Michel Petitjean
email: petitjean.chiral@gmail.com

About Chirality in Minkowski Spacetime

Keywords: chirality definition, symmetry, Mikowski spacetime, isometries PACS: 03.30.+p, 02.20.-a MSC: 14J33, 58D19

In this paper, we show that Lorentz boosts are direct isometries according to the recent mathematical definitions of direct and indirect isometries and of chirality, working for any metric space. Here, these definitions are extended to the Minkowski spacetime. We also show that the composition of parity inversion and time reversal is an indirect isometry, which is the opposite of what could be expected in Euclidean spaces. It is expected that the extended mathematical definition of chirality presented here can contribute to the unification of several definitions of chirality in space and in spacetime, and that it helps clarify the ubiquitous concept of chirality.

Introduction

The issue of chirality is of great importance in many fields of science, including real-life applications and the arts [START_REF] Ecker | Chiral Perturbation Theory[END_REF][START_REF] Mezey | New Developments in Molecular Chirality[END_REF][START_REF]Chirality in Drug Research[END_REF][START_REF] Brown | Chirality in Drug Design and Synthesis[END_REF][START_REF] Wagnière | On Chirality and the Universal Asymmetry[END_REF][START_REF] Cintas | Biochirality, Origins, Evolution and Molecular Recognition[END_REF][START_REF]Progress in Biological Chirality[END_REF][START_REF] Bouissou | Asymmetric Exchanges[END_REF]. The term, "chirality" was introduced in 1894 by Lord Kelvin [START_REF] Lord | The Molecular Tactics of a Crystal[END_REF]: "I call any geometrical figure, or group of points, chiral, and say that it has chirality if its image in a plane mirror, ideally realized, cannot be brought to coincide with itself". Although it is still useful and it is the main one used in chemistry [START_REF] Prelog | Chirality in chemistry[END_REF], this definition is physical (if not intuitive), rather than mathematical, and it implicitly assumes the existence of an Euclidean space. There were attempts to enhance it [START_REF] Whyte | [END_REF] and extend it to the classical spacetime [START_REF] Barron | True and false chirality and parity violation[END_REF][START_REF] Barron | True and false chirality and absolute enantioselection[END_REF]. Alas, none of these definitions are mathematical, and the one of Barron was severely criticized [START_REF] Mislow | Molecular chirality[END_REF]. It was noticed that, in chemistry, the use of the term "chirality" often engenders confusion and ambiguity [START_REF] Cintas | Chirality and chemical processes: A few afterthoughts[END_REF]. Even worse, in the proceedings of a four-week-long school organized in 2000 by the Clay Mathematics Institute, two distinct approaches were found to understand mirror symmetry, without a clear connection between physical and mathematical methods of proof [START_REF] Hori | Mirror Symmetry[END_REF], and even the notion of what one means by "proof" of mirror symmetry differs between the two fields.

A definition of chirality should rely first on a definition of symmetry. However, there are several mathematical definitions of symmetry in the literature, depending on which field it applies, such as geometry, the probability theory, functional analysis, and graph theory. Then, a unifying one was published in 2007 [START_REF] Petitjean | A definition of symmetry[END_REF], giving rise to a mathematical definition of chirality, which recovers the one of Lord Kelvin and extends it for any metric space [START_REF] Petitjean | Chirality in metric spaces[END_REF] (a preliminary version was published in [START_REF] Petitjean | Chirality in metric spaces[END_REF]).

We further show that these two definitions (symmetry and chirality) work in the Minkowski spacetime, despite the fact that the Minkowski metric used in special relativity is not a true metric, because it is not positive definite.

Results

For clarity, we summarize the symmetry definition given in [START_REF] Petitjean | A definition of symmetry[END_REF] as follows. An object is a function having its input argument in a metric space, E equipped with a distance function δ, and an isometry is an element of the group of all bijections of E onto E preserving δ (at least, the identity is so). An object is symmetric when it is invariant upon an isometry which is not the identity.

Proposition 1. The symmetry definition in [START_REF] Petitjean | A definition of symmetry[END_REF], summarized above, is extended to the Minkowski space.

The chirality definition given in [START_REF] Petitjean | Chirality in metric spaces[END_REF] (and based on [START_REF] Petitjean | A definition of symmetry[END_REF]), is summarized as follows. An isometry is direct when it can be written as the composition of squared isometries (at least the identity is so); if not, it is an indirect isometry. An object is chiral when it has no symmetry due to an indirect isometry.

Proposition 2. The chirality definition in [START_REF] Petitjean | Chirality in metric spaces[END_REF], summarized above, is extended to the Minkowski space.

Proposition 1 is valid because none of the properties of δ were used in [START_REF] Petitjean | A definition of symmetry[END_REF] and because the isometries of Minkowski spacetime are known to be a group acting on this spacetime: it is the Poincaré group. These isometries preserve inter-event intervals. Proposition 2 is valid because Proposition 1 is valid, and under the condition that all its isometries of the Minkowski space can be classified as being direct or indirect. We perform here this classification.

