Thomas Buhse

José-Manuel Cruz

María E Noble-Terán

Jean-Claude Micheau
email: micheau@chimie.ups-tlse.fr

Non-stochastic spontaneous symmetry breaking: What kind of weak asymmetric forces is propelling?

Keywords: chirality, mirror-symmetry breaking, non-stochastic behaviour, crystallisation xxxxxxx, ©JIMIS, Creative

We give a survey over recently published laboratory experiments on mirror-image chemical processes -inclusively new studies -that are mainly related to aggregation and crystallisation effects and have in common the observation of unexpected non-stochastic behaviour. While the outcome of these experiments should be rationally predicted as stochastic, i.e. to produce a statistically equal amount of left-handed and right-handed molecules or entities, the laboratory observations demonstrate in various cases exactly the contrary: one enantiomeric form arises persistently in significantly higher amounts than the other. These processes describe non-stochastic mirrorsymmetry breaking and are related to far-from-equilibrium autocatalytic dynamics. The puzzling non-stochastic effects are often attributed to the presence of unknown chiral impurities mainly coming from the bio-organic environment. However, there is some indication that the impurity supposition not always holds, so that other influences such as the parity-violating energy difference (PVED) or other unspecified physical factors in combination with nonlinear kinetics should be additionally taken into account.

I INTRODUCTION

The word chiral was coined by Lord Kelvin in 1894. It is derived from the Greek word kheir that stands for hand. In chemistry, chirality is most often related to molecules bearing an asymmetric carbon atom. From the stereochemical point of view, if the four substituents of a carbon atom are all different, the corresponding structure is non-superimposable onto its mirror image, i.e. the molecule is chiral and occurs in two enantiomeric forms [START_REF] Dutta | Enantiomer surface chemistry: conglomerate versus racemate formation on surfaces[END_REF]]. As polarized light passes through a solution of chiral molecules, the rotation of the plane of polarization may be either clockwise (shifted to the right) or anticlockwise (shifted to the left) depending on which enantiomer, dextrorotatory (d) or levorotatory (l), is dominant. The enantiomeric excess (ee) reflects the degree of amount to which a sample contains one enantiomer over the other. ee = (l -d) / (l + d)

(1)

For a racemic mixture ee = 0, while for an enantiomerically pure substance ee = 1. Most of the organic biomolecules such as amino acids and sugars are chiral and more interestingly, they are enantiomerically pure, occurring in only one of the two possible enantiomeric forms. Such a bias for a single handedness is viewed as among the most noteworthy natural features. Moreover, it is generally accepted that absolute chiral homogeneity was essential for the origin of life and that self-replication and biopolymerization would have been impossible without it [START_REF] Bonner | Origins of Life and Evolution of the Biosphere[END_REF]; [START_REF] Siegel | Homochiral imperative of molecular evolution[END_REF]; [START_REF] Podlech | Origin of organic molecules and biomolecular homochirality[END_REF]; [START_REF] Cintas | Chirality of living systems: a helping hand from crystals and oligopeptides[END_REF]; [START_REF] Weissbuch | Crystalline architectures as templates of relevance to the origins of homochirality[END_REF]].

The occurrence of biomolecules as only one of the two possible enantiomers, left-handed or right-handed, is an appealing example of mirror-symmetry breaking that has long captivated the scientific community. Mirror-symmetry breaking occurs when a non-equilibrium physicochemical process such as crystallization, aggregation or chiral autocatalysis spontaneously generates a large excess of one enantiomer over the other [START_REF] Viedma | Selective chiral symmetry breaking during crystallization: parity violation or cryptochiral environment in control?[END_REF]].

The origin of homochirality of the biomolecules is still a puzzling question. Abiotic theories consider both chance and determinate mechanisms. Among the determinate mechanisms, an energy difference between enantiomers has to be considered. This difference, also called parity-violating energy difference (PVED), is produced by the weak force, one of the four fundamental forces: electromagnetic, weak, strong, and gravity. Although extremely tiny (about 10 -16 kT), this force is universal and acting consistently in the same direction everywhere and continuously [START_REF] Macdermott | Chiroptical signatures of life and fundamental physics[END_REF]]. According to [START_REF] Kondepudi | Weak neutral current and the origin of biomolecular chirality[END_REF] such a difference could be sufficient to drive an amplification model system to homochirality, provided a sufficient waiting time during the amplification process. Moreover, some CHEMISTS LIKE ASYMMETRY TOO….! e must accept that chiral structures are the rule…" egel, Chi ral ity 1998, 10, 24.

to be closely related to the origin of life.

ame highly enantioenriched has been an several mechanisms have focused on organic compounds, including circularly such as quartz, 89 chiral crystallization of us absolute asymmetric synthesis, 93 et c.

imbalance can be introduced via these tion 45,94,95,96,97,98,99,100,101,102 is required to cal organic compounds.

l evolution of biological homochirality. -14 -anic compounds, 90,91,92 spontaneous absolute asymmetric synthesis, 93 et c.

Although the initial enantiomeric imbalance can be introduced via these a suitable asymmetric amplification 45,94,95,96,97,98,99,100,101,102 is required to ingle-handedness as seen in biological organic compounds.

Simplified scenario of the chemical evolution of biological homochirality.

ntioned, asymmetric autocatalysis is a powerful reaction to enhance the slight ic imbalance in molecules based on asymmetric autocatalysis toward ing enantioenrichment during consecutive reactions. Thus, if the initial low symmetric autocatalyst is introduced by an external chiral factor, we can

BIOLOGICAL HOMOCHIRALITY

correlations between the biological handedness and the sign of the PVED are supported by various gas-phase quantum calculations.

On the other hand, as it has been pointed-out by stochastic analysis [START_REF] Lente | Stochastic analysis of the parity-violating energy differences between enantiomers and its implications for the origin of biological chirality[END_REF]; [START_REF] Avalos | Homochirality and chemical evolution: new vistas and reflections on recent models[END_REF]], random choice could be also a solution because as it is claimed, the indiscriminate fluctuations could be sufficient to wipe out any tiny bias. In this last case, we do not need PVED to account for the emergence of biological homochirality, the observed handedness of life is fortuitous.

Here, we report a series of recently published laboratory experiments, which seem to behave in a way similar to the biological homochirality: they give rise to spontaneous mirror-symmetry breaking, they exhibit a chiral sign selection and a particular handedness excess, in other words, they are unexpectedly non-stochastic.

