178]. This combination can be useful when the number of RDCs or RCSAs are not sufficient one their own. For memory, five anisotropic data (at least) are requested for a C 1 -symmetry molecule. Such approach allows also to testing different combinations of data in order to improve the selection of a correct structure when the Q-factor is weakly discriminating. This strategy was tested in 2001 by the Orsay's group, combining 2 H-RQCs and ( 1 H-1 H)-RDCs as independent input data when 2 H-RQCs data were not enough to determine the orientational order by their own [174]. In this case, the computational program "SHAPE" initially developed by P. Diehl et al. [179] was suitably modified to handle both types of input data.

| The problem of the absolute configuration

In contrast to the relative configuration determination problem, assign the absolute configuration (AC) of enantiomers signals (R/S, M/P, Δ/Λ) with a single stereogenic center as well as enantiotopic elements (pro-R/pro-S) in prochiral molecules using anisotropic NMR in CLCs alone remains an unresolved issue since a no direct We can, however, mention the attractive description of a mean torque potential sensitive to the M/P chirality of helical solutes aligned in helical-particles based anisotropic media leading to the absolute enantiorecognition, when combined with simulation of order parameters using 191,192]. So far, a MD simulation involving PBLG using MD are rare in literature, and the results obtained are rather fragile in particular because the MD can be performed with a sufficient long time to correctly evaluate the effect of conformational dynamic of all side chains along the a-helix on the enantiodiscriminating mechanisms. This prerequisite needs a formidable calculation power, even for a small number of units considered.

Nevertheless, an empirical approach combining NAD 2D-NMR and a PBLG-based mesophase was proposed for assigning the AC of small chiral molecules [193]. The key idea of this approach lies on the fact that recognition phenomena of molecular shape play an important role in global ordering/differentiation mechanisms.

As a consequence, the relative magnitude of molecular ordering for each enantiomer, and their, is expected to be similar for a series of iso-structural chiral compounds dissolved in a given CLC, assuming the same experimental conditions (concentration and T). The strategy comprises three steps: i) the optimization of spectral enantiodiscriminations for the compound X1, ii) the analysis of NAD QUOSY 2D spectra for a series of n (enantio-enriched) mixtures (X2, X3, ... , Xn) of isostructural chiral molecules for which the AC of the major enantiomer is known, and iii) the assignment of the absolute stereochemistry of NAD signals of X1 from the analysis of NAD 2D spectra of reference molecules. This empirical strategy was first explored to determine the AC (unknown by chemists) of the major enantiomer in a prepared enantio-enriched mixture of 1,1,1-trifluoromethyl-2,3-epoxypropane (X1) synthesized in both racemic and enantiopure series. As depicted in Figure 39, the analysis of spectral 2D fingerprints of enantioenriched mixtures shows that the correlation between the fingerprints and enantiomeric shapes is the same for the various mixtures tested while the relative position (inner/outer) of 2 H-QDs can be correlated to the major/minor enantiomer (namely the higher/lower peak intensity) for the three 2 H sites [START_REF] Eliel | Stereochemistry of organic compounds[END_REF][START_REF] Emsley | NMR in anisotropic systems, theory[END_REF](4). Based on these repetitive spectral patterns it has been possible to conclude that the mixture X1 is most probably enriched in the R enantiomer, according to the Cahn-Ingold-Prelog (CIP) rules.

IX. STUDIES OF DYNAMIC SYSTEMS: A QUALITATIVE APPROACH

| The case of tetrahydrofuran

Fast exchange regime conditions on the NMR timescale lead to a fast exchange between conformations along the conformational pathway of a given molecule, meaning that the activation energy of this process is low compared to the thermal energy [194,195]. From the qualitative point of view, the NMR picture is blurried and allows to interpret the experimental data using a fictious molecular structure. This phenomenon is well known when dealing with the 1 H signal of methyl rotors for instance as all three constitutive protons usually appear with the same chemical shift due to fast conformational motion between them, or with cyclohexane where a 13 C single line (accounting for six carbon atoms) is found on the proton decoupled carbon-13 NMR spectra. In these two examples, the 1 H and 13 C nuclei are "dynamically" isochronous, although instantaneously they are (partly) anisochronous. In CLCs, same dynamic situation may occur and modulates the instantaneous spectrum one might record and individuate for each confrmation if NMR would be infinitely fast respect to conformtaional motion. The conformations of C 2v -symmetry and the C s -symmetry (depicted in Figure 40a,b) allow to understand the chemical equivalence between hydrogen (deuterium) atoms located on α and β positions of oxygen, that are expected in an isotropic solvent as well as in an ALC. Thus, in the C s -symmetry structure, sites α 1 and α 4 (resp. β 1 and β 4 ) as well sites α 2 and α 3 (resp. β 2 and β 3 ) refer to enantiotopic positions of THF, while in the C 2v -symmetry structure, all four hydrogen atoms (positions I and II) on each α and β site are isochronous. Therefore in any achiral environment, it is not possible to distinguish between both symmetries. The situation is, however, different when THF is dissolved in a CLC. If C s -symmetry conformers would have been observed in CLCs, two quadrupolar doublets for the couple α 1 and α 4 positions in the A conformer, equivalent to α 2 and α 3 positions in the B conformer by symmetry, centered on quite the same chemical shift 𝛿 ! would have been observed on NAD NMR. For the same reason, two 2 H-QDs for β 1 and β 4 positions in the A conformer, equals to β 2 and β 3 positions in the B conformer, all centered on quite the same chemical shift 𝛿 ! would have been observed. Experimental ANAD 2D spectrum in the CLC shows only two pairs of 2 H-QDs, the centered on 𝛿 ! and 𝛿 ! (see Figure 41). These results are only consistent with the C 2v -symmetry structure observed on the NMR timescale and confirmed by the disappearing of doubling of 2 H-QDs in the ALC while their splitting for the α and β sites is equal to the average value measured in the chiral mesophase. Both outcomes unambiguously insure that the high-energy level conformation of C 2v symmetry explain satisfactorily the experimental data observed due to the thermal conformational motion, too fast on the NMR timescale compared to conformational equilibrium, although this usual C 2v -symmetry do not thermodynamically exists. For THF, according to the literature, the activation energy barrier is small (ca 9 kJ/mol) [197], and so the conformational exchange cannot be slow down enough to increase the contribution of the C s -symmetry conformers, when decreasing the sample temperature even at very low values; only the pseudorotation conformational pathway has to be retained.

| The cis-decalin

In the world of apolar hydrocarbons, the cis-decalin (CDC) (see Figure 42a) is a special issue because this molecule can be described as chiral without any stereogenic center and possesses a two-fold rotational symmetry axis. Interestingly, this molecule is submitted to an interconversion between two limit conformers that constitutes a pair of C 2 -symmetry enantiomers. As a consequence, its orientational behavior in a CLC is highly dependent on the temperature used for the measure, while different structures are involved to interpret the experimental data qualitatively according to T. Experimentally, CDC was studied at different temperatures using deuterium (perdeuterated compound) as well as proton-decoupled carbon 198]. In each case, the NMR spectra obtained result from averaged NMR observables along the conformational pathway. However, in the fast exchange regime, a high-or low-energy specific conformation may be involved to describe qualitatively all experimental anisotropic NMR results.

In Figure 43 is presented the variation of 2 H spectrum of CDC in a PBLG phase as a function of temperature between 230 and 360 K. The coalescence is obtained at 300 K. As expected, the presence of a single highresolution 2 H-QD at 300 K originates from deuterium sites 9 and 10 on the bridgehead because, whatever the temperature, they exhibit no kinetic averaging between the enantiomeric conformational forms 1a and 1b [196].

Figure 43 61.4 MHz NAD NMR spectra of cis-decalin recorded at various temperatures. At 300 K, he single high-resolution doublet can be safely assigned to the deuterons 9 and 10 on the bridgehead because, whatever the temperature, they exhibit no kinetic averaging between the conformational forms. As seen in Figures 43 and44, the analysis and the interpretation of 2 H and 13 C 1D and 2D NMR spectra at high temperature (360 K) are coherent with a high-energy, C 2v -symmetry conformation, while at low temperature (270 K), chiral differentiation between the C 2 -symmetry invertomers are observed by 2 H NMR. The analysis of CDC oriented in the PBLG-based ALC properly confirms the interpration of results in the CLC as shown in Figure 44b, where the spectral discrimination of enantiotopic directions vanished in the racemic phase. Thus from 13 C NMR (see Figures 44c,d 195,196,197]. Thus, the analysis of NMR data (versus T) allows understand the molecular internal motions as well as access experimentally to dynamic parameters such as exchange or interconversion rate constants, which themselves depend on the magnitude of the barrier to interconversion, Δ𝐻 ‡ [195, 196, 198].

The dependence of the exchange/interconversion rate constant, k, on temperature is described by Eyring's equation as [196,198,199,200]:

𝑘 = !" ! ! ! exp - !! ‡ !" with Δ𝐺 ‡ = Δ𝐻 ‡ -𝑇Δ𝑆 ‡ , ( 11
)
where R is the gaz constant, N A is the Avogadro constant, h is Planck constant and T is the absolute temperature expressed in K.

For such investigations, 2 H-{ 1 H} NMR studies of (deuterated) molecules dissolved in aligned solvents is very efficient because it presents three advantages: i) simple high-resolution 1D spectra dominated by the 2 H quadrupolar interaction are obtained, ii) the coalescence phenomenon is clearly identified on spectra, iii) the spectral separations between exchanging 2 H anisotropic signals can be much larger than those observed in isotropic 2 H or 13 C NMR, thus allowing a much dynamic process range to be studied and iv) when the aligned solvent is chiral, it becomes possible to study the various conformational dynamic phenomena involving enantiomeric molecular forms or enantiotopic elements in prochiral molecules, spectrally undistinguished in [201,202].

ALCs

To illustrate our purpose, here is presented the case of (±)-1-bromo-2-methyl-3-deuterio-5-(1'naphthyl)benzene (BMNB), a chiral atropoisomer orthosubstituted biaryl (see Figure 45a). This monodeuterated aromatic compound has been investigated by 2 H-{ 1 H} 1D NMR in PBLG/CHCl 3 as a function of temperature.