The Poincaré group is the semi-direct product of the Lorentz group and the translation group in space and time [START_REF] Moshe | Representations of the Lorentz Group and Their Applications to the Gravitational Field[END_REF][START_REF] Müller-Kirsten | Introduction to Supersymmetry[END_REF]. The translation group is an abelian Lie group, and it is a normal subgroup of the Poincaré group. The Poincaré group has four components [START_REF] Müller-Kirsten | Introduction to Supersymmetry[END_REF], and the Lorentz group also has four components [START_REF] Moshe | Representations of the Lorentz Group and Their Applications to the Gravitational Field[END_REF]. Lorentz transformations are linear isometries, leaving the origin fixed. The Lorentz group contains the spatial orthogonal transformations (i.e., the rotations and their compositions with the spatial parity inversion), time reversal, and the boosts, which connects two uniformly moving bodies. Translations and rotations are direct isometries. Time reversal and compositions of spatial rotations with the parity inversion are indirect isometries. Hereafter, we show that boosts are direct isometries and that the composition of spatial indirect isometries with time reversal is an indirect isometry. Then, we complete the classification of all Minkowski isometries as being either direct or indirect-an object in the Minkowski space is chiral if no indirect isometry leaves it invariant.

Methods

Conventionally, we retain the signature (+,+,+,-) for the Minkowski metric. Vectors in R 3 are column vectors. A point in the spacetime is a 4-vector. Its spatial component x is a vector of R 3 , and its fourth component is the time, denoted ct, where c is the light speed.A single quote at the right of a vector or a matrix indicates a transposition. The identity matrix operating on spatial vectors is I, and the identity matrix acting on 4-vectors is I 4 . We denote by v the module of the velocity of a frame moving in the direction of the unit vector u (so, u u = 1). We set β = v/c and γ = (1

-β 2 ) -1/2 . Because v ∈ [0; c[, we have β ∈ [0; 1[ and γ > 1.
The matrix B, associated to a Lorentz boost in the direction u, can be written as below (this is immediately deduced from Equation (5) in [START_REF] Ungar | Thomas rotation and the parametrization of the Lorentz transformation group[END_REF], or from Equation (1) in [START_REF] Kennedy | Thomas rotation: a Lorentz matrix approach[END_REF]):

B = I + (γ -1)uu -γβu -γβu γ (1)
Let w be any fixed unit vector of R 3 orthogonal to u, and u × w the cross product of u and w. We define the rotation matrix R u , which depends only on u:

R u = u/ √ 2 w u × w u/ √ 2 -1/ √ 2 0 0 1/ √ 2 (2)
We establish first the following Lemma:

Lemma 1. R u contains the eigenvectors of B, with the respective eigenvalues Proof. Remember that γ = (1β 2 ) -1/2 and check that BR u = R u D.

Now comes the following theorem:

Theorem 1. Any Lorentz boost with parameter β in a given direction u is the square of two boosts in the direction u, each one of parameter λ = (1

-1 -β 2 )/β. It is so that λ ∈ [0; β[.

Proof.

From Lemma 1, B = R u DR u , where the rotation R u depends only on u, not on β. Thus, any boost in the direction u with relative velocity λ can be factorized as R u ∆R u , where the respective diagonal elements of ∆ are written 1+λ 1-λ , 1, 1, and 1-λ 1+λ . We look for a boost B λ , such that

B = B 2 λ . So, B = R u ∆R u R u ∆R u , that is, B = R u ∆ 2 R u . Equating the two expressions of B, we get ∆ 2 = D. The unique solution is such that ( 1+λ 1-λ ) 2 = 1+β 1-β , from which λ 2 -2 β + 1 = 0. Only one solution λ does not exceed 1: it is λ = (1 -1 -β 2 )/β. It can be easily checked that λ ∈ [0; β[.

Corollary 1.

A boost is a direct symmetry of the Lorentz group.

Proof. Apply Theorem 1: this matches the definition of direct isometries given in [START_REF] Petitjean | Chirality in metric spaces[END_REF]. Now we look at the composition of spatial indirect isometries with time reversal, and we prove the following Theorem: Theorem 2. The composition of any spatial indirect isometry with time reversal is an indirect isometry of the Lorentz group.

Proof. We consider an indirect spatial isometry

P Q = Q 0 0 1
, where the orthonormal matrix Q is such that Q Q = I and det(Q) = -1. The time reversal is T = I 0 0 -1 . The operators P Q and T commute, and

P Q • T = Q 0 0 -1 , with det(P Q • T) = +1.
Let us focus on its lower right diagonal element, -1, and assume that P Q • T is a direct isometry of the Lorentz group. If so, it is a product of squared elements of the group.We require that this product has its lower right diagonal element equal to -1. If T is not part of the product, this latter one would contain only boosts and spatial isometries.

Without the presence of T, all group elements in the product have their lower right diagonal element greater or equal to +1, and the product cannot get a negative sign at this place (even the product of two boosts cannot do that, because it is the product of one boost by a spatial rotation [START_REF] Ungar | Thomas rotation and the parametrization of the Lorentz transformation group[END_REF]). An odd number of T is needed in the product to generate this negative sign. Notice that the number of indirect spatial isometries is even in the product, because each square necessarily contains an even number of them, and we would get a negative determinant for the product, a contradiction. Thus, P Q • T is an indirect isometry of the Lorentz group.

The commutative composition of parity reversal with time reversal is

P -I • T = -I 0 0 -1 .
For clarity, P -I is denoted P, and P -I • T is denoted PT.

Corollary 2. Each of the operators T, P, and PT is a mirror in the Minkowski space. The subgroup of the Lorentz group generated by the operators I 4 , T, P, and PT is isomorphic to the Klein four-group.

Proof. Apply Theorem 2. The fact that T, P, and PT are mirrors is an immediate consequence of Definition 3 in [START_REF] Petitjean | Chirality in metric spaces[END_REF]. The composition of these mirrors are: P • T = T • P = PT, PT • P = P • PT = T, and PT • T = T • PT = P: this matches the definition of the abelian Klein four-group [START_REF] Whitelaw | Introduction to Abstract Algebra[END_REF].