Before going further in the analysis of these chiral bias, let us develop some classical elements related to chirality and chemistry.

II CHIRALITY and CHEMISTRY

Classical asymmetric synthesis

The preparation of enantiopure chiral compounds is highly desirable in the chemical industry, especially in the pharmaceutical sector. Enantiomers often smell and taste different and, moreover, can have dramatic differences in their biological action. The formation of a specific enantiomer is often performed by asymmetric synthesis.

In the early years, such synthesis was difficult to achieve, because left-handed and righthanded enantiomers tended to be produced in equal amounts if the process was not sufficiently directed, leading to a racemic mixture.

Enantioselective synthesis can be achieved by using chiral catalysts that favour the formation of one enantiomer over the opposite. Since the pioneering works of Noyori (2008), [START_REF] Knowles | Asymmetric Hydrogenations (Nobel Lecture)[END_REF], [START_REF] Sharpless | Searching for new reactivity (Nobel Lecture)[END_REF] and [START_REF] Kagan | Nonlinear Effects in Asymmetric Catalysis: A Personal Account[END_REF], catalytic asymmetric synthesis has reached a mature age and high yields and high ee are nowadays more easily reached.

CHIRALITY & CHEMISTRY

A + B P*

A + B P* C* troduction

Asymmetric autocatalysis is a reaction in which a chiral product acts as a chiral yst for its own production. The structure of the chiral catalyst and the product is the . The process is a catalytic automultiplication of a chiral compound (Fig. 1). On the other hand, general experimental expectancy and thermodynamic theory shows that when a chiral compound is formed from achiral precursors in the absence of enantiomerically pure chiral compounds or asymmetric physical fields, a racemic mixture is obtained [START_REF] Barabas | Violation of distribution symmetry in statistical evaluation of absolute enantioselective synthesis[END_REF]]. While this statement applies to catalytic reactions, it does not still hold true when dealing with autocatalytic reactions. This is the case of so-called spontaneous absolute asymmetric synthesis where chiral asymmetry is generated through an autocatalytic process. Hence, spontaneous absolute asymmetric synthesis goes along with mirror-symmetry breaking.

Frank's mechanism

In 1953, Frank suggested a mechanism for spontaneous mirror-symmetry breaking [START_REF] Frank | On spontaneous asymmetric synthesis[END_REF]]. The so-called Frank model is composed of an enantioselective quadratic autocatalysis, coupled to a mutual inhibition. An achiral reactant A turns into chiral products, R and S.

The system is open such that the achiral reactant A is supplied from outside and its concentration is kept fixed. The purpose of the mutual inhibition (racemate elimination) is to increase the ee [START_REF] De Min | Chiral resolution, asymmetric synthesis and amplification of enantiomeric excess[END_REF]]. On the other side, the matter lost during the elimination is replenished by its own reproduction through the autocatalytic steps. These conditions lead to the dominance of the enantiomer originally in slight excess and in the disappearance of its antipode. Let us now develop briefly the mathematics of the Frank model [START_REF] Saito | Colloquium: Homochirality: Symmetry breaking in systems driven far from equilibrium[END_REF]]. The time evolution of the concentrations of the two enantiomers R and S is described by the following differential equations:

d[R]/dt = k a [A][R] -k i [R][S] and d[S]/dt = k a [A][S] -k i [S][R] (2a & 2b)
where k a and k i are the autocatalytic and the inhibition rate constants, respectively.

In an open system, [A] is kept constant since it is assumed that the achiral substrate A is supplied continuously. It can be simply seen that the steady state is racemic:

[R]ss = [S]ss = k a [A]/k i (3) 1 2 3 4 5 6 7 8 9

I ntrod uction

Asymmetric autocatalysis is a reaction in which a chiral product acts as a chiral catalyst for its own production. The structure of the chiral catalyst and the product is the same. The process is a catalytic automultiplication of a chiral compound (Fig. 1).

Fig. 1.

The concept of asymmetric autocatalysis compared with the usual asymmetric catalysis.

From the standpoint of synthesis, asymmetric autocatalysis has the following advantages over the usual asymmetric catalysis:

(1) The process is automultiplication, in other words, self-replication of a chiral compound with high efficiency. Automultiplication is an essential feature of life.

(2) In the usual asymmetric catalysis, the loss and deterioration of catalyst are unavoidable. In sharp contrast, in asymmetric autocatalysis, the amount of catalyst increases because the product becomes the catalyst. Moreover, there is no deterioration of catalytic activity because the new catalyst, i.e., product, is formed continuously.

(3) In the usual asymmetric catalysis, the separation of product from catalyst is necessary. On the other hand, in asymmetric autocatalysis, because the structure of the catalyst and the product is the same, it is not necessary to separate the product from the catalyst.

In this report, we describe how we discovered the asymmetric autocatalysis of

I ntrod uction

Asymmetric autocatalysis is a reaction in which a chiral product acts as a chiral catalyst for its own production. The structure of the chiral catalyst and the product is the same. The process is a catalytic automultiplication of a chiral compound (Fig. 1).

Fig. 1.

The concept of asymmetric autocatalysis compared with the usual asymmetric catalysis.

From the standpoint of synthesis, asymmetric autocatalysis has the following advantages over the usual asymmetric catalysis:

(1) The process is automultiplication, in other words, self-replication of a chiral compound with high efficiency. Automultiplication is an essential feature of life.

(2) In the usual asymmetric catalysis, the loss and deterioration of catalyst are unavoidable. In sharp contrast, in asymmetric autocatalysis, the amount of catalyst increases because the product becomes the catalyst. Moreover, there is no deterioration of catalytic activity because the new catalyst, i.e., product, is formed continuously.

(3) In the usual asymmetric catalysis, the separation of product from catalyst is However, this racemic steady state is unstable. From the rate equations it is easily demonstrated that the difference between the enantiomer concentrations, [R] -[S], diverges exponentially:

R + S RS A + R 2R 2S A + S T i m e 0 2 0 4 0 6 0 8 0 1 0 0 C o n c e n t r a t i o n 0 . 0 0 0 1 0 . 0 0 1 0 . 0 1 0 . 1 1 R S S
d([R] -[S])/dt = k a [A]([R] -[S]) (4) then ([R] -[S]) = ([R] 0 -[S] 0 )exp(k a [A]t) = α(t) (5)
where [R] 0 and [S] 0 represent the initial concentrations of R and S, respectively.