The spectral variation versus T is shown in Figure 45b; note the coalescence at 250 K. From data measured in simulated NMR spectra and the analysis of the Eyring plot, namely the natural logarithm of kN A h/RT against 1/T above and below the coalescence temperature [201, 202], the activation parameters ΔH ‡ , ΔS ‡ , and subsequently ΔG ‡ (T), can be extracted from the slope (-ΔH ‡ /R) and the y-intercept of plot (ΔS ‡ /R).

Noteworty is the rate constant, k, and the free energy of activation, Δ𝐺 ‡ , at the spectral coalescence temperature, T c , (denoted herefater Δ𝐺 ‡ 𝑇 ! ) can easily be deduced from the measurement of the half-difference of quadrupolar splittings, 190]. Indeed, at this particular temperature, assuming identical time constant (𝑇 ! * ) for the FID decay of both exchanging deuterons, we can write:

ΔΔ𝜈 ! = Δ𝜈 ! ! -Δ𝜈 ! ! /2, in the 2 H spectrum below T c [
𝑘 = 𝜋 × !" ! ! ! - !" ! ! ! (12)
and

Δ𝐺 ‡ 𝑇 ! = 𝑅𝑇 ! × ln !" ! ! ! × ! ! !! ! ! ! !! ! ! (13)
In practice, the anisotropic 2 H-{ 1 H} / NAD-{ 1 H} spectra measured at T c can be very different depending on the signs of Δν Q for deuterons A and B as well as the magnitude of

Δ𝜈 ! ! -Δ𝜈 ! ! compared to Δ𝜈 ! ! and Δ𝜈 ! ! [192].
In the case of BMNB, the value of k is equal to 3.2 × 10 2 s -1 while the activation parameters were found to be equal to Δ𝐻 ‡ = 44.7 ± 0.5 kJ.mol -1 and Δ𝑆 ‡ = -18 ± 2 J.mol -1 .K -1 .

| Monitoring enzymatic reaction in DNA-based oriented solvents

From the study of norbonene in PBLG in which both prostereogenic faces of this prochiral molecule are spectrally The presence of conformational degrees of freedom in flexible molecules, meaning large, low frequency torsional molecular motions, may strongly complicates the interpretation of the anisotropic NMR data compared to the case of rigid compounds [210, 211]. Indeed, in the case of flexible analytes one has to deal with the effect of two simultaneous motional averagings: the averaging process due to the tumbling motion of the compound (as for rigid compounds) and the averaging process due to conformational flexibility, implying the existence of large number of different rotamers and conformations, each with its own potential energy and stability. Any flexible molecule therefore exists in more than one conformation each with its own probability and its own Saupe ordering matrix. If the conformer interconversion rate is in slow exchange NMR limit, a separate spectrum is expected for each conformer. The analysis of these NMR spectra yields separated conformer geometries and order tensors, while conformer probabilities are obtained from relative integrated intensities. This case is therefore as in conventional isotropic solvents quite easy to be dealt with. Most cases of interest, however, are in the fast exchange NMR limit, and a single, statistically averaged spectrum is measured in that situation.

In previous sections, we have presented various qualitative interpretations of experimental results obtained with dynamical molecules, but that was intimely connected to the conditions of fast or slow exchange regime. To go further, we describe in this last section of this review a quantitative approach developed in the latest decade and used on pharmaceutical targets. Indeed, the spatial arrangement adopted by bioactive flexible compounds affects the interactions they create with endogenous ligands as well as specific systems for the controlled delivery of active agents, influencing hence their biological activity, pharmacokinetic properties and metabolic degradation pathways [212, 213, 214]. The elucidation of 3D structures and conformational equilibrium is a preliminary step for rationalizing the relationship between conformation and drug activity and can help for drug design, screening processes and pharmaceutical formulation development.

NMR investigations made in liquid environments are interesting since it is the state of matter where molecules generally interact with endogeneous ligands and, if bioactive, play their role in the living organism.

Over the years, J-couplings and nOe measurements have provided key clues about the conformation the drug can adopt in solution [215]. However, these NMR parameters are quintessentially short range, thus these methods have limitations for obtaining connectivity information between atoms which are far apart and they are often insufficient for an unequivocal structural or conformational determination, in particular when these measurements average over two conformers or more [START_REF] De Alba | NMR dipolar couplings for the structure determination of biopolymers in solution[END_REF]216,217].

As an alternative and/or complementary strategy, the combination of NMR using liquid-crystalline solvents and the AP-DPD approach (see below) has been developed and exploited to analyze molecules of pharmaceutical interest and possessing single and multirotors. Two families of drugs were investigated: i) the profens (from flurbiprofen to naproxen), ii) the salycilates (phenylsalicylic acid to diflunisal). As illustrations, the AP-DPD approach combined to NMR in CLC will be only examplified on the case of diflunisal, a single rotor molecule, the case of naproxen possessing two independent rotors and finally the (S)-ibuprofen as model for coupled multirotors molecules (see Figure 47). To explicitely consider both the orientational motion and the internal rotation motion, it is possible to define a singlet orientational-conformational distribution function, 𝑃 !" Ω, 𝜙 , assuming as generally done, the conformational probability of a molecule is independent on the probability of its neighbours. Here, for a sake of clarity and simplicity, the conformational problem will be only examplified for the specific case of the dipolar interaction. In an uniaxial liquid-crystalline medium where the director aligns along the B 0 axis (see Figure 2), the experimental RDC between the ith and jth nuclei of a flexible molecule, 𝐷 !"

!"# , can be written as [218]:

𝐷 !" !"# ∝ 𝑃 !" 𝛺, 𝜙 !" 𝛺, 𝜙 𝜙 (14)
where the function 𝑃 !" 𝛺, 𝜙 describes the properly normalised orientational-conformational probability distribution function of the solute in the partially ordered environment. In other words, it represents the probability of finding the mesophase director Ω-oriented in the solute-fixed frame when the probe-molecule is in its 𝜙 -conformation in the aligned solution.

Following Eq. 14, 𝐷 !" 𝛺, 𝜙 is the RDC between i and j nuclei within the solute for a given conformation, 𝜙 , submitted to its own orientation 𝛺 respect to the director, n, of the mesophase. The quantity 𝑃 !" 𝛺, 𝜙 can be related to the total singlet orientational energy, 𝑈 !" 𝛺, 𝜙 , 𝑃 !" 𝛺, ϕ 𝑒𝑥𝑝 -

! !" !, ! ! !
, with 𝑇 the absolute temperature and 𝑘 ! the Boltzmann constant. In an anisotropic phase, the probability of finding a molecule with a particular conformation 𝜙 independent of its orientation, 𝑃 !" 𝜙 , is determined by summing over all the orientations 𝑃 !" 𝜙 𝑃 !" 𝛺, 𝜙

The mean-field approach allows to be built the term, 𝑈 !" Ω, 𝜙 , as a "mean-torque" potential by summing two contributions: 𝑈 !"# 𝛺, 𝜙 and 𝑈 !"# 𝜙 . Identically null in isotropic solvent 𝑈 !"# 𝛺, 𝜙 an intermolecular potential which depends on both orientational and internal angles, hence contains the conformation/orientation coupling. 𝑈 !"# 𝜙 is the internal rotational potential independent of orientation of the molecule: it depends only on the conformational state, and hence the bond rotational potentials. Following 𝑃 !" 𝛺, 𝜙 expressed as:

𝑃 !" 𝛺, 𝜙 𝑃 !"# 𝜙 × 𝑒𝑥𝑝 - ! !"# !, ! ! ! , (15)
where the distribution function 𝑃

!"# 𝜙 𝑒𝑥𝑝 !! !"# ! ! !
represents the target of any conformational study in liquids. P !"# 𝜙 is free from any possible conformational effects induced by the orientational ordering of LC, and describes the conformational distribution of the solute in a "virtual" solvent possessing the same physical chemistry properties as the oriented solvent used, with the exclusion of its ability to induce an orientational order.

Note that 𝑃 !"# 𝜙 and 𝑃 !" 𝜙 are fundamentally not expected to be identical, and are in general not equal, with a difference between them increasing as the orientational order increases.

In order to simplify the treatment data, it is quite usual to assume that internal vibrations and overall reorientational (tumbling) motions of the molecule are decoupled, because dependence of 𝑆 𝛺 on the vibrational state of the molecule is believed to be small enough to be neglected. It would be tempting to assume that the dependence of Saupe order parameters on 𝜙 can also be neglected, as often done when dipolar couplings are used to study protein structures; it is however likely to lead to major errors in the function 𝑃 !" 𝜙 and the structures derived when there is an appreciable degree of intramolecular motion present in the molecule.

The experimental observables involve products of conformational and factors and so they cannot be determined individually. Therefore, either the Saupe order matrix 𝑆 !" (one for each conformer)

or the conformer populations should be known. Without any reliable equation relating the 𝑆 !! elements of different conformers, models has to be tested in order to treat the conformational-orientational problem and extract useful information from the experimental data. These models allow: i) to shape the conformational distribution function, i.e a set of potential/probability parameters, ii) to describe, in some way, the conformational dependence of the order parameters. Even so, the number of unknowns versus the number of independent variables is often critical and a huge amount of experimental data would be necessary to describe all degrees freedom. According to this approach, the 𝑈 !"# 𝛺, 𝜙 written as:

𝑈 !"# 𝛺, 𝜙 = - 𝜀 !,! 𝜙 𝐶 !,! 𝛺 !!!;!!!,±! , (16)
where 𝐶 !,! Ω are modified spherical harmonics, 𝜀 !,! 𝜙 are elements of a suitable conformation-dependent solute-solvent interaction tensor expressed with respect to the principal axes for each conformation 𝜙 . Their chemical meaning is related to strength of the interaction between the molecular mean field and the molecule in its conformation 𝜙 .