Thus, the improper antichronous Lorentz transformation PT is a mirror in Minkowski spacetime. This result may be thought of as being counterintuitive because det(PT) = +1, but we recall that we work in R 4 equipped with the Minkowski metric, not with the Euclidean metric.

Discussion and Conclusions

We remember that: (i) an isometry is a direct one if it can be written as the composition of squared isometries (thus the composition of direct isometries is a direct isometry), and (ii) any composition of isometries involving one indirect isometry is an indirect isometry. So far, in the Lorentz group, we have identified:

•

Direct isometries: spatial rotations and boosts.

•

Mirrors (involutions): parity inversion P, time reversal T, and their commutative composition, PT.

The compositions of the symmetry operators above generate the full Lorentz group. The spatial translations and the time translations being direct isometries of the Poincaré group, we are now able to classify any isometry of the Poincaré group as being direct or indirect.Thus, we have defined chirality in the Minkowski spacetime-an object is chiral if no indirect symmetry leaves it invariant. If it is not chiral, it is achiral.

We outline that our point is not to redefine symmetries: they are well-known, and the definition we used corresponds to what is usually encountered in physics. The situation is different for chirality. The word chirality, (together with chiral and achiral), has different definitions, depending on the context in which it was used. When time is discarded, the definitions are mainly based on Lord Kelvin's one, and applied both to rigid objects and to flexible objects, such as molecules conformers (it is a complex topic, far outside the scope of this paper). In spacetime, Lord Kelvin's definition no longer works. In particle physics, chirality is related to helicity, that is, it depends on the sign of the projection of the spin vector onto the momentum vector, but there are other uses, such as chiral superfields [START_REF] Hübsch | Linear and chiral superfields are usefully inequivalent[END_REF], chiral algebra, and chiral homology. We do not aim to unify all these terminologies. However, to get some compatibility between concepts of chirality in space and in spacetime, in our opinion, the very first step was to extend our definition of chirality to the spacetime of Minkowski, as we did here. This is mainly because there is a consensus about the validity of special relativity. Extensions to these other fields are outside the scope of this paper, and are to be considered further. Another extension is quantitative chirality, that is, measuring how less or more chiral objects are by themselves, not by their interactions [START_REF] Petitjean | Chirality and symmetry measures: A transdisciplinary review[END_REF]. They were considered only in the Euclidean case [START_REF] Petitjean | Chiral mixtures[END_REF] (see also [START_REF] Petitjean | On the root mean square quantitative chirality and quantitative symmetry measures[END_REF] and Section 2.9 in [START_REF] Petitjean | Chirality and symmetry measures: A transdisciplinary review[END_REF]), and the underlying mathematical model was extended to a theory of docking [START_REF] Petitjean | From shape similarity to shape somplementarity: Toward a docking theory[END_REF]. However, extending this quantitative chirality theory to objects in spacetime is difficult.

Funding: This research received no external funding.

  = R u DR u , D being the diagonal matrix which contains the respective eigenvalues of B; det(B) = 1.

Symmetry 2019, 11, 1320; doi:10.3390/sym11101320 www.mdpi.com/journal/symmetry

Acknowledgments: I am very grateful to Tidjani Negadi for useful discussions and for having provided me several helpful documents.

Conflicts of Interest:

The authors declare no conflict of interest.


Claire Fleurisson

Erica Benedetti
email: erica.benedetti@parisdescartes.fr

Laurent Micouin
email: laurent.micouin@parisdescartes.fr

diamines derived from bicyclic hydrazines: Synthesis and applications

Keywords: cyclic cis-1,3-diamines, bicyclic hydrazines, fragments RNA ligands and probes, N-N bond cleavage, hydroxylation, fluorination, reductive amination 1

Cyclic cis-1,3-diamines are versatile building blocks frequently found in natural molecules or biologically active compounds. In comparison with widely studied 1,2-diamines, and despite their chemical similarity, 1,3diamines have been investigated less intensively probably because of a lack of general synthetic procedures giving access to these compounds with good levels of chemo-, regio-and stereocontrol. In this account we will give a general overview of the biological interest of cyclic cis-1,3-diamines. We will then describe the synthesis and potential applications of these compounds with a particular focus on the work realized in our laboratory. 1 Introduction 2

Biological relevance of the cis-1,3-diamine motif 3

Classical synthetic strategies towards cis-1,3-diamines 4 N-N bond cleavage of bicyclic hydrazines: a versatile method to access cis-1,3-diamines 4.1

Preparation of five-membered cyclic cis-1,3-diaminoalcohols 4.2

Access to fluorinated 1,3-cis-diaminocyclopentanes 4.3 Synthesis of cis-1,3-diaminocyclohexitols 4.4

Formation of cyclic cis-3,5-diamino-piperidines 5

Applications of cyclic cis-1,3-diamines 5.1 Small molecular RNA binders 5.2 Fluorinated 1,3 diamino cyclopentanes as NMR probes 6

Concluding remarks

Introduction

Although cyclic 1,2-diamine moieties are recognized as important motives commonly used as building blocks in synthetic chemistry or as ligands in catalysis, 1 their cyclic 1,3diamine analogues have received much less attention. [START_REF]diamines[END_REF] As a matter of fact, there is still a need for diversity-oriented general procedures providing practical access to cyclic 1,3diamines with good regio-and stereo-control. This lack of general methods, compared to the variety of strategies developed to access the cyclic 1,2-diamine motif, limits considerably the applications of these compounds in different research fields. Even today, 1,3-diamine containing substrates can therefore be considered important targets in organic chemistry.