If there is at the beginning a small chiral bias (

[R] 0 -[S] 0 ) ≠ 0), ([R] -[S]
) is amplified until the withdrawal of the minor enantiomer.

On the other side, from the same rate equations, the evolution of the enantiomeric ratio [R]/[S] can also be reached in the following manner (from 2a and 2b):

(d[R]/[R] -d[S]/[S])/dt = k i ([R] -[S]) (6) then d(ln([R]/[S]))/dt = k i ([R] -[S]) (7) or d(ln([R]/[|S]))/dt = k i ([R] 0 -[S] 0 )exp(k a [A]t) (8)
which can be integrated to:

[R]/[S] = ([R] 0 /[S] 0 )exp{(k i /k a )([R] 0 -[S] 0 )[exp(k a [A]t) -1)]} = β(t) (9)
These two analytical functions ([R] -[S]) = α(t) and [R]/[S] = β(t) completely define the mode of evolution of the Frank model, which appears intrinsically unstable.

The enantiomeric excess is:

ee(t) = (β -1)/(β + 1) (10)
and the total chiral matter is:

([R] + [S]) = αee (11) When t  , α  , ee  1.
Various comparable mathematical models, where autocatalytic loops with amplifying (autocatalytic) and inhibitory roles provide positive and negative feedback, were subsequently proposed [START_REF] Decker | The origin of molecular asymmetry through the amplification of "stochastic information" (noise) in bioids, open systems which can exist in several steady states[END_REF]; [START_REF] Bonner | Origins of Life and Evolution of the Biosphere[END_REF]; Iwamoto ( 2002 The theoretical Frank model has stimulated numerous organic chemists aiming at the discovery of the ultimate asymmetric synthesis [START_REF] Stich | Necessary conditions for the emergence of homochirality via autocatalytic self-replication[END_REF]]. Until 1995, no chemical reaction has so far been invented or discovered to generate enantioenrichment according to that model [START_REF] Avalos | Homochirality and chemical evolution: new vistas and reflections on recent models[END_REF]].

Soai's reaction

The autocatalytic Soai reaction [START_REF] Soai | Asymmetric autocatalysis and amplification of enantiomeric excess of a chiral molecule[END_REF]] provided the first experimental proof-ofconcept that amplification of enantiomeric excess could arise from a small initial imbalance via autocatalysis [START_REF] Hein | Chemical and physical models for the emergence of biological homochirality[END_REF]]. The reaction consists of the asymmetric addition of diisopropylzinc to achiral alkynyl pyrimidine aldehydes in which the resulting chiral pyrimidyl zinc alkoxides serve as catalysts for their own formation. The Soai group performed extensive studies on this reaction, revealing many appealing details and evidencing a sound ee amplification power. For instance, amplification can be reproducibly performed using a large variety of chiral sources such as organic or inorganic chiral crystals and even the subtle chiral effect provided by isotopic substitution [START_REF] Soai | Asymmetric autocatalysis of pyrimidyl alkanol and related compounds. Self-replication, amplification of chirality and implication for the origin of biological enantioenriched chirality[END_REF]].

Virtually every described chiral additive becomes successful, thus providing ee amplification in conjunction with an almost quantitative yield. This extreme sensitivity originates from another extraordinary feature: namely, the Soai reaction is able to display spontaneous mirrorsymmetry breaking by yielding enantiomerically enriched pyrimidyl alkanols even in the absence of any external chiral influence and starting with entirely achiral conditions. It then appears that the Soai reaction corresponds to a kinetically controlled absolute asymmetric synthesis that can be described within the framework of a This spontaneous symmetry breaking could be rationalized by the fact that in the beginning of the reaction the two enantiomeric products are submitted to small statistical fluctuations giving rise to a tiny ee [START_REF] Mills | Some aspects of stereochemistry[END_REF]]. Such small ee is then powerfully amplified by the asymmetric autocatalysis. Under these conditions a stochastic distribution of the R-and S-pyrimidyl alcohols, i.e. the product enantiomers, is expected.

To check this hypothesis, a systematic analysis of the Soai results has been carried-out by [START_REF] Barabas | Violation of distribution symmetry in statistical evaluation of absolute enantioselective synthesis[END_REF][START_REF] Barabas | Stochastic and empirical models of the absolute asymmetric synthesis by the Soai-autocatalysis[END_REF]. Careful data analysis has been performed on the data of two independent articles by the Soai group. The first provides 37 experiments "without adding chiral substances" [START_REF] Soai | Asymmetric synthesis of pyrimidyl alkanol without adding chiral substances by the addition of diisopropylzinc to pyrimidine-5-carbaldehyde in conjunction with asymmetric autocatalysis[END_REF]] and the second reports 84 experiments "in the presence of achiral silica gel" [START_REF] Kawasaki | Spontaneous absolute asymmetric synthesis in the presence of achiral silica gel in conjunction with asymmetric autocatalysis[END_REF]], i.e. both series of experiments were performed under achiral initial conditions. Although the first series of experiments shows some interesting tendencies, the authors could not arrive at conclusive statistical results. On the contrary, the second experimental series, affording 39 R-and 45 S-excesses, has been shown to represent two distinct and non-symmetric sets with different non-Gaussian distributions. In this case, a clear S preference was found with ee S = 73.4 ± 23.6 and ee R = 59.3 ± 24.02. [START_REF] Singleton | Enantioselective synthesis without discrete optically active additives[END_REF] and [START_REF] Gridnev | Reflections on spontaneous asymmetric synthesis by amplifying autocatalysis[END_REF] also reported an S-biased non-random distribution of enantiomers even without any chiral inductor added.

The reason for the observed S preference could be either chemical (chiral impurities of unknown origin) or physical (e.g. PVED or other chiral physical factors), without being able in the present time to draw the necessary conclusions from this analysis. Choosing between these two possibilities requires additional experimental and theoretical efforts. However, if the Spreference holds true, it would have a far-going significance [START_REF] Barabas | Stochastic and empirical models of the absolute asymmetric synthesis by the Soai-autocatalysis[END_REF]] that could be identified in other areas, i.e. beyond the scope of pure molecular chemistry.