The peculiar feature of the AP method is that these conformationally dependent interaction tensors 𝜀 !,! 𝜙 can be written as a sum of conformationally independent contributions from each rigid fragment j in the molecule,

𝜀 !,! 𝜙 𝜀 !,! ! ! ! 𝐷 !,! ! 𝛺 ! ! .
Here, j runs on the number of fragments and 𝛺 ! ! is the set of Euler angles that describes the orientation of the j-fragment in the molecule with respect to fixed axes in a rigid fragment of the molecule, via the Wigner rotation matrix 𝐷 !,! ! 𝛺 ! ! . In this way, with 𝜀 !,! ! as adjustable terms, it is possible to express the continuous dependence of the order parameters on the set of internal coordinates. The number of unknowns, 𝜀 !,! ! , is strongly reduced when chosen according to the local symmetry of the fragments within the molecule. There will be for instance only one contribution, 𝜀 !,! ! , for a fragment with cylindrical symmetry and two, 𝜀 !,! ! and 𝜀 !,! ! , from a fragment with lower symmetry, provided that the principal axes for 𝜀 !,! ! are known.

Description of 𝑼 𝐢𝐧𝐭 𝝓 and DPD approach. To fully describe 𝑃 !" 𝜙 P !"# 𝜙 , the conventional way is to set an appropriate model for 𝑈 !"# 𝜙 . Models for 𝑈 !"# 𝜙 may vary, and are often specific to the molecule under consideration and the amount of experimental data available from which they may be characterized. An alternative approach is based on the Direct Probability Description (DPD) of the conformational distribution [224]. In this case, 𝑃 !" 𝜙 is modelled without any assumptions on 𝑈 !"# 𝜙 . According to this model, 𝑃 !" 𝜙 is written as a sum (for interdependent conformations) or a product (for independent states) of Gaussian functions whose related parameters can then be varied independently, but with the constraint that 𝑃 !" 𝜙 is normalized.

Knowing 𝑃 !" 𝜙 , it becomes straightforward to determine 𝑃 !"# 𝜙 . Compared with the conventional Fourier expansion approach, the DPD method has particular advantages: i) a smaller number of parameters is useful to give a correct description of the probability function, ii) each parameter in the expansion influences only one peculiar aspect of the probability distribution (positions, width and height of maxima), and iii) conformational parameters are usually less correlated. This makes very easy to describe even complicated conformational probability density functions (even those needed to deal with cooperative motions). Usually, a good starting approximation for the shape of 𝑃 !"# 𝜙 is obtained from in silico calculations using any appropriate theoretical calculations software and method.

Technically speaking, the determination of structure, order and conformational distribution of the molecule is obtained by fitting a set of calculated dipolar couplings (obtained by a trial set of orientational, geometrical and conformational parameters) against the experimental data, while iterating on a set of unknown parameters till their optimized values. Such unknowns can be collected into three categories: i) the geometrical parameters (distances and angles), ii) the orientational parameters, whose number varies with the molecule and model used to fit the data, and iii) the potential parameters, including those parameters required to shape the internal potential. Dipolar couplings are iteratively calculated for each selected conformation, averaged on the whole distribution, and then fitted against their experimental counterpart, until their best agreement (see Figure 49). Using weakly orienting polypeptide vibrational effects do not affect the values of 1 H- Indeed, no significant difference was observed on the structures and proportions of conformers obtained by allowing a geometry optimization at each step of the conformational sampling, and by the rigid PES scan described. Following the solute symmetry, a couple of two non-equally probable conformers is found (see Figure 50). Starting from the shape of conformational distribution found in vacuo, 𝑃 !"# 𝜑 can be directly modelled as a sum of four Gaussian functions by the DPD, namely:

𝑃 !"# 𝜑 ∝ 𝐴 ! 2 • 𝑒𝑥𝑝 - 𝜑 -𝜑 ! !"# ! 2ℎ ! ! + 𝐴 ! 2 • 𝑒𝑥𝑝 - 𝜑 -𝜋 + 𝜑 ! !"# ! 2ℎ ! ! (17)
+ ! ! ! • 𝑒𝑥𝑝 - !! ! ! !! ! !"# ! !! ! ! + ! ! ! • 𝑒𝑥𝑝 - !! ! ! ! ! ! !"# ! !! ! ! ,
where 𝜑 ! !"# and 𝜑 ! !"# are the most probable values of the torsion angle, 𝐴 ! and 𝐴 ! = 1 -𝐴 ! are the relative weights of the two couples and, finally, ℎ ! and ℎ ! give the width at half maximum height of each Gaussian. In the case of DFL, ten unknowns (six 𝜀 !,! ! and four potential terms) have to be determined using forty independent 𝐷 !" !"# (twenty one for the fluorinated ring, nine for the salicylic ring, 10 between the two rings). In this peculiar case, NMR methodology was performed in order to properly be able to measure and individuate couplings involving fluorine nuclei [230]. After optimization, a good agreement between 𝐷 !" !"# and 𝐷 !" !"#! has been obtained.

| Conformational distribution of multi-rotors molecules

Analysis of (S)-naproxen. (S)-naproxen is a commonly used as non-steroidal anti-inflammatory drugs (NSAIDs) belonging to the family of profens. From the NMR point of view, this chiral molecule is a 14-spin system constituted by two non-coupled torsional degrees of freedom (ϕ and θ) and three fragments interconnected, denoted "M" (methoxy-group), "N" (naphtalenyl group), and "P" (profenic group) were denoted for the conformational analysis, as shown in the Figure 51a. As for single-rotor molecules, the AP-DPD method combined to anisotropic NMR data (PBLG) allows to determine the molecular conformational distribution [232].

In the case of independent rotors (here two), as previously stated, it is quite trivial to define 𝑃 !"# 𝜃, 𝜑 = 𝑃 !"# 𝜃 × 𝑃 !"# 𝜑 once the isotropic single rotor probability function determined by analog processing described for diflunisal preceedingly. Analysis of (S)-ibuprofen (S)-ibuprofen is a three-rotor chiral molecule whose the conformational surface can be described according to the three dihedral angles (ϕ 1 , ϕ 2 and θ) as displayed in Figure 52 [233]. For this solute, the torsional degrees of freedom ϕ 1 and ϕ 2 are coupled while θ is independent. Compared to previous cases, this last system is more complex to handle and request a larger number of anisotropic data. In order to increase this number, two PBLG-based anisotropic samples with different concentrations in polypetide were prepared (9.0 and 11.7 wt%) and data were combined. As for the others cases, 3D structure of ibuprofen was first investigated using DFT calculations with the widely employed hybrid method functional B3LYP and the basis set 6-31++G**.

Assuming the independence of the rotations of the two para-substituents, rigid and relaxed PES scans were performed separately for the torsion angle θ for the propanoic acid fragment, over the 0° -360° range with a 3°step sampling, and for the torsion angles φ 1 and φ 2 , for the isobutyl chain, over the 0° -360° range with a 20°-step sampling, allowing to obtain the in vacuo theoretical conformational distributions 𝑃 !"#$ 𝜃 and 𝑃 !"#$ 𝜑 ! , 𝜑 ! , respectively [232]. Concerning the dihedral angle θ, it was possible to extract 24 independent dipolar couplings to characterize 𝑃 !"# 𝜃 : they were reproduced with a good agreement and a satisfactory RMS error of 0.37 Hz, and two equally populated maxima at 13° ± 1° (conformation I) and 193° ± 1° (conformation II), indicating the quasiplanarity of the hydrogen atom directly bound to the stereogenic carbon atom were found.

Concerning both cooperative motions defined by 1 and ϕ 2 , a good fitting of the experimental data of each sample could be achieved considering an equilibrium between four predominant gauche conformers ( 23 Finally, for the sake of completness, it is noteworthy to stress out a promising alternative approach combining MD simulations and RDC-based orientational tensorial constraints (MDOC) in order to overcome the use of theoretical models. Latest developments were successfully obtained, in particular, on DFL, based on the experimental data reported in Ref.

XI. CONCLUSION AND PROSPECTS

Proposing tools to reveal the complexity of molecular stereogenicity concept is a fascinating investigation field and a continuous challenging task for spectroscopists. Many new aspects and original approaches have been explored and applied in France, but also in other international research groups, with the intensive work or thus covering all the aspects of enantiomorphism (enantiomerism and enantiotopism). Advantageously, the advent of routine high magnetic fields has made possible the observation of nuclei with very low sensitivity with acceptable S/N ratios such as deuterium at the natural abundance level.

In this comprehensive compendium, we have shown, through illustrative examples, the various possibilities to probe chirality and prochirality based on the same concept, namely the induction of diastereomorphic interactions inside an enantiopur oriented environments. The key advantage of this methodology is therefore the astonishing ability of polypeptide helices to interact enantioselectively with almost type of enantio-objects (whatever their chemical nature and their structure) to give sufficiently large orientational differences exploitable in various analytical domains: 3D structure, configuration and conformational analysis, internal dynamics, natural isotopic distribution determination, and finally the molecular stereochemistry if the alignment system is chiral and enantiopur.

For almost all of the NMR methods designed, enantiodifferentiation and quantitative determination of enantiomeric purity is experimentally possible even if each of them possesses both advantages and drawbacks in terms of spectral complexity, signal sensitivity, quality of enantiodiscrimination and accuracy in measurements of ee values. Selecting the most adequate anisotropic NMR tool and spy nucleus for a given compound will depend on both the chemical and NMR features but also the available amount of material. This choice, however, does not exclude the examination of other nuclei of the molecule when the first results obtained are not convincing.

Finally, the valuable advantage lies on the labelling or any chemical modification of the chiral material is needed and the NMR requirements are not different compared to those for other isotropic methods.

Pioneered from 1967, the efficient development of anisotropic enantiodiscrimination NMR tools really start with the use of weaking weakly oriented system made of polymeric-based, as PBLG, leading continuous new methods demonstrating the vitality of this exciting approach to solve new analytical problems.

Numerous research axes exploiting the enantiodisciminating potential of NMR in CLC are still open, both from a methodological and application point of view. In particular, the recent developments of the dissolution dynamic nuclear polarization (DNP) that could potentially be applied in anisotropic media to improve the sensitivity of low-abundance nuclei such as 13 C or 2 H when analyzing natural products with sub-ten milligram quantities, seems to be a formidable challenge.

An active work on the conformational analysis of flexible enantiomers in PBLG phase remains to do, in particular with the analysis a mixture of enantiomers (in as single sample) and the comparison of their respective in-solution conformational distribution. This important aspect of topics is under investigation and a complete work will be reported soon.

Also, as already mentionned above, important developments remain to perform concerning the computational predition of solute orientational orders (in polypeptide based system as PBLG) based on molecular dynamics approaches. Several papers have been reported on the subject, so far [START_REF] Serhan | Analysis of NAD 2D-NMR spectra of saturated fatty acids in polypeptide aligning media by experimental and modeling approaches[END_REF]180,181,182,183,184,185,186].