Since the early 2000s, our group has been interested in the design of efficient stereoselective routes to 1,3-diamines, with a particular attention to five-and six-membered cyclic compounds showing cis-substitution patterns (Figure 1). Our original goal was to explore the chemical space of these motives, regardless of their potential applications. The challenge was mostly synthetic : how to install a 1,3 diamine motif onto a functionalized five or six-membered ring, in a simple and versatile manner, with a control of the relative, and/or absolute configuration of the stereogenic centers of these densely functionalized scaffolds. Application of these synthetic methods for the design of compounds with interesting properties (mostly biological ones) has also been the focus of our studies during the last ten years. In this account article we will firstly present the biological relevance of cyclic cis-1,3-diamines. We will also briefly describe the few synthetic approaches that can be followed to access these systems. We will then report our efforts towards the development of new general methods for preparing such derivatives on synthetically useful scales via a key strategy involving the N-N bond cleavage of bicyclic hydrazines. We will finally describe the use of the cyclic cis-1,3-diamine compounds synthesized in our laboratory as RNA binders and probes.

Biological relevance of cyclic cis-1,3-diamines

Various substrates incorporating five-membered cyclic cis-1,3 diamine moieties have been reported to possess interesting biological activities. 3 Some selected examples are described in Figure 2, and include compounds showing antifungal, 4 analgesic or anti-inflammatory activities 5 as well as potential therapeutic agents against different human diseases. Among them figure a melanin-concentrating hormone (MCH) receptor antagonist that may be used for the treatment of obesity and other metabolic syndromes, 6 mammalian Janus Kinase 1 (JAK1) inhibitors which could be employed to cure patients affected by myelofibrosis, 7 and a neuronal nitric oxide synthase (NOSs) inhibitor that may help to prevent neurodegenerative conditions such as rheumatoid arthritis, Alzheimer's disease, and Parkinson's disease. 8 Pyrimidine carbocyclic nucleosides have finally been reported to possess interesting tumor cell growth inhibitory activities (Figure 2). 9

Figure 2 Biologically active compounds incorporating five-membered cyclic cis-1,3-diamine moieties.

Six-membered cyclic cis-1,3-diamine-containing molecules may also present intriguing biological properties, and many of these compounds are, for this reason, described in patent applications. 10 Few examples of such substrates are depicted in Figure 3, and include a dipeptidyl peptidase-4 (DPP-4) inhibitor useful for the treatment of type 2 diabetes, 11 a protein kinase modulator, 12 as well as a spleen tyrosine kinase (Syk) inhibitor that may serve to treat inflammatory and autoimmune disorders. 13 In addition, sugar derivatives showing the six-membered cyclic cis-1,3-diamine motif have been used to develop stimuli-responsive liposomes for drug delivery (Figure 3). 14 Aminoglycosides irreversibly bind to the 16S ribosomal RNA of prokaryotes and prevent bacterial protein synthesis. Extensive studies including NMR investigations 15 and X-ray crystallographic studies 16 have highlighted the key interactions provided by the cis 1,3 diamine motif present in all these natural products. Several analogues of the aminoglycoside antibiotics, and in particular of their central core (2-DOS, Figure 4), have been reported over the years. As their parent compounds, these derivatives can show interesting antibacterial properties (Figure 4). 17

Classical synthetic strategies towards cis-1,3diamines

Over the years, cyclic cis-1,3-diamines have been prepared following various approaches. In many cases, different strategies have been used to introduce the amine functionalities onto the desired compounds, one after the other, using linear pathways that require two or more consecutive synthetic steps. The most commonly used methods to achieve these syntheses include epoxide ring-opening reactions, [START_REF] Clark | For the preparation of five-membered rings[END_REF] nucleophilic substitutions, 3b,19 palladium-catalyzed allylic aminations, 20 oxime reductions, 19c,21 Curtius rearrangements, 22 intramolecular Henry reactions, 23 ring opening of cyclic sulfates, 24 and the reduction of aromatic nitro compounds 25 (Figure 5). Interestingly, only few of these methods enable to simultaneously install the two amino groups in a single synthetic step.

Scheme 1 Examples of epoxide ring-opening reactions

Epoxide ring-opening reactions in the presence of hydrazine, 26 benzylamine 27 or sodium azide 28 can for example be used to form six membered cyclic cis-1,3-diamines after a simple reduction of the reaction products (Scheme 1). Nucleophilic substitutions have been successfully employed to introduce two nitrogen-containing functional groups at once, both on five-29 and six-membered cyclic compounds (Scheme 2). 30

Scheme 2 Examples of nucleophilic substitutions

Examples of oxime reductions 31 and Curtius rearrangements 32 have also been reported as ways to functionalize both five-and six-membered rings with two amino groups at the same time (Scheme 3). In some cases, however, the former method does not allow to control the selectivity of the reactions and mixtures of stereoisomers (cis-an trans-1,3 diamines) are obtained. 31 Scheme 3 Selected examples of oxime reductions and Curtius Rearrangements.

Six-membered cyclic cis-1,3-diamines can finally be prepared starting from di-nitrobenzenes simply by converting the two NO2 groups into the corresponding amine functions and concomitantly dearomatizing the aromatic group to form a fully saturated ring (Scheme 4). 25 Quite harsh conditions are however required to perform the two reductions in a single synthetic step. Other nitrogen-containing aromatic compounds (e.g. imines or 1,3-benzenediamines) can be submitted to a reductive dearomatization as well. 25a,33

Scheme 4 Example of nitroarene reduction

All of these transformations generally proceed with low levels of stereocontrol. Mixtures of cis-and trans-1,3 diamines are therefore frequently obtained in moderate to good yields as main reaction products.