III CRYSTALLIZATIONS and AGGREGATIONS

A spontaneous resolution can occur during conglomerate crystallization. Roughly 10% of racemic chiral compounds crystallize as conglomerates when an enantiomer of a substance has a much greater affinity for the same enantiomer than for the opposite one. It results a mechanical mixture of enantiomerically pure crystals, for instance (RRR..) crystals and (SSS..) crystals. On the contrary, the 90% remaining substances crystallize mainly as racemates, for instance (RSRSR…) crystals or identically (SRSRS…) crystals.

A careful literature survey on conglomerate crystallization shows that, during spontaneous resolution, a number of researchers have observed anomalies in the behaviour of enantiomers [START_REF] Shinitzky | Comments in a discussion: differential rates of D-and L-tyrosine crystallization[END_REF]]. They do not fit the expectation of stochastic (random or 50/50 coin tossing) distributions. In order to illustrate this idea, let us describe some representative examples.

the CuCl/triallylamine coordination system

When neat tri-allyl amine (taa) and CuCl were allowed to react without stirring at room temperature, monoclinic crystals of helical [Cu 4 Cl 4 (taa) 4 ] were obtained by [START_REF] Vestergren | Non-stochastic homochiral helix crystallization: cryptochirality in control?[END_REF]. Out of five batches, three led to P-1 crystals and two M-1 crystals, indicating a stochastic behaviour. Conversely, when all CuCl was dissolved by heating before crystallization, only P-1 helices were obtained as it was demonstrated from the CD spectra of bulk samples.

The authors conclude that more experiments are obviously needed to confirm the reality of these non-stochastic homochiral crystallizations. On the other side, if they are aware that their reactions take place in a cryptochiral environment, they also recognize that if many compounds can be shown to exhibit the same behaviour, the observed chiral bias might have had something to do with a prebiotic mechanism linked to the origin of homochirality.

the cobalt thiophenic acid bipyridine coordination system

Another remarkable example has been found in the field of organometallic complexes. Namely a non-random distribution of chiral cobalt phosphonate from achiral precursors in the absence of chiral inducers has been described by [START_REF] Gao | Particular handedness excess through symmetrybreaking crystallization of a 3D cobalt phosphonate[END_REF]. The cobalt thiophenic acid bipyridine complex exhibits a 3D chiral metal phosphonate framework built from achiral ligands in the absence of any chiral source. It is argued that both the Cotton effect displayed in the CD spectra and the structural refinements (Flack parameter) of randomly selected 26 crystals demonstrate that the chiral complex is enantioenriched with left-handed helical chains. It is explained that the handedness of the complex originates from the preferential conformation adopted by the thiophenic acid ligand. Therefore, the formation of left-handedness of bulk crystals over right-handedness, could be related to this preferential chiral conformation. It is assumed that such an unusual phenomenon is comparable to the biological homochirality, but it is recognized that it is the consequence of the combined action of many factors and its cause is still unidentified.

the 2-propyl-1H-benzimidazole system

Even more astonishing are the cases where one of the two possible enantiomorphic crystals was absent. The curious case of 2-propyl-1H-benzimidazole (2PrBzIm) in the solid state has been described by [START_REF] Quesada-Moreno | The curious case of 2-propyl-1H-benzimidazole in the solid state: an experimental and theoretical study[END_REF]. While this compound is achiral by fast rotation about the propyl single bond, it exhibits a conformational chirality and crystallizes as a conglomerate. The careful analysis of the solid state VCD spectra was supported by DFT computed structural models delivering wave number assignments. It was found that whatever the preparation method (commercial samples, synthesized samples, recrystallized from various solvents, produced by sublimation ...) all samples gave identical VCD spectra, validating the same handedness in all assessed crystals. Moreover, any attempt to get the opposite chirality, which was successful with some other similar imidazole, was always vain with 2PrBzIm.

The authors believe it is imperative to publish these questioning results even if they seem contrary to the 2 nd law. They confess that they have not been able to identify any hidden cryptochiral environment.

the diprotonated tetrakis-(4-sulfonatophenyl) porphyrin system

It is recognized that aggregation, like crystallization, behaves as an experimental autocatalysis [START_REF] Plasson | Emergence of homochirality in far-fromequilibrium systems: mechanisms and role in prebiotic chemistry[END_REF]]. Being prominent in the field of chiral aggregation, the sign selection on the J-aggregates of diprotonated tetrakis-(4-sulfonatophenyl) porphyrin (H 4 TPPS 4 2-

) should also be mentioned [START_REF] Hachemi | Chiral sign selection on the J-aggregates of diprotonated tetrakis-(4-sulfonatophenyl) porphyrin by traces of unidentified chiral contaminants present in the ultra-pure water used as solvent[END_REF]]. When the J-aggregates of H 4 TPPS 4 2-are formed, they show circular dichroic spectra.

The unexpected result is that after several hundred independent experiments, the number of CD spectra corresponding to negative signals have appeared with a frequency approximately less than 0.4%. It is then assumed that the recurring formation of these chiral J-aggregates is the result of an unidentified biological contamination in the ultra-pure water used as a solvent and that the chiral contaminant exhibits some character of a cationic amphiphile. Therefore, chiral additives were used to confirm this hypothesis. The two additives enantiomers were Rand S-N,N-dimethyl-N-hexadecyl-N-(1-phenyl)ethylammonium bromide. Under the addition of the R-enantiomer, the same CD spectra as that of the blank but with higher dichroic values was observed. On the contrary, the addition of the enantiomer S-gave as expected an opposite CD spectrum. However, this effect only occurred above a critical concentration of the S- enantiomer depending on the water quality. For the best quality, the limit can be estimated around 20nM (10ppb). In conclusion, there is a competitive effect of the S-and an additive effect of R-enantiomer with some chiral contaminant which is present even in the ultra-pure water. The chiral impurity probably consists of fragments of microorganism cell membranes or protein components. On the other hand, without taking into account the presence of such impurities, it has been shown that spontaneous symmetry breaking can be produced in an uncharged achiral porphyrin system by small temperature gradients that act as asymmetrical physical perturbations [START_REF] Mineo | Supramolecular chirality induced by a weak thermal force[END_REF][START_REF] Mineo | New evidence about the spontaneous symmetry breaking: action of an asymmetric weak heat source[END_REF]]. Conglomerate crystallisation or chiral J-aggregates formation is not limited to organic or organometallic complexes, some simple salts such as sodium chlorate undergo also conglomerate crystallisation.

the paradigmatic case of NaClO 3

The easy to handle sodium chlorate (NaClO 3 ) exists in solution as achiral dissociated ions or small clusters without any fixed chirality, but crystallises as two enantiomeric chiral crystals in the cubic space group P2 1 3 [Viedma (2005a)]. The stable chiral cubic unit includes four chlorate ions arranged in a helical architecture. Sodium chlorate is polymorphic as it can also exist transiently as an achiral metastable monoclinic phase [START_REF] Niinomi | Achiral metastable crystals of sodium chlorate forming prior to chiral crystals in solution growth[END_REF]]. One may easily identify the handedness of individual NaClO 3 crystals between crossed polarizers by its rotation of linearly polarized light [START_REF] Mcbride | Spontaneous resolution by stirred crystallization[END_REF]], where the two enantiomeric forms can be distinguished as dark and bright crystals.