However contrasted results were obtained depending on the model/parameter used, and authors remain more or less cautious about the relevance of predicted results. The difficulty for fine-grained predictions originates in particular from the complexity of the conformational dynamics of the polymer side and the subsequent elaboration of a robust model keeping physically and chemically significant. Additionnally, in the case of flexible molecules, it is far from trivial to take into account also to a good approximation the whole solute conformational space interacting into (at least) a three-bodies system in case of lyotropic media (solute, polymer, co-solvent)

[180].

Finally, the ultimate challenge (and probably the most difficult one) will be the promotion of this methodology to the large community of chemists (non-expert NMR users) and the demonstration of the advantages of anisotropic NMR compared to isotropic NMR methods used in routine.

XII. ACKNOWLEDGMENTS

Authors would like to warmly thank Prof. Jacques Courtieu who initiated the first developments of NMR spectroscopy on CLCs in France, Prof. Marcello Longeri for fruitful scientific discussions and finally Profs. Aahron Loewenstein, James W. Emsley and Zeev Luz, for their active contributions over two decades.

Also thank you to all international (Brazil, Canada, England, Finland, Germany, India, Israel, Italy, Poland, USA, …) scientific partners, all national (CEA, ENSCP, ICSN, Universities of Lille, Nantes, Paris VI, Paris VII, Orléans, Rennes, Strasbourg, …) collaborators and ICMMO's colleagues who successfully contributed to all the work done, with the help of all students and visitors of our laboratory who also participated, as well.

Figure 3

 3 Figure 3 Schematic principle of the spectral enantiodiscrimination in CLCs based on an absolute difference of: (a) ( 13 C)-RCSA (⏐νi anisoνi iso ⏐), (b) ( 1 H-1 H)-RDC (AX spin) with ⏐Dij ⏐> ⏐Jij ⏐ and (c) ( 2 H)-RQC (spin I = 1), expected in (left) achiral liquids, (middle) in ALCs and (right) in CLCs. The various 1D signals are not plotted to scale, and the S and R assignments are arbitrary. Figure partially adapted from Refs. [4] and [6] with permission.

Figure 4 A

 4 Figure 4 A schematic description of the main homo-and heteronuclear ( 1 H-1 H or 13 C-1 H pairs) experiments used for simplifying the analysis of anisotropic spin-1/2 spectra ( 1 H/ 13 C) without (top) and with selective excitation (bottom).

Figure 5

 5 Figure 5 Schematic description of homo-and hetero-nuclear 2 H-QUOSY 2D/3D experiments: (a) echo-type schemes (Qresolved sequences and δ-resolved), (b) COSY-type scheme (Q-COSY), (c) double quantum-type scheme (Q-DQ), (d) Heteronuclear HETCOR-type sequence (CDCOM or NASDAC or 2D Refocussed-NASDAC) and (e) homonuclear 3D-type sequence (3D Q-DQ). Figure adapted from Ref. [23] with permission.

  Figure 5 Schematic description of homo-and hetero-nuclear 2 H-QUOSY 2D/3D experiments: (a) echo-type schemes (Qresolved sequences and δ-resolved), (b) COSY-type scheme (Q-COSY), (c) double quantum-type scheme (Q-DQ), (d) Heteronuclear HETCOR-type sequence (CDCOM or NASDAC or 2D Refocussed-NASDAC) and (e) homonuclear 3D-type sequence (3D Q-DQ). Figure adapted from Ref. [23] with permission.

Figure 6

 6 Figure 6 (a) Some examples of structures of homopolypeptide-based helical polymers with a L-configuration for the repeating unit (PBLA, PPLA, PBLG, PSMBLG, PELG, PBPMLG, PCBLL). (b) Other examples of non-polypeptide helical polymers: (L/D)-Val-based PLA, (R/S)-PPEMG, (L/D)-Ala-based PPI. In these polymers, the stereogenic center is only located on the sidechain.

  different polypeptides. When chemically compatible, the mixture of two polypeptides

  provides a useful overview of the main LLCs for biomolecules and their use as ordered media [95]. Polynucleotide systems (DNA). Another type of helically chiral polymers (rod-like molecules) can be found with aqueous solution of polynucleotide systems as deoxyribonucleic acid (DNA) strands (a double helix) as first demonstrated in 2009 [97]. At suitable concentration, DNA/water systems can form cholesteric phases [98] that initially used to measure RDCs in biomolecules [97]. In practice, only chiral LLCs made of short DNAfragments (150-300 base pairs) has shown interesting enantiodiscrimination potentialities when combined with 2 H NMR of isotopically enriched molecules (as chiral amino acids) [99]. Clearly, those DNA-based systems may therefore provide convenient oriented media for chirality NMR studies, but here again, compared to polypeptides LLCs, their preparation (control of pH, ionic balance, sample homogeneity, …) is more difficult, and requires the fragmentation of (sonication step by microwave) of commercially available material [99].

Figure 7

 7 Figure 7 (a) 1 H 1D spectrum (400.1 MHz) of (R)-1,2-epoxypropane (25% ee) recorded in (bottom) isotropic phase (CDCl3) and (top) anisotropic chiral phase (PBLG/CHCl3). An assignment of the lines belonging to each enantiomer is shown for the site H 3 . (b) 1 H GSERF 2D experiment applied on the propylene oxide to edit the total couplings for each enantiomer. Figure partially adapted from Ref. [29] with permission.

[ 102 ]

 102 , ii) the intramolecular hydroamination catalysed by ate and neutral rare-earth complexes [103], iii) the aza-Michael additions of O-benzylhydroxylamine to N-alkenoyloxazolidinones catalysed by samarium iodobinaphtholate [104], iv) the monofluorination reactions on propargylic compounds (see Figure 8a) [26], or v) the asymmetric reactions leading to chiral monophosphine oxides and boranes (see Figure 8b) [105], vi) address the dynamic kinetic resolution of racemic α-chloro-β-ketoesters and α-chloro-β-ketophosphonates through ruthenium-mediated asymmetric hydrogenation [106].

Figure 8

 8 Figure 8 (a) 100.3 MHz 13 C-{ 1 H} signals of the ethynyl carbon atoms (sp carbon) of 2-fluoro-3-nonyne in racemic (top) and enantioenriched (bottom) series in PBLG/CHCl3. (b) signals of the 1-methyl-1-r-oxo-2-cis,5-trans-diphenyl-phospholane in PCBLL/CHCl3. In these two xexamples, the chiral discrimination on the basis of a difference of 13 C-X total couplings (X = 19 F and 31 P, respectively). Figure adapted from Refs. [26] and [105] with permission.

Figure 9

 9 Figure 9 (a) Comparison of three 13 C-{ 1 H} 1D signals of the N-(2-methyl-2-hydroxy-1-((2-methylphenyl)chromiumtricarbonyl) propyl)-N-benzyl-hydroxylamine, a chiral (η 6 -arene) chromium tricarbonyl complex recorded in PBLG in racemic (top) and enantiopure series (bottom). (b) Four enantiodisciminated 13 C-{ 1 H} 1D signals of (±)-S-methyl-S-P-tolyl-N-tosylsulfoximine. Figure partially adapted from Refs.[START_REF] Lesot | Natural abundance deuterium NMR spectroscopy in polypeptide liquid crystals as a new and incisive means for enantiodifferentiation of chiral hydrocarbons[END_REF] and[START_REF] Madiot | Highly enantioselective propargylic monofluorination established by carbon-13 and fluorine-19 NMR in chiral liquid crystals[END_REF] with permission.

  C-1 H 2D experiments) removes all long-range 13 C-1 H couplings (J and D) [32]. In the same family of sequence, we can also mentionned the CLIP/CLAP HSQC 2D experiments which allow the total couplings, 1 T CH , to be extracted with high resolution from the indirect dimension F 1 [111].

Figure 11 (

 11 Figure 11 (a) Expanded region (centered on C-10 and C-11 carbon sites) of 13 C-1 H T-resolved 2D spectrum of (±)-2,2'dibromo-6-biphenylcarboxylic acid. (b) 1 H-J-HSQC-BIRD 2D spectrum of (±)-propylene oxide. (c) Zoom of the 1 H-13 C HETCOR 2D spectrum of (±)-2-bromopropionic acid located on the region of the methyl group. All spectra were recoded in the PBLG mesophase. Figure adapted from Refs. [28], [32] and [109] with permission.

  C and 1 H decoupling can provide interesting maps where two sets of spectral information are correlated while enantiodisciminations are simultaneously observed on a difference of T HH (F 1 ) and T CH (F 2 ) [112]. A nice example of such spectral situation is presented in Figure 11c. 4.3 | Deuterium NMR 1D/2D NMR on enriched molecules. Disregarding the high sensitivity of quadrupolar interaction to a difference of local order parameters, S C-D (see Eq. 7), the main advantage interests to use selectively deuterated chiral analytes are the simplicity of the spectral analysis due a limited number of deuterons and the absence of total J/D couplings 2 H-1 H ( 1 H decoupling). The exploratory, systematic study of analytical potential of 2 H-{ 1 H} 1D NMR in polypeptide-based CLCs was initiated with the case of (±)-1-phenylethanol (PE) deuterated on the stereogenic center, [54], an interesting model molecule featured by polar groups (OH able to form hydrogen bonds) and a strong shape anisotropy of substituent around the asymmetric carbon atom (see below) [113].From this first result almost all classes of organic chiral compounds were tested, including the case of compounds chiral by virtue of the isotopic substitution, and enantiomers with a large variety of structures and functional groups, such as alcohol, amines, carboxylic acids, esters, ethers, epoxides, tosylates, chlorides, bromides, …, and even cyclic hydrocarbons were successfully spectrally discriminated[114, 115, 116, 117, 118, 118, 120]. In majority of case, the method revealed to be efficient whatever the molecular structure: rigid, semi-rigid or flexible, with however significant variations in RQC difference (|ΔΔν Q |), which generally depends on the distance of deuterated site to the sterogenic center in more or less flexible compounds [116].