Taking the above mentioned examples into account, it appears clear that cyclic cis-1,3-diamines, and in particular fivemembered rings, can hardly be synthesized in a straightforward manner with high levels of relative or absolute stereocontrol. The development of new rapid and versatile processes leading to the stereoselective formation of this class of compounds therefore still remains an important challenge for synthetic organic chemists.

4.

N-N bond cleavage of bicyclic hydrazines: a versatile method to access cis-1,3-diamines A less common but intriguing approach to prepare cyclic cis-1,3-diamines is based on the reductive cleavage of the N-N bond in bicyclic systems containing an hydrazine subunit. These key compounds can easily be synthesized through a Diels-Alder cycloaddition starting from cyclopentadiene and using differently substituted diazo compounds as dienophile counterparts (Scheme 5).

Scheme 5 Access to cyclic cis-1,3-diamines using Diels-Alder reactions and N-N bond cleavages as key synthetic steps.

First reported in the eighties, 34 this method allows the formation of the two C-N bonds in a single synthetic step with a total control of their relative (syn) configuration. By appropriate functionalization of key Diels-Alder cycloadducts, over the past few years, we have been able to adapt this procedure to the preparation of both five-and six-membered cyclic cis-1,3-diamines. In the next sections we will give an overview of the different scaffolds that were synthesized in our group using this convenient strategy.

Preparation of five-membered cyclic cis-1,3aminoalcohols

Various five-membered cyclic cis-1,3-diaminoalcohols have been successfully synthesized in few steps starting from readily available compounds and using the nitrogen-nitrogen bond cleavage of hydrazines described above as a key reaction. A first series of compounds was prepared through the epoxidation of the bicyclic adduct 21a, easily obtained via a Diels-Alder cycloaddition between cyclopentadiene and dibenzyl azodicarboxylate (Scheme 6). 35 The acid-catalyzed rearrangement of epoxide 22a, and subsequent nucleophilic attack, led to the formation of substituted bicyclic hydrazines 23a-d in a regio-and stereoselective manner (Scheme 6). 36 Different reaction conditions were used to perform this transformation. Treatment of the epoxide 22a with sulfuric acid in a non-nucleophilic solvent (trifluoroethanol) allowed the functionalization of the bicyclic system with an additional hydroxyl group. A methoxy substituent was added onto the molecule under similar conditions when methanol was employed as the solvent. Lewis acid activated reactions in the presence of organoaluminum reagents led to the formation of new carbon-halogen or carbon-carbon bonds. All the observed skeletal rearrangements allowed the control of the relative configuration of four contiguous stereocenters in a single operation, and were supposed to proceed via the formation of an aziridinium intermediate. 36 As a last synthetic step, hydrogenolysis over platinum oxide under acidic conditions of the hydrazine bond afforded the desired aminocyclopentanes 24a,b in almost quantitative yield (Scheme 6).

Scheme 6 Synthesis of cyclic cis-1,3-diaminocyclitols through epoxidation, skeletal rearrangement and reductive cleavage of bicyclic hydrazines.

The hydroboration of different Diels-Alder adducts (21a-d Scheme 7) was also investigated with the aim to prepare a second series of cyclic cis-1,3-diaminoalcohols. Treatment of the bicyclic compounds with BH3•THF followed by oxidative workup afforded the corresponding hydroxylated racemic products 25a-d in moderate to good yields (Scheme 7). A reductive cleavage of the hydrazine bond was then used as the key step to access hydroxylated N-protected cyclic cis-1,3diamines (Scheme 7). 37 Scheme 7 Synthesis of cyclic cis-1,3-diamines through hydroboration and reductive cleavage of bicyclic hydrazines.

Asymmetric hydroborations were studied as well. In this case, a reversal of the enantioselectivity of the reaction was observed in the presence of chiral diphosphine ligands and Rhor Ir-based catalysts. 37 Treatment of meso hydrazine 21a with catecholborane (CatBH), [Rh(COD)Cl]2, and (S,S)-BDPP in DME afforded the corresponding hydroxylated product 25a in 90% yield and 84% ee. The absolute configuration of this compound (1R,4R,5R) was confirmed by X-ray analysis after reductive cleavage of the N-N bond and subsequent protection of the final diamino alcohol as p-bromo dibenzamide (Scheme 8). 37 Interestingly, when [Ir(COD)Cl]2 was employed as catalyst to promote the hydroboration, the hydroxylated derivative of opposite configuration (1S,4S,5S) was isolated as the major product (Scheme 8). Differently substituted meso hydrazines behave similarly under the same experimental conditions. 37 In an effort to rationalize the reaction mechanism, two different insertion pathways were invoked to explain the systematic reversal of enantioselectivity observed with the two transition metal complexes. While the use of [Rh(COD)Cl]2 was supposed to involve a key Rh-H migratory insertion step, we suggested that a metal-boron insertion could be favored when using the [Ir(COD)Cl]2 catalyst. 37 Indeed, alcohols of opposite absolute configuration were obtained while promoting the reaction with [Rh(COD)Cl]2, simply by replacing the (S,S)-BDPP ligand with (S,S)-DIOP (Scheme 9). 37 However, a racemic product was isolated in low yield in the presence of BINAP (Scheme 9). 37a P,N-ligands could be used in place of diphosphines with the [Ir(COD)Cl]2 catalyst, and, in this case, the desired hydroxylated product 25a was isolated in a better yield and in a higher enantiomeric excess (Scheme 10). 38 Scheme 10 Asymmetric hydroborations compound 21a with [Ir(COD)Cl]2 and P,N-ligands As confirmed by NMR analysis, the hydroboration of bicyclic hydrazines exclusively led to the formation of exo alcohols. 37 The inaccessibility of the endo face, probably hindered by a pseudo axial carbamate, was confirmed by many unsuccessful trials of alcohol inversion by SN2 type mechanisms. To obtain an all-cis diamino alcohol, the enantioenriched hydroxylated hydrazine 25a was converted to the corresponding ketone by Swern oxidation (Scheme 11). Subsequent reduction of the carbonyl group, followed by N-N bond cleavage and protection of the free amino groups as N-tert-butyl-carbamates led to the formation of the enantioenriched compound 29 in a good overall yield (Scheme 11). 39 Scheme 11 Synthesis of an all-cis diamino alcohol