Kipping and Pope (1898) measured the handedness of more than three thousands NaClO 3 crystals after quiescent crystallization and found that their D-vs L-crystal counting fell well within the statistical error of an unbiased 50/50 distribution as they have anticipated. Later, [START_REF] Kondepudi | Chiral symmetry breaking in sodium chlorate crystallization[END_REF] confirmed these results.

On the contrary, when evaporating aqueous NaClO 3 solutions were subjected to left-handed electrons from an 90 Sr source, an asymmetric crystal distribution favouring D-crystals was obtained. Inversely, exposure to energetic right-handed positrons from an 22 Na source yielded mainly L-crystals. Without providing a more detailed mechanism, it was concluded that the spin polarized energetic particles are able to produce chiral nucleating sites and that the chirality of the radiation is correlated with the chirality of the created crystals [START_REF] Mahurin | Effect of beta radiation on the crystallization of sodium chlorate from water: a new type of asymmetric synthesis[END_REF]; [START_REF] Pagni | Asymmetric synthesis of optically active sodium chlorate and bromate crystals[END_REF]].

ISSN: xxxxxxx, ©JIMIS, Creative Co Revue en libre accès : j i mi s . e p i s c i e n c e s . o r g

Volume : N -Année : AAAA, DOI : XXXX -000

mirror symmetry breaking in stirred solutions

When the NaClO 3 solution is stirred during crystallization, spontaneous mirror-symmetry breaking occurs [START_REF] Kondepudi | Chiral symmetry breaking in sodium chlorate crystallization[END_REF]]. D-crystals predominated in 18 batches and Lcrystals in 14. Mirror-symmetry breaking has been explained by the rapid production of secondary nuclei from a primary nucleus after its collision with the stirrer or by crystal-crystal collission. This process is autocatalytic as new-born small crystals play the role of seeds to promote the creation of more crystals of the same handedness [START_REF] Mcbride | Spontaneous resolution by stirred crystallization[END_REF]].

Viedma ripening

The so-called Viedma deracemization is an undeniable improvement of the Kondepudi experiment since it has been demonstrated to be applicable to many other systems [START_REF] Blackmond | Chiral amnesia" as a driving force for solid-phase homochirality[END_REF]; Noorduin (2008aNoorduin ( , 2008b[START_REF] Noorduin | Enantioselective symmetry breaking directed by the order of process steps[END_REF]; [START_REF] Viedma | Evolution of solid phase homochirality for a proteinogenic amino acid[END_REF]; [START_REF] Hein | Pasteur's tweezers revisited: on the mechanism of attrition-enhanced deracemization and resolution of chiral conglomerate solids[END_REF]; Spix et al. (2016a)]. Under Viedma's attrition-grinding process, the smaller NaClO 3 crystals dissolve faster than the larger ones because their surface-to-volume ratio is relatively high and thus their surface energy is high as well. The larger crystals grow by consuming smaller crystals via the saturated solution to give an Ostwald-like situation whose limit is the smallest possible surface-to-volume ratio. Repeated crystallization, grinding and dissolution where the chiral NaClO 3 crystals again transform into the achiral NaClO 3 solute, finally triggers an output of a homochiral crystal population [START_REF] Amabilino | Spontaneous deracemization[END_REF]; [START_REF] Crusats | Homochirality as a consequence of thermodynamic equilibrium?[END_REF]]. Because the suggested rationalization is related to the concept of Ostwald ripening [START_REF] Noorduin | From Ostwald ripening to single chirality[END_REF]], this phenomenon has been referred to "Viedma ripening" [START_REF] Viedma | Experimental evidence of chiral symmetry breaking in crystallization from primary nucleation[END_REF][Viedma ( , 2005a))].

In Viedma's NaClO 3 experiments [START_REF] Viedma | Selective chiral symmetry breaking during crystallization: parity violation or cryptochiral environment in control?[END_REF]], the starting L-and D-crystals (racemic mixture) were the whole and "natural" population generated in a quiescent evaporation process. After attrition-grinding for 24h, Viedma found a clear-cut, non-stochastic predominance of D-crystals in repeated experiments. Again, such non-random chiral bias could be of chemical or physical origin. The author suspects the presence of optically active impurities that may either promote the formation of D-nuclei or disfavour the growth of Lcrystals by selective surface adsorption. However, there is no indication about the possible nature of such impurity. Ultrasound-enhanced grinding has been successfully applied to the Viedma deracemization of NaClO 3 crystals. From 37 different crystallizations, 18 produced L-crystals and 19 were almost total D-crystals. A total of 19907 crystals were harvested. While 8473 were left-handed and 11434 right-handed, i.e. 15% more of D-crystals, it was concluded "as that reported previously, in the absence of outside perturbations, statistically equal numbers of L-and Dcrystals were obtained in each crystallization" [START_REF] Song | Ultrasonic field induced chiral symmetry breaking of NaClO 3 crystallization[END_REF]].

Aerosol crystallization

In the same way, the asymmetric chiral growth of micron-size NaClO 3 crystals in water aerosols has been described by [START_REF] Osuna-Esteban | Asymmetric chiral growth of micron-size NaClO 3 crystals in water aerosols[END_REF]. They have shown that similarly to the Viedma ripening or ultrasonic irradiation, their autocatalytic aerosol-liquid cycle is able to amplify any tiny initial imbalance up to total chiral purity in a single step process. Even in the absence of added catalysts, it was found that some undefined ambient contamination leads to preferential D-homochiral crystallizations in a ratio of 4 to 1. This ratio is more or less reversed (up to 1 to 2) by adding only 2 ppb of D-Phe. However, the effect of other additives (L-Phe and L-and D-Tyr) is less conclusive about the involvement of natural L-amino acids as chiral contaminants.