Figure 12

 12 Figure 12 presents four typical examples of 2 H-{ 1 H} 1D spectrum of monodeuterated flexible and rigid enantiomers with or without polar chemical functions or atoms, including the difficult case of enantiomers of chiral deuterofluoroalkanes with generic formula C m H 2m+1 C*D(F)C n H 2n+1 [120].In this last example, the spectral enantiodifferences, although smaller, are observed both on a difference of 2 H-RQCs as well as on the total 2 H-19 F couplings, leading in fine to two dedoubled 2 H-QDs, (see Figure12d). The analysis of these four selected examples illustrates the versatility of the method in terms of type of enantiomers that can be differentiated by the chiral polymer as well as the magnitude of 2 H enantiodiscriminations. Clearly, the deuteration position (on the stereogenic center or more or less far from it) is not necessarily crucial. However, the position can affect the quality of the enantiodiscriminations (ΔΔν Q ), depending on the molecular dissymmetry "probed" at a given nuclear site (see below) [116, 118].

Figure 12

 12 Figure 12 Four illustrative examples of monodeuterated chiral structures enantiodiscriminated by 2 H-{ 1 H} NMR (38.4 MHz): (a) (±)-2-deutero-octan-2-ol in PBLG/CH2Cl2, (b) (±)-2-methyl-2-(4-methylbenzene)-3-oxo-3-deutero-propanoic acid, methyl ester, (c) (±)-endo-4-deuterobicyclo[3.2.1]oct-2-ene and (d) (±)-3-deutero-3-fluorohexane. Figure adapted from Refs. [114], [116], [118] and [120] with permission. Exchangeable deuterons. Discrimination of deuterated chiral enatiomers prepared in situ by 2 H NMR in CLCs is possible in the case of compounds bearing a labile hydrogen (OH or NH) since their isotopic labelling can be achieved by a simple exchange in presence of MeOD or D 2 O [116, 118]. First example was reported with NH group of amides and amines of drug candidates [27] at room temperature) for which the labile site was in a

Figure 13

 13 Figure 13 Example of spectral enantiodiscrimination on (deuterated) hydroxyl group of (±)-[O-2 H]-1-phenylethanol in PBLG by 2 H-{ 1 H} NMR(61.4 MHz). Figure adapted from ref. [121] with permission.

Figure 14

 14 Figure 14 Two CDCOM 2D maps recorded at 9.4 T (100.3 MHz (F2) and 61.4 MHz (F1)) showing the simultaneous enantiodiscrimination for both 13 C and 2 H: (a) (±)-[1-2 H]-1-octyn-3-ol (chiral molecule) and (b) [ 2 H11]-diphenylmethanol (prochiral molecule), both recorded in the PBLG phase. Figure adapted from Refs. [122] with permission.

Figure 15

 15 Figure 15 (a) Example of ANAD-{ 1 H} spectrum of (±)-hept-3-yn-2-ol in PBLG/CHCl3. (b) Zoom on the tilted Q-resolved Fz 2D spectrum showing the discrimination on the group methyl 1 (two 2 H-QDs) and group methylene 6,6' (4 2 H-QDs). Figure partially adapted from Ref. [42] with permission.

  ) [124]. The analyses of the NAD signals of the methyl of precursor and those associated with one of the diastereotopic deuterons of the methylene group 3 for product, both in racemic and enantioenriched series, have established unambiguously that the reaction was a racemisation-free process (ee over 95%) (see Figure 16b-e). In another context, NAD 2D NMR in CLC was also successfully applied for controlling the enantiopurity of monomethylated chiral fatty acids, prepared by a zinc-mediated cross-coupling, prior to be incorporated into the trehalose core preparation of an analogue of Mycobacterium tuberculosis tetra-O-acylated sulfolipid [125].

Figure 16 .

 16 Figure 16. (a) Structure of dolastatine-14. (b-e): 61.4 MHz NAD signals of (b and c) methyl group 1 of 1 and (d and e) one of two deuterons in methylene group 3 of 2 in racemic (top) and enriched series (bottom) in PBLG/CHCl3. Figure partially adapted from Refs. [124] with permission.

  for efficiently analyzing all classes of chiral hydrocarbons: apolar (rigid and flexible) hydrocarbons, such as bicyclic and acyclic alkenes, alkanes, alkynes, aromatics, including the cases of chiral molecules with no stereogenic carbon, with large resonances [21, 126]. The effectiveness of this approach comes from efficient shape recognition mechanisms both involved in the orientation and enantiodiscrimination phenomena of the solute interacting with the chiral polymer. This first 2 H-RQC-based enantiodiscriminations observed of (±)-MH in PBLG was nicely confirmed recently by replacing in the polypeptide-based CLC by another helical polymer made of poly(arylacetylene) (see Figure 17b) where the stereogenic center is located on the side chain [127].

Figure 17 2 H

 172 Figure 17 2 H-RQC-based enantiodiscriminations observed with (±)-MH observed in (a) PBLG and (b) valine-derived polyacetylene (PLA) mesophases (NAD sub-spectra extracted from Q-COSY and Q-resolved Fz map, respectively. Figure partially adapted from Refs. [126] and [127] with permission.

the 2 H

 2 -QDs on the same ANAD QUOSY map is shown in Figure 18a in case of the (±)-2-(fluoromethyl)oxirane dissolved in PCBLL/DMF [129]. Thus, on this Q-COSY Fz map, recorded in around 5 hours with m R or S = 0.20 mmol of analyte, all deuterons coupled with the 19 F nucleus (methylene group 1,1') are characterized by 2 H-QDs which each component is dedoubled by the 2 H-19 F total coupling. This extra splitting appears parallel to the main diagonal of the map (dotted line) and allows a direct measurement of the total 2 H-19 F spin-spin couplings ( n T DF = n J DF + 2 n D DF ). In this example that the two bonds ( 2 H-19 F)-RDCs are slightly different for each isomer whereas those between the 19 F nucleus and the deuteron in the stereogenic carbon atom (site 2) is null by compensation with scalar couplings ( 3 J DF = -2 3 D DF ).

Figure 18 (

 18 Figure 18 (a) Two examples of ANAD QUOSY maps recorded in PBLG (Q-COSY Fz and Refocussed-NASDAC experiment) showing the detection of 2 H-X enantio-isotopomers where X is a fluorine-19 nucleus ((±)-2-(fluoromethyl)-oxirane) or a carbon-13 nucleus (benzyl alcohol). Note in (a) that the 2 H-19 F total couplings, TDF, are slightly different for each enantiomer.

Figure

  Figure adapted from Refs. [45] and [128] with permission.

  [START_REF] Lesot | Enantiotopic discrimination in the NMR spectrum of prochiral solutes in chiral liquid crystals[END_REF].Thus, first examples of enantioseparations of exotic chirality in polypeptide phases have been observed by case, two 2 H-QDs were detected (see Figure19a). The spectral differentiation of enantiomers (Λ and Δ) of deuterated metallic complexes that are chiral by virtue of helical chirality has been reported in PBLG (see Figure19b)[130, 131].

Figure 19

 19 Figure 19 Two examples of enantiodiscrimination involving "atypical chirality" revealed by 2 H-{ 1 H} NMR (61.4 and 46.0 MHz), on isotopically enriched molecules: (a) the (R/S)-2-deutero-2,3-pentadiene-1-ol, (b) the (Λ/Δ)-perdeuterated tris(diimine)ruthenium(II) complexe. The spectrum (b) corresponds to the F1 projection of a Q-COSY 2D map. Both analytes are dissolved in a PBLG phase. Figure adapted from Refs. [129] and [130] with permission.

Figure 20 2 H

 202 Figure 202 H enantiodiscrimination of (±)-nonamethoxy-CTV observed on: (a) the dideuterated methylene goups located in the cone, (b) the methyl of methoxy-groups (tilted NAD Q-resolved Fz map), both recorded in PBLG/DMF (320 K). Figure adapted from Ref. [132] with permission.

  C-RCSA differences (recorded at 9.4 T) as in the case of atroposiomers[START_REF] Meddour | Proton-decoupled carbon-13 NMR spectroscopy in a lyotropic chiral nematic solvent as an analytical tool for the measurement of the enantiomeric excess[END_REF][START_REF] Lesot | Natural abundance deuterium NMR spectroscopy in polypeptide liquid crystals as a new and incisive means for enantiodifferentiation of chiral hydrocarbons[END_REF][START_REF] Berdagué | Multi-nuclear NMR of axially chiral biaryls in polypeptide orienting solvents: spectral discriminations and enantiorecognition mechanisms[END_REF]. Thus, it has been demonstrated that enantiomers of chiral ortho-trisubstituted biphenyls (see Figure21a), as well as binaphthyls (the 2,2-dihydroxy-1,1-binaphthyl and 2,2-dimethyl-1,1-binaphthyl) (see Figures21b, 21c). As seen enantiodifferences up to values of 17 Hz have been discrimination, but only affects its magnitude experimentally measured, allowing an easy measurement of possible ee. The intercomparison of the two last examples are interesting and shows the role of the hydroxyl group in the discrimination mechanisms, and finally on the magnitude to the spectral difference.Thus, it appears here that the replacement of a polar functional group (such as OH) by a nonpolar one (CH 3 ) does not avoid the mechanisms of enantodiscrimination.