In order to prepare dihydroxylated products, bicyclic hydrazine 21a was finally engaged in a reaction with osmium tetroxide to afford hydrazine 30 in 90 % yield (Scheme 12). The reductive cleavage of the N-N bond of this compound successfully leaded to the formation of diaminodiol 31, which could be isolated in a quantitative yield (Scheme 12).

Scheme 12 Synthesis of diaminodiol 31

Access to fluorinated 1,3-cisdiaminocyclopentanes

The introduction of a fluorine atom onto organic compounds can significantly influence their physicochemical properties and is nowadays considered as a classical strategy to tune their biological activities. 40 In this context, we naturally became interested in the possibility to develop practical ways to prepare fluorine-containing cyclic cis-1,3-diamines in an enantio-and/or diastereoselective manner by using the reductive cleavage of hydrazine as a key step. One of the objectives was to tune the acidity of the amines in order to modify their affinity with RNA (see infra), another one was to use fluorinated diamines as tools for 19 F NMR.

Fluorinated 1,3-cis-diaminocyclopentanes were successfully synthesized starting from racemic or enantioenriched hydroxylated bicyclic hydrazine 25a. 39 Treatment of this compound with N,N-diethylaminosulfur trifluoride (DAST) led to the formation of the corresponding fluorinated bicycle 32, which was isolated as a single diastereomer (Scheme 13). The formation of a transient aziridinium was taken into account to explain the retention of the relative configuration observed after the reaction. The formation of this transient meso species was confirmed when a racemic product was isolated starting from the enantioenriched hydrazine (ee = 84%, Scheme 13). N-N bond cleavage of hydrazine rac-32 ultimately afforded the racemic fluorinated cis-1,3-diamine 33 in 80% yield (Scheme 13).

Scheme 13 Fluorination of bicyclic hydrazine 25a

With the aim to avoid the racemisation process and to isolate enantioenriched product ()-33, the all-cis Boc protected amino-alcohol 29 (non-racemic compound with a 1R,2S,4R absolute configuration) was next submitted to the fluorination reaction with DAST. This step proceeded with a complete inversion of the configuration at the carbon atom bearing the leaving group, 39 any anchimeric participation of the nitrogen protecting group being precluded by the cis relationship existing between the -OH and -NHBoc functions (Scheme 14).

An alternative synthetic pathway was envisaged to access an all-cis enantioenriched fluorinated compound. The hydroxylated bicyclic hydrazine 25a was reductively cleaved and the two amines were protected with four different protective groups (Scheme 15). The fluorination of these derivatives was then studied. The nature of the protective groups on the nitrogen atoms was found to significantly influence the outcome of this transformation. Indeed, Boc and tosyl groups actively participated to the reaction, leading to the formation of oxazolidinone 34 and meso aziridine 36 as the major products (Scheme 16). On the contrary, tetrabenzyl derivative 26e was converted into the corresponding racemic trans fluorinated compound 38 through a transient meso aziridinium intermediate (Scheme 16). Only 2,4-dinitrophenyl (DNP) groups allowed the fluorination reaction to proceed with full inversion of configuration at the stereocenter bearing the leaving group. The enantioenriched compound 39 could be obtained in this case, albeit in a low 21% yield (Scheme 16).

Scheme 16 Fluorination reactions of differently-protected cyclic diamino compounds.

A sequence of deprotection/protection/deprotection was lastly required to isolate the pure fluorinated diamino compound 40 (hydrochloride form) starting from compound 39 (Scheme 17).

39

Scheme 17 Access to pure fluorinated free amino compound 40 Fluorinated cyclic cis-1,3-diamines bearing a hydroxyl group at different positions can also be prepared starting from the bicyclic hydrazines. 39 A 1,2,3,4-tetrasubstituted cyclopentane could indeed be synthesized via the dihydroxylation of compound 21a, followed by a reductive cleavage of its hydrazine moiety and subsequent protection of the resulting free amines with DNP groups (Scheme 18). O-monobenzylation of diol 41, and consequent reaction with DAST delivered the protected fluorinated tetrasubstituted cyclopentane 42 as a single diastereomer (Scheme 18). A final sequence of deprotection/protection/deprotection reactions enabled to obtain product 43 in its hydrochloride form (Scheme 18).