Petri dish experiments

On our side, we have carried out 16 series of Petri dish experiments, i.e. the slow evaporation of saturated quiescent NaClO 3 solutions in open vessels without any chiral additives, in which more than 10,000 crystals were counted. In these Petri dish experiments, primary nucleation occurs mainly at the air-water interface. Moreover, as stirring or mechanical crushing were excluded, the influence of secondary nucleation was negligible and large-sized crystals were generally obtained. After crystal counting, we recognized a persistent chiral bias towards Dcrystals. We still do not know the cause of this chiral bias. However, it is interesting to note that the same D-shifted bias has been also acknowledged before in two different laboratories by two different experimental set-ups [START_REF] Veintemillas-Verdaguer | The effect of stirring on sodium chlorate crystallization under symmetry breaking conditions[END_REF]; [START_REF] Viedma | Selective chiral symmetry breaking during crystallization: parity violation or cryptochiral environment in control?[END_REF]]. Being fully aware that such bias is normally ascribed to the occurrence of heterogeneous nucleation on impurities present in the system and coming from the bio-organic world, we have undertaken a series of experiments by intentionally adding chiral "impurities". However, it was impossible to reallocate the D-bias even after careful addition of a number of chiral additives such as D-mannitol, D-and L-amino acids, D-and L-arabitol, D-and L-quartz, fungal spores, lactobacillus, protein extracts or residues from microfilters that were used to clean the aqueous NaClO 3 solutions.

IV DISCUSSION

All autocatalytic chiral amplifications fit effectively the classical Frank nonlinear kinetic model where the major enantiomer catalyses its own production while it suppresses that of its mirror image [START_REF] Mcbride | Did life grind to a start?[END_REF]; [START_REF] Cintas | On the physical basis of asymmetry and homochirality[END_REF]]. These far-fromequilibrium thermodynamics situations lead to a bifurcation in which chiral fluctuations at the bifurcation point impel the system stochastically to one or the other sign. However, under the influence of some chiral forces, it can be biased deterministically to one sign and not to the other [START_REF] Crusats | Homochirality as a consequence of thermodynamic equilibrium?[END_REF]]. The nature of such chiral forces is still matter of a hearty debate.

Looking at the absolute asymmetric synthesis literature, several papers can be found, where the question of a possible existence of a chiral bias is addressed. Three main conclusions are encountered.

The first one, and the easier to defend interpretation, is to argue, that there will be finally a random distribution, i.e. the system tends to converge to a random distribution after a sufficiently large number of experiments [START_REF] Pincock | Probability distribution of enantiomorphous forms in spontaneous generation of optically active substances[END_REF]; [START_REF] Kondepudi | Chiral symmetry breaking in stirred crystallization of 1,1'-binaphthyl melt[END_REF][START_REF] Kondepudi | Chiral autocatalysis, spontaneous symmetry breaking, and stochastic behaviour[END_REF]]. Also, two recent prominent papers can be cited [START_REF] Kawasaki | Spontaneous formation and amplification of an enantioenriched a-amino nitrile: a chiral precursor for Strecker amino acid synthesis[END_REF]; [START_REF] Kaji | Asymmetric synthesis of an amino acid derivative from achiral aroyl acrylamide by reversible Michael addition and preferential crystallization[END_REF]], they are both related to the spontaneous formation and asymmetric synthesis of enantioenriched -amino acid precursors through preferential dynamic crystallisation of a conglomerate without the addition of any chiral substances. Both articles claim a quasistochastic distribution, i.e. that both enantiomers appeared randomly in each crystallization with almost the same probability.

Other recent papers asserting a "small bias" do not provide a sufficient number of experiments to be validated by robust statistical analysis [START_REF] Ziach | Mirror symmetry breaking upon spontaneous crystallization from a dynamic combinatorial library of macrocyclic imines[END_REF]; Steendam et al. (2015a)].

More profound analysis is demanded since there are many situations where seeding has been used for the purpose of mechanistic studies [START_REF] Steendam | Controlling the effect of chiral impurities on Viedma ripening[END_REF]; [START_REF] Gherase | Experimental and theoretical study of the emergence of single chirality in attrition-enhanced deracemization[END_REF]; Steendam et al. (2015b); Spix et al. (2016b)]. Additionally, the inability to control the handedness of the product motivates synthetic chemists to orientate the output chirality by seeding, therefore genuine absolute asymmetric synthesis is barely performed.

Much more interesting are the systems that exhibit a clear non-stochastic chiral outcome.

Looking carefully at the literature, it appears that the few afore mentioned examples are not the only ones. In a number of chiral crystallization experiments, preferential symmetry breaking was observed. For instance, potassium silicododecatungstate seems to afford always dextro crystals exclusively [START_REF] Hachemi | Chiral sign selection on the J-aggregates of diprotonated tetrakis-(4-sulfonatophenyl) porphyrin by traces of unidentified chiral contaminants present in the ultra-pure water used as solvent[END_REF]]. Other older examples include the spontaneous formation of optically active methyl ethyl allyl anilinium iodide [START_REF] Havinga | Spontaneous formation of optically active substances[END_REF]] and the revisitation of the sodium ammonium tartrate experiment by [START_REF] Thiemann | Is there an energy difference between eantiomorphic structures?[END_REF]. Few years later, [START_REF] Shinitzky | Unexpected differences between Dand L-tyrosine lead to chiral enhancement in racemic mixtures[END_REF] investigated the solubility of D-and L-tyrosine in water. Amazingly, D-tyrosine crystallised faster than L-tyrosine and the supernatant was enriched in L-tyrosine. However, the difference between the crystallization rates of the tyrosine enantiomers disappeared in D 2 O. The authors concluded that these results seem to contravene the generally accepted consensus that there are no physical or chemical differences between enantiomers other than their well-known interaction with polarized light and that it could be the manifestation of the PVED.

It is not astonishing that such an unexpected discovery has been questioned and criticized. [START_REF] Lahav | Parity violating energetic difference and enantiomorphous crystals -caveats; reinvestigation of tyrosine crystallization[END_REF] have shown that commercial L-amino acids are generally contaminated by small amounts of other amino acids and that these impurities greatly shape the growth and dissolution of enantiomorphous crystals. They suggest to keep in mind the role played by impurities in the early stages of crystal nucleation because it is expected that such effects could be of several orders of magnitude larger than the awaited PVED.