Figure 21

 21 Figure[START_REF] Lesot | Natural abundance deuterium NMR spectroscopy in polypeptide liquid crystals as a new and incisive means for enantiodifferentiation of chiral hydrocarbons[END_REF] Four examples of atypical chirality (atropoisomerism) revealed by13 C-{ 1 H} NMR (100.3 MHz) in the PBLG mesophase. Figure adapted from Refs.[START_REF] Meddour | Proton-decoupled carbon-13 NMR spectroscopy in a lyotropic chiral nematic solvent as an analytical tool for the measurement of the enantiomeric excess[END_REF],[START_REF] Lesot | Natural abundance deuterium NMR spectroscopy in polypeptide liquid crystals as a new and incisive means for enantiodifferentiation of chiral hydrocarbons[END_REF] and[START_REF] Berdagué | Multi-nuclear NMR of axially chiral biaryls in polypeptide orienting solvents: spectral discriminations and enantiorecognition mechanisms[END_REF] with permission.

  as crypto-optically active compounds, was first introduced byMislow and Bickart [133]. De facto, the determination of the absolute configuration of these cryptochiral compounds remains analytically extremely challenging. A typical example is found with the 5-ethyl-5-propylundecane (termed trivially as (n-butyl)ethyl(nhexyl)(n-propyl)methane), a chiral saturated hydrocarbon with a central, stereogenic quaternary carbon atom and bearing similar substituents. Neither enantiomeric form has any observable optical rotation, as demonstrated byWynberg et al. some 35 years ago [134]. The underlying reason for the lack of optical rotation is the lack of heteroatom(s), π-electrons, or chromophores, as well as the very small difference between the four substituents on the asymmetric carbon atom or the specific electronic properties (mass distribution) of the chiral molecule[135].Excluding carbon isotopologues, the "smallest" cryptochiral compound possessing an all-carbon quaternary stereogenic center is the appropriately deuterated neopentane ( 2 H 6 -NP): C(CH 3 )(CH 2 D),(CHD 2 )(CD 3 ) [132, 136]. It is the prototype of a crypto-optically active molecule that owes its chirality exclusively to an asymmetric distribution of the masses of the nuclei, whilst having virtually the same electron distribution as all-1 Hneopentane. As a matter of fact, within the commonly envoked Born-Oppenheimer (BO) approximation, 2 H 6 -1 is electronically achiral and computations of its optical rotation would return a zero value [137]. The experimental assignment of the absolute configurations of the enantiomers of 2 H 6 -NP therefore poses a formidable challenge but was accomplished through matching of the experimentally determined Raman Optical Activity (ROA) spectra and Vibrational Circular Dichroism (VCD) with density functional theory (DFT) computations [136, 137]. From the NMR point of view, it should be noticed that the 2 H-{ 1 H} NMR in PBLG phase failed to discriminate the enantiomers 2 H 6 -NP, while: i) this tool allowed the determination of the e.r. of its deuterated ωene cyclopropanic precursor (axial chirality) (see Figure 22a), ii) the isotropic 2 H-{ 1 H} NMR confirmed the structure of 2 H 6 -NP (see Figure 22b) [136]. Clearly in the case of 2 H 6 -NP, the overall shape recognition mechanisms are of PBLG was insufficient to differentiate the extremely small topological differences between the mono-, di-, and trideuterated asymmetric methyl groups around the central, stereogenic carbon atom.

Figure 22 ( 1 | 1 .

 2211 Figure 22 (a) Determination of the enantiomeric ratio (e.r.) of cyclopropanic precursor of 2 H6-NP by 2 H-{ 1 H} NMR (14 T / cryogenic probe) in PBLG (upper panel: racemic series; lower panel: enantiomerically-enriched series). (b) 2 H assignment (δ and J) of 2 H6-NP recorded in pyridine. Note that the scalar coupling patterns associated with CDH2 and CD2H groups when 1 H decoupling is off (lower panel). Figure adapted from Ref. [136] with permission.

Figure 23 .

 23 Figure 23. Schematic representation of changes expected in the orientation and equivalence of the principal axes system (PAS) of the Saupe ordering tensor in the four allowed groups, ongoing from achiral to chiral LC. The variation of the length of the double arrows represents the change in the magnitude of the Sαβ. Figure from Ref. [5] with permission.

Figure 24

 24 Figure 24 presents four examples of the discrimination of enantio-isotopomers in a CLC using NAD NMR. All these experimental outcomes validated definitely the theoretical work showing that only the C s , C 2v , S 4 and D 2d molecular structures undergo a reduction of their effective symmetry in a CLC, thus leading to the differenciation of their enantiopic elements. As a last illustrating example, we can mention the interesting case of 1,2-diiodoferrocene, a C s -symmetry prochiral molecule in average (see Figure 25). Actually this molecule possesses both enantiotopic C-H directions (not bonded on the same carbon atom) as well as homotopic direction due to the fast rotation of the cyclopentadienyl ring around the axis joining its center and the Fe atom. As in case of ethanol, the enantiotopic C-H directions are expected to be oriented differently in average in a CLC, and so be spectrally discriminated. Experimentaly, this enantiodiscrimination has been clearly observed, as expected, by ANAD NMR in PBLG [142]. Note that this enantiotopic discrimination phenomenon in prochiral molecules have been also detected with other chiral oriented systems than those made of polypeptide. Thus we can mention, for instance, the case of oriented systems made polyacetylene [128] or polysaccharide gels [143].

Figure 24

 24 Figure 24 Four typical examples of Cs, C2v, D2d and S4 molecules possessing enantiotopic sites (red/blue) and their associated NAD-{ 1 H} enantiodiscriminated signal: (a) 1,1'-dimethyloxyrane (Cs), (b) quadricyclane (C2v), (c) spiropentane (D2d), (d) icosane (S4). In each structure, the pairs of enantiotopic sites are shown with a red or blue color. In spectra, enantiopic signals are marked by an open or filled circle. Figure from Refs. [5] and [144] with permission.

Figure 25

 25 Figure 25 Tilted Q-COSY Fz 2D spectrum of 1,2-diiodide ferrocene (Cs symmetry in average) recorded in PBLG and where enantiotopic sites (noted b and b') are spectrally discriminated. Figure adapted from Ref. [142] with permission.

  This observation was explained by the different incorporation mechanisms of hydrogens into the chain during the elongation of fatty acids via the fatty acid synthetase (FAS) enzyme [158].

  the Δ 9 and Δ 12 desaturase) leading to the biosynthesis of the ML in Fusarium Lateritium, (a fungus species) [158], ii) the possible difference of enzymatic bioconversion processes of ML to MV involved in the case Euphorbia lagascae and Vernonia galamensis, plants [159].

Figure 28 (

 28 Figure 28 (a) Isotropic 2 H 1D spectrum of methyl linoleate. (b) Series of NAD 1D sub-spectra of ML in PBLG/pyridine extracted from the anisotropic 2 H tilted Q-COSY Fz map. The 2 H-QD of methyl 18 is not shown. (c) Variation of eie(%) vs. the methylene groups. Figure adapted from ref. [157] with permission.

Figure 29

 29 Figure[START_REF] Giraud | Analyses of enantiomeric mixtures using chiral liquid crystals[END_REF] Atomic numbering of TB along with the stereochemical relationships between the different hydrogenated sites of the glyceryl part(sites i, j, k) and methylene (2 and 3) and methyl (4) groups of the three acyl chains (a, b and c) of TB. (b) Computed 3D structure of the ''all-trans" conformer of TB. (c) Series of 1D NAD subspectra extracted from the 92.1 MHz tilted Q-COSY Fz spectrum of TB in PBLG/Py. The assignment of 2 H-QDs in relation to the various diastero-and enantiotopic positions is arbitrary. Figure from Ref. [160] with permission.

  )-benzyl-5-iodo-4methylhex-2-enoate, induced by samarium diiodid and according with the assumed mechanisms shown in Figure 30b [161]. According to this mechanism, if the opening of intermediate B into A is slower than the reduction oif B into C, no racemisation is expected. In order to facilitate the study, a dideurated probe (-CD 2 -) was chemically introduced without biasing the original composition through a LiAlD 4 reduction of the esters groups, while introducing a (non-zero) geminal 2 H-2 H couplings (dipolar) on each stereoisomer in an oriented solvent.

Figure 30 (

 30 Figure 30 (a) Structure of 'anti'-(E)-benzyl 5-iodo-4-methylhex-2-enoate. (b) Assumed mechanisms for the intramolecular cyclisation. (c) Structure of the meso (M) and active forms (A and B) of deuterated cyclopropanic derivatives. The enantiotopic deuterons in the meso are referred to as M1 and M2. Figure adapted from Ref. [161] with permission.

Figure 31 (

 31 Figure 31 (a) Spectroscopic principle for grouping the 2 H-QDs observed in a mixture made of dideuterated, meso-isomer and A,B enantiomeric pair in CLC and ALC. Coupling correlations are only observed between doublets originating from deuterons belonging to the same stereoisomer. (c) Experimental and deconvolution 2 H-{ 1 H} spectra recorded in PBLG. Figure adapted from Ref. [161] with permission.

  molecules: the quadricyclane. Quadricyclane is a fully carbonated polycyclic prochiral molecule of C 2v symmetry that could be formed by photochemically induced [2 + 2] cyclisation of norbornadiene (NBN) (see Figure 34a) [142]. There have been numerous discussions about the geometry of the formed cycle 2-3-5-6, due to large contraints undergo by norbornadiene during the chemical transformation. Indeed, molecular mechanics method MM2 model predicts distances of 147.7 and 149.1 pm for the C2-C6 (C3-C5) and C2-C3 (C5-C6) bonds, respectively, while the semi-empirical PM3 model (or ab initio models) clearly foresees a lower σbond character leading to longer internuclear bonds (152.8 and 154.8 pm, respectively).

Figure 33 (

 33 Figure 33 (a) Structures of both isomers to be individuated. (b) Structures obtained by molecular modelling of the cis and trans compounds. (c) Correlation plot between experimental RDC (D Exptl. / D Calc.), and the back-calculated RDC values, D Calc., for the two geometrical structures considered. The ideal results are points are aligned on the line D Exptl. = D Calc.

Figure 34 (

 34 Figure 34 (a) Structure of quadricyclane. (b) Flowchart showing the three-step analytical procedure adopted for the analysis of weakly ordered organic prochiral molecules through ANAD NMR. Figure from Ref. [142] with permission.

Figure 35 (

 35 Figure 35 (a) Molecular structure and numbering of FCH. (b) Simplified flowchart showing the principle of molecular structural analysis using the integrated computational program "MSpin-RQC" from the 2 H-RQCs extracted from ANAD 2D-NMR experiments The correlation plot ( 2 H-RQC Calc. vs. 2 H-RQC Exptl.

Figure 36 8 . 1 |

 3681 Figure 36 The principal molecular axis system (Sx', Sy', Sz') of the diagonalized Saupe matrix, the inertia tensor axes, (Ix', Iy', Iz'), and the Saupe tensor surface representation (red and green surfaces indicate positive and negative 2 H-RQCs, respectively) for (S)-FCH and (R)-FCH oriented in PBLG/CHCl3. Figure adapted from ref. [163] with permission.

  2 H-Finally the selection of the right stereoisomers is obtained unambigously when a large difference between the first and the second smallest Q exists. The schematic procedure is schematically depicted in Figure 38.