Scheme 18 Synthesis of 1,2,3,4-tetrasubstituted cyclopentane 43 A 1,2,4,5-tetrasubstituted cyclopentane was finally obtained starting from bicycic hydrazine 23a. This compound was prepared via the epoxidation of derivative 21a and subsequent skeletal rearrangement as described in section 4.1 (Scheme 6). The treatment of hydrazine 23a with DAST delivered product 44 in a chemo-and diastereoselective manner. To explain the remarkable selectivity of the fluorination reaction, a mechanism involving the activation of both hydroxyl groups was postulated. The restricted conformational freedom of the carbamates caused by the presence of the hydroxymethylene group was supposed to force the nitrogen lone pairs of the hydrazine to be either in a pseudo equatorial or in a pseudo endo axial position (intermediate A, Scheme 19). Stereoelectronic effects were therefore assumed to favor the departure of the leaving group located on the ethylene bridge. Steric hindrance generated by the second activated hydroxyl group (intermediate B, Scheme 19) was finally invoked to account for the excellent regioselectivity of the final ring-opening step (Scheme 19). Control experiments were realized to support the proposed mechanism. 39 Scheme 19 Synthesis of 1,2,4,5-tetrasubstituted cyclopentane 45

In the last synthetic step, compound 44 was successfully submitted to hydrogenolysis to isolate fluorinated cyclic cis-1,3-diamine 45 in quantitative yield (Scheme 19).

Synthesis of cis-1,3-diaminocyclohexitols

The synthetic potential of diazabicyclo[2.2.1] heptane scaffold is not limited to the synthesis of 5-membered rings. It is indeed possible to access to 6-membered rings by a one atom homologation playing with the reactivity of the double bond.

Recently, the versatile reactivity of hydrazine 21a has been exploited in our laboratory to access various cis-1,3diaminocyclohexitols. 41 Bicyclic compound 21a was either submitted to ozonolysis or engaged in a two-step dihydroxylation-oxidative cleavage sequence to form meso dialdehyde 46 in a straightforward fashion (Scheme 20). The cyclization of this derivative by treatment with nitromethane was then investigated. Different conditions were screened to perform the double nitroaldol condensation. No reaction was observed in the presence of NaOH (Scheme 20). The desired product 47 could be formed in low yield when using sodium alkoxide (Scheme 20). In this case, the basic conditions were supposed to promote a cleavage of the benzylcarbamates, as traces of benzyl alcohol were detected by 1 H NMR of the crude reaction mixture. Compound 47 was finally obtained in a high yield when running the reaction with trimethylamine, 41 and using nitromethane as the solvent. Interestingly, cyclic product 47 was formed as a single diastereomer, although six possible isomers could theoretically be obtained through the condensation. The ring-homologation of bicyclic hydrazine 21a thus enabled to control the relative configuration of five contiguous stereogenic centers in a single operation. The relative configuration of derivative 47 was determined through NMR studies ( 1 H and 13 C experiments, correlation and NOE spectroscopy). 41 To explain the observed selectivity, dialdehyde 46 was supposed to react by keeping a Cs symmetrical conformation, and the cyclisation was thought to proceed under thermodynamic control.

Template for SYNLETT © Thieme Stuttgart • New York 2020-11-13 page 8 of 16

Scheme 20 Nitroaldol condensation reactions

Hydrogenolysis under acidic conditions led to the full reduction of bicycle 47, and delivered cis-1,3diaminocyclohexitol 48 in a quantitative yield (Scheme 21). The nitro function of derivative 47 could also be reduced in a selective manner by treatment with zinc in acetic acid (Scheme 21). Reductive aminations with aldehydes or ketones could be used to functionalize product 49. The N-N hydrazine bond of the resulting bicycles was finally cleaved to obtain Nsubstituted triaminocyclohexitols 50a,b in a good overall yield (Scheme 21).

Scheme 21 Functionalization of bicyclic hydrazine 47

Acid-catalyzed skeletal rearrangements were also investigated starting from hydrazine 47. 41 Despite our many efforts, the expected hydroxylated product could never been obtained. However, a brominated adduct was isolated when the reaction was conducted in the presence of a bromine source (Scheme 22). The formation of a transient azetidinium followed by a selective ring opening was proposed to explain the formation of the rearranged product 51. Hydrogenolysis of this derivative finally afforded diamine 52 in a quantitative yield as its hydrochloride salt (Scheme 22).

Scheme 22 Synthesis and reactivity of brominated derivative 51

Formation of cyclic cis-3,5-diamino-piperidines

Meso cis-3,5-diaminopiperidines can be considered as valuable building blocks for the preparation of 2-deoxystreptamine (DOS) analogues since libraries of compounds can rapidly be prepared simply by functionalizing the piperidine nitrogen atom as reported by Hermann and co-workers (Scheme 23). 42 Scheme 23 Importance of the meso cis-3,5-diaminopiperidine scaffold However, previously reported syntheses of this class of compounds rely on tedious high-pressure hydrogenation steps.

In addition, the selective derivatization of the piperidine nitrogen atom in the presence of cis-3,5-diamino groups also revealed to be problematic under certain experimental conditions. In our ongoing work on the preparation of cyclic cis-1,3 diamines, we considered the possibility overcome these synthetic limitations by exploiting the reactivity of bicyclic hydrazines to develop a new general approach to access Nsubstituted cis-3,5-diaminopiperidines.