After the influential warnings by Lahav, various authors became more cautious in their conclusions, invoking nearly always a "chiral impurity", a "chiral substance" or a "cryptochiral environment" [START_REF] Cheung | Complete asymmetric amplification of ethylenediammonium sulfate using an abrasion/grinding technique[END_REF]; [START_REF] Steendam | Emergence of single-molecular chirality from achiral reactants[END_REF]].

Independently, [START_REF] Kojo | Racemic D,L-asparagine causes enantiomeric excess of other coexisting racemic D,L-amino acids during recrystallization: a hypothesis accounting for the origin of L-amino acids in the biosphere[END_REF] have shown that during the recrystallization of D,L-amino acid mixtures with excess D,L-asparagine, amino acids with the same configuration as asparagine (valine, arginine, aspartic acid, glutamine, histidine, leucine, methionine, phenylalanine, serine, tryptophane and tyrosine) were preferentially resolved. This result indicates that it is the nature of a mixture of racemic amino acids to produce a spontaneous high enantiomeric excess. Moreover, after consideration of the observed unexpected chiral bias, it was claimed that enantioselective crystallization of racemic amino acids induced by spontaneous resolution of D,L-asparagine assisted the selection of L-amino acids.

As already pointed-out by [START_REF] Wald | The origin of optical activity[END_REF] many years ago, crystallization is a puzzling issue because it is very sensitive to seeding and external contamination. Almost in every laboratory exists the feeling that after a first time-consuming and tricky crystallization, the following will proceed much more easily. Any attempt to establish which kind of impurity is accountable for the initial symmetry breaking is challenging. To the best of our knowledge, this question has not yet been well addressed [START_REF] Steendam | Controlling the effect of chiral impurities on Viedma ripening[END_REF]]. Regarding the wide variety of physicochemical situations where a chiral symmetry breaking is occurring (molecules, solvent, involvement of crystals or aggregates) makes the assumption of the involvement of a same impurity, for instance that of biological origin, unlikely. One must remember, for instance, that the S preference in the Soai reaction was also observed independently by [START_REF] Singleton | Enantioselective synthesis without discrete optically active additives[END_REF] and by [START_REF] Gridnev | Reflections on spontaneous asymmetric synthesis by amplifying autocatalysis[END_REF]. The same situation also occurred for the D-preference in the NaClO 3 system observed by different research groups. It is also interesting to note that probably in all of these cases the research was not directed to discover such non-stochastic behaviour, in contrary, the puzzling results emerged as a "side effect" leaving the researcher with open questions to find and explain the possible causes.

One the other side, we must also be aware that, although very weak, the existence of a PVED between enantiomers is theoretically certain, although its experimental confirmation is still pending. According to [START_REF] Tranter | Parity-violating energy difference of chiral minerals and the origin of homochirality[END_REF], the parity-violating energy difference has been calculated for a number of fundamental biomolecules. In each case, the natural enantiomer found in terrestrial biochemistry was shown to be stabilized over its unnatural enantiomer. If we apply the PVED, not to a unique molecule, but to a transition state in the framework of a kinetic approach, its tiny value appears now multiplied by "n", the huge number of elementary units in a macroscopic crystal lattice. Such an assumption, associated with Kondepudi's idea [START_REF] Kondepudi | Weak neutral current and the origin of biomolecular chirality[END_REF]] that even in a randomly fluctuating environment a Frank-like non-equilibrium mechanism can be shifted left or right by an infinitesimal chiral bias such as those occurring from the PVED, would revamp the chiral weak force hypothesis.

V CONCLUSION

In this survey, we have gathered a selection of articles about chiral symmetry breaking, i.e. related to all situations where an achiral input gives rise to a chirally asymmetric output. The common point of all these systems is the presence of an autocatalytic-like aggregation process coupled with a mutual inhibition step or perhaps with a higher-order autocatalytic process in which mutual inhibition becomes no longer essential.

A first key question arises: is the tiny ee emerging from parity violation sufficient or not to overcome stochastic fluctuations? [START_REF] Kondepudi | Weak neutral current and the origin of biomolecular chirality[END_REF] say "yes, but it will be lengthy", [START_REF] Lente | Stochastic analysis of the parity-violating energy differences between enantiomers and its implications for the origin of biological chirality[END_REF] responds "no" on the basis of a stochastic analysis ignoring the amplification power of a Frank-like mechanism, while [START_REF] Cruz | Noise-Induced enantioselection in chiral autocatalysis[END_REF] underline the possible constructive role of noise for the enantioselection in the presence of tiny chiral influences.

The second key question is addressed to the experimentalists: Have you eliminated all obscure sources of dissymmetric influences? The unanimous answer is: it is a daunting if not insurmountable task because undetected chiral impurities may induce favourable nucleation of one of the enantiomers. Since unwanted impurities control the chiral outcome, some experimentalists artificially neutralize their effect by adding a known amount of a "neutralizing" chiral additive. For our part, we encourage the experimentalists to consider that the manifestation of PVED is an assumption that cannot be ruled-out without caution and to unveil their observations.

Beyond these fascinating but still speculative questions remain the importance of chirality.

Chirality is now a top-class subject for academic research because it is related to the problem of origin of life and exobiology, but on a much more practical point of view, chirality is also very crucial for our everyday life. Nowadays, the access to pure enantiomers is a considerable scientific and economical issue. Most enantiomers of racemic drugs exhibit marked differences in biological activities. For instance, the pain reliever S-naproxen binds to the enzyme cyclooxygenase and inhibits its action in the synthesis of prostaglandins, while its enantiomer R-naproxen does not inhibit the cyclooxygenase and is toxic to the liver. Nowadays, many drugs are still used as racemates with their side effects. This problem is probably due not only to the synthetic or separation difficulty, but also to the production costs because asymmetric synthesis or resolution of racemic mixtures are generally expensive and may turn out to be tedious. If new chiral separation and synthesis methods will be developed with low cost, the number of racemic drugs could reduce drastically [START_REF] Nguyen | Chiral drugs: an overview[END_REF]]. Therefore, it is an imperative to promote the topic subject of chiral purity in scholar research as well as in the industry.