Figure 37 (

 37 Figure 37 (a) Molecular structure, atomic labels and R/S descriptors of artemisinin associated with the (known) absolute configuration. (b) DFT-optimized 3D structures labeling the α/β (a/b) positions of diastereotopic protons/deuterons. (c) Part of the 92.1 MHz NAD-{ 1 H} Q-resolved Fz map (tilted and then symmetrized). Figure adapted from ref. [177] with permission.

Figure 38

 38 Figure 38 Simplified principle of fitting of experimental 2 H-RQC dataset to a possible set of possible structures. Figure adapted from ref. [177] with permission.

  to do so [180]. In fact, as the enantiodiscrimination process originates from a difference in orientational ordering of enantiomers or enantiotopic elements, the a priori assignment of AC of a given signal can be only envisaged via ab initio methods (as Molecular Dynamic (MD) computational simulations) able to predict molecular alignment and ordering differences from the evaluation of interactions between a given analyte and a helically chiral system[180, 181, 182, 183, 184, 185, 186]. Although some achievements were obtained for predicting orientation behavior of analytes interacting with achiral, nematic orienting solvents [187, 189], or even polypeptide-based achiral systems (PBG) [75], the case of two enantiomers interacting with helical-polymer based CLCs remains still problematic despite some recent progresses [180

Figure 39 (

 39 Figure 39 (a) Structure of enantiomeric shapes denoted A and B and nature of substituents (X1 to X6) on the epoxyde ring. The correspondence between the A shape and the stereodescriptors R/S is given. (b) Four examples of 92.1 MHz NAD spectral 2D fingerprints associated with the three epoxyde ring deuteron sites in compounds 1, 2, 4 and 6 (ee = 33%). Figures adapted from ref. [193] with permission.

  Figure 40a, which is however quite unlikely, according to twell-established thermodynamical rules of stability 5membered rings. Following its conformational pathway, this high-energy C 2v -symmetry representation of THF might actually results from a fast motionally averaged equilibrium of two low-energy C s -symmetry conformers (see Figure 40b) or alternatively from the pseudorotation conformational equilibrium shown in Figure 40c.

Figure 40 (

 40 Figure 40 (a) High-energy C2v-symmetry conformation of THF and associated atoms numbering. (b) Cs-symmetry conformation of THF. (c) Schematic description of the pseudorotation conformational pathway of THF. Figure partially adapted from ref. [196] with permission.

Figure 41

 41 Figure 41 61.4 MHz NAD Q-COSY 2D spectrum of THF dissolved in a PBLG/CDCl 3 solvent recorded at room temperature. Figure adapted from ref. [196] with permission

Figure 42 (

 42 Figure 42 (a) C2-symmetry conformers of cis-decalin and associated position numbering; (b) C2v-symmetry high-energy conformation of cis-decalin. Figure adapted from ref. [196] and [198] with permission.

  Figure43 61.4 MHz NAD NMR spectra of cis-decalin recorded at various temperatures. At 300 K, he single high-resolution doublet can be safely assigned to the deuterons 9 and 10 on the bridgehead because, whatever the temperature, they exhibit no kinetic averaging between the conformational forms. Figure adapted from ref.[196] with permission.

  ), one single line for each homotopic pair and enantiotopic carbon atom in ALC are expected, meaning one for C-9 et C-10, one for C-5, C-1, C-8, C-4 and one for C-6, C-2, C-7, C-3, whereas in CLC are expected five single lines at maximum, still one for C-9 et C-10, but one for C-5 et C-1 different from the one associated with C-6 et C-2, and one for C-7 et C-3 different from the one associated with C-8 et C-4. For same reasons, a maximum of five 2 H-QDs are expected in 2 H NMR in the CLC, D-9/D-10, D-5, D-4, D-1, D-8, D-5', D-4', D-1', D-8', D-6, D-2, D-7, D-3, D-6', D-2', D-7', D-3, and nine of 2 H-QDs in CLCs, two for each enantiotopic pairs and one for the homotopic pair D-9/D-10.

Figure 44 (

 44 Figure 44 (a and b) 61.5 MHz NAD NMR tilted Q-COSY 2D spectra of CDC in PBLG/CHCl3 and racemic PBG/CHCl3 at 360 K. (c and d) 9.4T 2 H-13 C HETCOR correlation spectra of CDC at 243 and 356 K. Figure adapted from refs. [196] and [198] with permission.9.3| Determination of the activation barrier energy by anisotropic 2 H NMR

  distinguished and each atomic site belonging to the same face individuated [139], it was established that 2 H-{ 1 H} NMR in CLC may provide a key analytical tool to follow/examine, for instance, the specific approach of an enzyme toward a preferential site or face of a ligand [141]. This information is very useful for understanding the enzymatic racemization mechanism that is a crucial step for designing possible inhibitors of associated enzyme [203]. To provide an alternative analytical method to well-established methods (circular dichroism (CD) or UV spectroscopy) or more advanced tools (chiral capillary electrophoresis) [204, 207, 206], in 2013 has been demonstrated the ability the 2 H-{ 1 H} NMR in CLC to follow simultaneously, in situ and in the concentrations of a couple of enantiomers in presence of an enzyme versus time, in order to determine kinetic parameters and in a second step to study the effect of possible inhibitors [207].

Figure 45 (

 45 Figure45 (a) Enantiomeric conformers associated with the (±)-1-bromo-2-methyl-3-deuterio-5-(1'-naphthyl) benzene. (b) Experimental 2 H-{ 1 H} 1D-NMR spectra(61.4 MHz) in PBLG versus five temperatures and showing the coalescence. Figure adapted from Ref. [202] with permission.The basic idea is to follow the distinct 2 H NMR signals of both deuterated enantiomers dissolved in the CLC in the presence of an active enzyme, and calculate the turnover numbers (k catL and k catD ) of the enzyme without the requirement of any chemical transformations of the substrate as needed for current analytical tools, excepted prior deuteration (see Figure46). The proof of principle has been first reported in the case of the alanine racemase (AR) which recognizes a chiral substrate, such as L-Alanine (L-ALA) and reversibly converts it into its enantiomer, i.e. D-Alanine (D-ALA) or vice versa. In this example, as alanine racemase (AR) is only watercompatible, chiral oriented lyotopic solutions of DNA fragments in water has been used as discriminating aligning medium [99, 207], while (D/L)-alanine was selectively deuterated on the methyl group in order to benefit of the quarupolar interaction as well as minimize the acquisition time of 2 H spectra during the reaction monitoring.

Figure

  Figure 46a presents the variation of 2 H-QDs of L-and D-Alanine-d 3 at different times after the introduction of the enzyme. Careful deconvolution of 2 H spectral lines allows to follow the concentration of D and L-isomers and their quantitative interconversion. Figure 46b shows the time-dependent concentrations, showing clearly the

Figure 46 (

 46 Figure 46 (a) Evolution of 2 H-{ 1 H} 1D signals (same scale) of (L)-and (D)-ALA-d3 recorded at different time intervals after the introduction of AR in the DNA-based mesophase. (b) Variation of ee (red dashed line) and experimental (black and grey continuous lines) and fitted (white dashed lines) concentrations (in mM) of (L)-and (D)-ALA-d3 as a function of reaction time.Figure from Ref. [207] with permission.

1 |

 1 Figure 46 (a) Evolution of 2 H-{ 1 H} 1D signals (same scale) of (L)-and (D)-ALA-d3 recorded at different time intervals after the introduction of AR in the DNA-based mesophase. (b) Variation of ee (red dashed line) and experimental (black and grey continuous lines) and fitted (white dashed lines) concentrations (in mM) of (L)-and (D)-ALA-d3 as a function of reaction time.Figure from Ref. [207] with permission. Finally, in the framework of in-situ reaction monitoring combining weakly orienting solvents and 2 H NMR, several anisotropic 2 H ultrafast 2D-NMR experiments (denoted "ADUF" spectroscopy) have been designed to follow extremely fast and/or multiple (cascade) chemical transformations for which the fast identification and reliable quantification of 2 H signals of chiral (or not) products (reactants, (un)stable intermediates, products, …) could be required. These particular homonuclear experiments replace the usual time encoding by a spatial encoding in QUOSY-like sequences (Q-COSY, Q-resolved and Q-DQ) allow recording anisotropic 2 H spectra in sub-second experimental times, allowing to enlarge the usefulness of the proposed technique [208, 210].

Figure 47

 47 Figure 47 Molecular structures of (a) diflunisal, (b) (S)-naproxen and (c) (S)-ibuprofen along with the various rotors (ϕI or θ)shown.

Figure 48 10 . 3 |

 48103 Figure 48 Schematic overview of the analytical strategy used describing the conformational distribution of flexibles solutes, involving the approach. The final target is 𝑃 !"# 𝜙 .

Figure 49 .

 49 Figure 49. Schematic description of the conformational analysis in a PBLG mesophase using the AP-DPD model: from the sample preparation (step 1) to the determination of the isotropic conformational distribution of the flexible solute studied (step 4).

  Within the AP model, for the specific case of DFL, the description of orientational interaction energy requires six fragment tensors 𝜀 !,! ! , as follows: i) 𝜀 !,! !"#$ ! and 𝜀 !,! !"#$ ! for the biaxial 𝐶 !! -symmetry fluorinated ring (ring A), ii) 𝜀 !,! !"#$ ! and 𝜀 !,! !"#$ ! , for the biaxial 𝐶 !! -symmetry salicylic ring (ring B), iii) 𝜀 !,! !!! for the monoaxial component along the C-F bond direction, iv) 𝜀 !,! !!! , for the monoaxial component along the C-C(OOH) bond direction (see Figure 44a).

Figure 50 (

 50 Figure 50 (a) AP modelisation of 𝑈 !"# 𝛺, 𝜙 applied to diflusional. (b) Variation of normalized potential energy as a function of φ1 obtained from B3LYP/6-31++G** calculations in vacuo. (c) Modelisation of 𝑃 !"# 𝜑 of DFL compared to its in vacuo theoretical torsional curve 𝑃 !!!" 𝜑 . Figure adapted from Ref. [230] with permission.

Figure 51 (

 51 Figure 51 (a and b) Decomposition of (S)-naproxen into three fragments and AP modelisation applied. (c and d) Shapes of functions 𝑃 !"# 𝜃 and 𝑃 !"# 𝜑 , respectively, compared to their in vacuo theoretical torsional curves couterparts. (e) Plot of 𝑃 !"# 𝜃, 𝜑 resulting from AP-DPD approach in the case of (S)-naproxen. adapted from Ref. [232] with permission.

  .25% each in terms of relative population) and two barely populated trans structures(3.5% each). The corresponding torsional probability distribution 𝑃 !"# 𝜑 ! , 𝜑 ! is reported in Figure52d. It shows a significant qualitative agreement with the theoretical probability distribution P !!!" 𝜑 ! , 𝜑 ! calculated from the potential energy values obtained by DFT calculations (see Figure52d) as well as theoretical data found in the literature although an upper shift of about 10° -15° can be noticed by comparing the 𝜑 ! value for gauche conformers. As a consequence of the symmetry of the whole molecular system, the twelve conformers obtained combining the spatial arrangement for the propionic fragment (I and II) and for the isobutyl chain (g 1 , g 1 *, g 2, g 2 *, t and t*) end up overlapping, so that only six independent minimum-energy structures are needed to describe exhaustively the conformational flexibility of Sibuprofen (see Figure52e).

Figure 52 (

 52 Figure 52 (a) Description of best fit agreement between experimental and calculated dipolar couplings of (S)-ibuprofen allowing to characterize the function 𝑃 !"# 𝜃 shown in (b). (c) Description of best fit agreement between experimental and calculated dipolar couplings allowing to characterize 𝑃 !"# 𝜑 ! , 𝜑 ! shown in graphs (d). (e) Representation of the six conformers of (S)-ibuprofen. Figure adapted from Ref. [233] with permission.

  183. Although a promising tool for the investigation of torsional angle distributions in flexible molecules, this approach requires quite high level/CPU time-consuming data computing and the use of the specific software (COSMOS) [183].

  remarkable achievements of G. Celebre, G. De Luca, R.R. Gil, C. Griesinger, B. Luy, G.E. Martin, A. Navarro-Vázquez, C.M. Thiele, M. Reggelin, R.T. Williamson, for instance, (non-exhaustive, alphabetic list), and all their collaborators.Based on the detection of residual order-dependent intramolecular NMR interactions (RCSA, RDC, RQC), multinuclear NMR of magnetically oriented-samples (weaking oriented CLCs) provides original, efficient, flexible and versatile analytical tools to extract a wide variety of global or local molecular information on solutes,

Table 1 .

 1 Number of order parameters the Saupe matrix for non-flexible molecules of different point group symmetries dissolved in an achiral and chiral uniaxial mesophase and location of the a, b, c axes

				Non-chiral mesophase	Chiral mesophase
		Molecular	Effective	Indep. order	Location of	Effective	Indep. order	Location of
	point group	molecular	parameters	a, b, c axes	molecular	parameters	a, b, c axes
		symmetry	point group	(Saupe elements)		point group	(Saupe elements)
	C 1 , C i	C 1 , C i	S aa , S bb -S cc	not special	C 1	S aa , S bb -S cc	not special
					S ac , S ab , S bc			S ac , S ab , S bc
	C s	a		C s	S ab , S bb -S cc	a ⊥ to the plane	C 1	S aa , S bb -S cc	not special
					S bc			S ac , S ab , S bc
	C 2 , C 2h	C 2 , C 2h	S ab , S bb -S cc	a // to the 2-fold	C 2	S ab , S bb -S cc	a // to to the 2-
					S bc	axis, b ⊥ to the		S bc	fold axis
						mirror plane
	C 2v	a	C 2v	S aa , S bb -S cc	a // to the 2-fold	C 2	S ab , S bb -S cc	a // to the 2-
						axis, b ⊥ to the		S bc	fold axis
						mirror plane
	S 4	a		S 4	S aa	a // p-fold axis	C 2	S ab , S bb -S cc	a // to the 2-
						or intersection			fold axis
						of mirror planes
	D 2 , D 2h	D 2 , D 2h	S aa	a,b,c // to the 2-	D 2	S aa , S bb -S cc	a,b,c // to 2-
					S bb -S cc	fold axes			fold axes
	D 2d	a	D 2d	S aa	a // p-fold axis	D 2	S aa , S bb -S cc	a,b,c // to 2-
						or intersection			fold axes
						of mirror planes
	C (with n = 3, 4, 6),	S aa	a // p-fold axis or intersection	C n or D n (with n > 3)
				S 6 ,		of mirror planes
				C ∞ , C ∞v ,
				C ∞h , D ∞h

n , C nv , C nh , D n , D nh , D nd , (with n = 3, 4, 6), S 6 , C ∞ , C ∞v , C ∞h , D ∞h C n , C nv , C nh , D n , D nh , D nd ,

  13

24, 25, 26, 27, 28].

  F, 31 P, 10 B, …, are possible nuclear probes to separate the signals of enantiomers or enantiotopic elements [In particular, the CF 3 groups are excellent probes for a spectral differentiation (two triplets observed on 19 F-{ 1 H}

spectra) on basis of ( 19 F-19 F)-RDCs difference (see below). For monofluorinated or organophosphored chiral compounds, enantiodiscriminations are expected on the basis of a difference of 19 F-RCSAs or 31 P-RCSAs, respectively, assuming that protons are decoupled

[START_REF] Madiot | Highly enantioselective propargylic monofluorination established by carbon-13 and fluorine-19 NMR in chiral liquid crystals[END_REF][START_REF] Phillips | The measurement of high enantiomeric excesses in chiral liquid crystals using 19 F NMR and exchangeable protons in 2 H NMR[END_REF][START_REF] Berdagué | Multi-nuclear NMR of axially chiral biaryls in polypeptide orienting solvents: spectral discriminations and enantiorecognition mechanisms[END_REF]

.

45]. Obviously

  , all 2 H 2D experiments design for recording anisotropic 2 H spectra at natural abundance can be applied to study poly-or perdeuterated solutes, as well.

2 H nuclei either in deuterated molecules ("CDCOM" experiment: Carbon-Deuterium Correlation in Oriented Media) [40] or at natural abundance level ("NASDAC" experiment: Natural Abundance Spectroscopy Deuterium And Carbon) [

  The last frontier of this game has been crossed in 2020 by exploiting the advantages (simple spectral data, magnitude of RQCs, dilute spins, …), of 2 H-RQCs extracted from ANAD NMR 2D spectra, assuming that the sensitivity is nowadays real drawback[177]. The performance of this last tool has been examined for two multi-cyclic chiral, natural compounds of pharmaceutical interest oriented in PBLG-based oriented samples: the strychnine, a poison, and artemisinin, an antimalarial drug used against Plasmodium malaria) as examples. Figure37shows the ANAD Q-resolved Fz 2D spectrum of artemisinin recorded within 15 hours with only(50 mg (MW: 282.3), ie a molar concentration for each monodeuterated isotopomer of about 35 mmol.l -1 ) where unambiguous 15 (over 16) 2 H-QDs can be experimentally measured and used.

1 H)-RDCs data [172, 173, 174], and more recently from 13 C-RCSA's data [175, 176].

  In the last decade, was developed a hydrid approach, combining the AP method and a direct probability distribution, DPD, for describing directly the internal potential as sum of Gaussians [224]. Here will be however only introduced and discussed the AP and DPD approaches useful for describing 𝑃 !" 𝛺, 𝜙 in the attempts done in CLCs, as schematically overviewed in Figure48. A more detailed description of this approach is available in references 225 and 226.The AP methods for modelling 𝑼 𝐞𝐱𝐭 𝜴, 𝝓 . Several ways of modeling 𝑈 !"# 𝛺, 𝜙 have been proposed; here is discussed only the so-called AP-Independent Fragment model the most commonly used approach to model

10.2 | Theoretical approaches describing the interdependent conformational-orientational problem.

Several such approaches have been considered. The simplest possible model, the Rotational Isomeric State (RIS) model assumes that only a restricted set of minimum-energy structures is populated [219]. For instance, the free rotation of methyl groups are described by the three statistically weighted staggered conformers. More realistic models allow for continuous bond rotations. Two main approaches, alone or combined, have been proposed in the past [220]: i) the Additive Potential (AP) method, an "approximate" approach whose aim is to reduce latest equations to manageable expressions by using some physically justifiable approximations [221], ii) the Maximum Entropy (ME) method, an "unbiased" method based on Information Theory [222, 223].

𝑈 !"# 𝛺, , originally introduced by Marčelja and subsequently modified by

Emsley et al. [221, 227]

.

228, 229].

  1 H or 1 H-13 C dipolar couplings, and hence, the computation of vibrational corrections is rarely considered [

H-{ 1 H} NMR with monodeuterated chiral allenics that exhibits axial chirality of C 1 symmetry [129]. In this

nitrile group in the PBLG chiral phase. This pattern disappears in the PBG-based ALC as displayed in Figure 26b [139]. The transformation of the RAS (a, b, c) into the PAS (a', b', c') can be viewed as a simple rotation around the b axis with a tilting angle, α, of +14° between the initial and final planes (see Figure 26a). From this experimental result, a more fundamental stereochemical issue about the definition of the concept of prochirality and enantiomorphism is raised here. Indeed, MLN can be regarded as a prochiral compound when interacting with the CLC, while for organic chemists (who generally define prochirality in terms of the practical possibility that an achiral molecule becomes chiral in the course of a single-step chemioselective chiral process), this molecule is not! Amazingly, the spectral differentiation in a CLC of enantiotopic directions in MLN, a C 2v symmetry molecule without a prostereogenic tetrahedral center has validated a stereochemical hypothesis made by Mislow and Raban in 1967 [145]. Indeed it had been speculated that "for molecules of the type "CXXYY"…, the two X groups as well as the Y groups are equivalent and cannot be distinguished in chiral or achiral circumstances. However, the relationships between X and Y groups are not all equivalent. The four X-Y relationships may be ordered into two enantiotopic sets of two equivalent relationships" The NMR result obtained with MLN validate also a more recent concept of stereogenicity defined by Fujita in 1990 who considers that "the compounds (CX 2 Y 2 ) can be regarded as prochirals, since the four edges (X-Y) construct an enantiospheric C 2v (C 1 )

orbit" [146, 147, 148].