For instance, it is worth noting that the Viedma ripening technique [START_REF] Viedma | Selective chiral symmetry breaking: when "Left" and "Right" cannot coexist and "Left" is the best option[END_REF]] could be matured as an efficient separation/deracemization methodology in a pharmaceutical perspective. Moreover, mimicking Soai's chiral autocatalysis, i.e. using the chiral product as part of the chiral catalyst, an enantioselective production of a key precursor to the HIV drug Efavirenz has been described allowing its manufacturing cost to be substantially reduced [START_REF] Chinkov | Asymmetric autocatalysis enables an improved synthesis of Efavirenz®[END_REF]].

Figure 1 .

 1 Figure 1. Molecule with an asymmetric carbon atom. The corresponding structure is nonsuperimposable onto its mirror image (left). Biopolymers are almost exclusively composed of L-amino acids (proteins) and D-sugars (nucleic acids) while their mirror images are missing in the natural world (right).

  s a powerful reaction to enhance the slight d on asymmetric autocatalysis toward secutive reactions. Thus, if the initial low ced by an external chiral factor, we can BIOLOGICAL HOMOCHIRALITY CHEMISTS LIKE ASYMMETRY TOO….!"…we must accept that chiral structures are the rule…" J.S.Siegel, Chiral ity 1998, 10, 24.

Figure 3 .

 3 Figure 3. Principle of the Frank model where the achiral reactant A generates autocatalytically the product enantiomers R and S. Mutual inhibtion leads to the formation of the heterochiral dimer RS.

  Biochi m. Biophys. Acta 1953, 1 1 , 459. ISSN: xxxxxxx, ©JIMIS, Creative Commons Revue en libre accès : j i mi s . e p i s c i e n c e s . o r g Volume : N -Année : AAAA, DOI : XXXX -000

  ); Plasson et al. (2007); Crusats et al. (2009); Ribó et al. (2014); Stich et al. (2016)].

Figure 4 .

 4 Figure 4. The autocatalytic Soai reaction featuring asymmetric amplification and spontaneous mirror-symmetry breaking.

  Frank-like network [Ribo et al. (2014); Noble-Teran et al. (2018)].

  ee ISSN: xxxxxxx, ©JIMIS, Creative Commons Revue en libre accès : j i mi s . e p i s c i e n c e s . o r g Volume : N -Année : AAAA, DOI : XXXX -000

Figure 5 .

 5 Figure 5. Solid-state CD spectra of 10 randomly selected crystals as well as the bulk sample (labeled in bold red). Note that there is only one (+) CD spectrum. [Published in: Chao-Ying Gao; Fei Wang; Hong-Rui Tian; Lei-Jiao Li; Jian Zhang; Zhong-Ming Sun; Inorg. Chem. 2016, 55, 537-539. DOI: 10.1021/acs. inorgchem.5b02187. Copyright © 2015 American Chemical Society].

Figure 6 .

 6 Figure 6. Recorded IR (upper) and VCD (lower) spectra for liquid and solid samples of 2PrBzIm. Note the persistent negative VCD signals. [Published in: María Mar Quesada-Moreno; Aurora J. Cruz-Cabeza; Juan Ramón Avilés-Moreno; Pilar Cabildo; Rosa M. Claramunt; Ibon Alkorta; José Elguero; Francisco J. Zúñiga; Juan Jesús López-González; J. Phys. Chem. A 2017, 121, 5665-5674. DOI: 10.1021/ acs.jpca.7b05220. Copyright © 2017 American Chemical Society].

  ISSN: xxxxxxx, ©JIMIS, Creative Co Revue en libre accès : j i mi s . e p i s c i e n c e s . o r g Volume : N -Année : AAAA, DOI : XXXX -000

Figure 7 .

 7 Figure 7. Characteristic CD spectra resulting from the competitive induction of S-1and R-1 in the CD spectra of H 4 TPPS 4 2-. [Published in: El Hachemi, Z., Escudero, C., Arteaga, O., Canillas, A., Crusats, J., Mancini, G., Purello, R., Sorrenti, A., D'Urso, A., Ribo, J.M. (2009). Chiral sign selection on the J-aggregates of diprotonated tetrakis-(4-sulfonatophenyl) porphyrin by traces of unidentified chiral contaminants present in the ultra-pure water used as solvent. Chirality 21 408-412.]

Figure 8 .

 8 Figure 8. The principle of the Viedma ripening in the case of achiral NaClO 3 molecules that crystallise as a conglomerate. [Sögütoglu, L.C., Steendam, R.R.E., Meekes, H., Vlieg, E., Rutjes, F.P.J.T.; Viedma ripening: a realible crystallisation method to reach single chirality. Chem. Soc. Rev. 2015, 44, 6723] -Published by The Royal Society of Chemistry.

Figure 9 .

 9 Figure 9. Counting of NaClO 3 crystals after 16 series of Petri dish experiments without chiral additives (more than 10,000 D-and L-crystals in total) showing a clear bias towards Dcrystals.

  , ©JIMIS, Creative Co Revue en libre accès : j i mi s . e p i s c i e n c e s . o r g Volume : N -Année : AAAA, DOI : XXXX -000

  [START_REF] Szabo-Nagy | Demonstration of the parity-violating energy difference between enantiomers[END_REF] claimed the observation of the parity-violating energy difference between enantiomers, namely the spontaneous formation of a non-random distribution of tris(1,2-ethanediamine) cobalt(III) and tris(1,2-ethanediamine) iridium(III) crystals. Instead of organic compounds, these metal complexes were selected because the PVED increases in proportion to the sixth power of the atomic number Z of the element in the asymmetry center: C (Z = 6), Co (Z = 27) and Ir (Z = 77). ISSN: xxxxxxx, ©JIMIS, Creative Co Revue en libre accès : j i mi s . e p i s c i e n c e s . o r g Volume : N -Année : AAAA, DOI : XXXX -000

  ISSN: xxxxxxx, ©JIMIS, Creative Co Revue en libre accès : j i mi s . e p i s c i e n c e s . o r g Volume : N -Année : AAAA, DOI : XXXX -000

Acknowledgements

We would like to dedicate this work to Professor Wolfram H.-P. Thiemann on his 80th birthday. This research was conducted and financially supported in the framework of the French-Mexican International Laboratory (LIA) LCMMC. We also acknowledge financial support by the grant CONACYT-ANR 188689.