Starting from hydrazine 21a, dihydroxylation and oxidative cleavage (or alternatively ozonolysis) were again preformed to prepare the relatively unstable key dialdehyde 46. This compound was then directly submitted to a series of double reductive amination reactions in the presence of various amine derivatives (Scheme 24). 43 Aliphatic or aromatic amines readily reacted in the presence of NaBH(OAc)3 under acidic reaction conditions. Diamines could also be engaged in this transformation. Remarkably, this method allowed the introduction of a stereocenter α to the piperidine nitrogen without racemization. 43 Scheme 24 Development of double reductive amination reactions

With the aim to access acylated triazabicyclo[3,2,1]octanes, product 53a, showing an allylic substituent at the piperidine nitrogen atom, was treated with palladiumtetrakis(triphenylphosphine) to form a NH-free bicyclic heterocycle (54, Scheme 25). This compound smoothly underwent acylation, peptidic coupling, or carbamoylation to afford the corresponding N-acylated products 55a,b, 56 and 57 in high yields (Scheme 25).

Scheme 25 Synthesis of acylated triazabicyclo[3,2,1]octanes

Differently functionalized bicyclic hydrazines 53, 55, and 56 were successfully submitted to hydrogenolysis in the presence of Pd-or Pt-based catalysts to generate the final cis-3,5diaminopiperidines 58a,f and 59a,b in good to quantitative yields (Scheme 26). In most cases, the reaction conditions had to be tuned in order to avoid catalyst poisoning or to prevent over-reduction of the (hetero)aromatic rings that frequently decorate the synthesized molecules.

Our strategy could finally be used to prepare Hermann's intermediate, 43 without any high pressure reduction step (Scheme 27). The free NH2-groups of product 58a were indeed easily protected by treatment with Boc2O. The resulting derivative 60 reacted with CAN in the last step of the synthetic sequence to yield the desired Boc-protected cis-3,5diaminopiperidine in 58% yield (44% overall yield from cyclopentadiene, Scheme 27).

Scheme 27 Synthesis of Hermann's intermediate

Applications of cyclic cis-1,3-diamines

All the synthetic transformations of bicyclic hydrazines described above enable a large coverage of chemical space around the cyclic 1,3 diamine motif. As briefly explained in section 2, this motif can be found in several classes of compounds possessing interesting biological properties. Among them, we focused our attention on aminoglycosides. The cyclic cis-1,3 diamines synthesized in our laboratory are structural mimetics of 2-deoxystreptamine (DOS), the central core of aminoglycosides. These natural products can be viewed as valuable ″RNA friendly″ molecules.

Scheme 26 Examples of hydrogenolysis of hydrazines 53, 55 and 56.

RNA is an essential biomolecule which is involved in the regulation of many cell processes. 44 As a result, RNA is nowadays considered as an attractive target in drug discovery. 45 In this context, the development of small molecules capable of binding specific RNA structures represent a major challenge for organic chemists. We therefore decided to explore the possibility to use our compounds as building blocks for the design of new RNA ligands and probes.

Small molecular RNA binders

Functional sites within the genomic RNA of HIV-1 have emerged as promising targets for new antiviral approaches. [START_REF] Luedtke | Small Molecule DNA and RNA Binders: FromSynthesis to Nucleic Acid Complexes[END_REF] tRNA Lys3 is the natural primer of HIV-1. In virions, tRNA Lys3 interacts with the genomic RNA and forms a complex that recruits the reverse transcriptase, the enzyme responsible for viral replication.

We become interested in identifying small compounds capable of interacting with tRNA Lys3 in order inhibit the reversetranscription initiation complex. [START_REF] Marquet | [END_REF] However, the rational design selective binders of tRNA Lys3 is a difficult task since shape is very similar to that of other RNA molecules. This is reflected in the fact that only low-affinity structure-specific peptide ligands could be identified by following a combinatorial approach. 48 Taking advantage of well-known crystallographic 49 and NMR data, 50 our group undertook a fragment based approach 51 to access new ligands of tRNA Lys3 . A NMR spectroscopic screening was performed with a focused library of organic compounds (prepared in-house or commercially available). 52 Binding of these compounds to the target induced detectable changes in the 1D NMR spectra of tRNA Lys3 (imino region, =10-15 ppm), and allowed us to identify racemic cyclic cis-1,3-diamine 61 (DACP) and kynuramine (62) as new RNA ligands (Figure 6). 2D HMQC experiments with 15 N-labeled tRNA allowed us to determine the NMR chemical shift mapping of the two ligands on tRNA Lys3 . At least two binding sites were observed for compound 61 (one in the T-stem and the other near the D-arm, the binding to the D arm being observed at higher concentration of the ligand). On the contrary, derivative 62 was found to specifically interact with the D-stem (Figure 6). Two nucleotides (G15 and m7G46) resulted to be affected by the binding of both compounds (Figure 6). NMR titrations were employed to calculate the apparent dissociation constants of the new RNA ligands (2 mM and 5 mM respectively).

A qualitative specificity study was also realized by analyzing the NMR spectroscopic footprints of compounds 61 and 62 with other nucleotide sequences (tRNAf Met and tRNAm Met of E. coli). Interestingly, derivative 62 showed some sequence-specific binding. 52 This result encouraged us to investigate the affinity between tRNA Lys3 and various kynuramine analogues. Despite showing a close resemblance with the structure of derivative 62, commercially available compounds 63-68 (Figure 7) were not detected to bind to tRNA Lys3 in chemical shift perturbations (CSP) experiments. The covalent linkage of kynuramine-like fragments to diaminecontaining compound 61 was investigated next in order to improve the binding affinity of our RNA ligands. Short aminoester were first used as linkers with the aim to design Darm binders. 52 Five representative linked products (73-77) were prepared in four steps starting from cyclic cis-1,3-diamine 26a (Scheme 28).

Under stoichiometric conditions, clear CPS was observed when comparing the NMR footprints of ligands 73, 74 and 77 on the target tRNA Lys3 and the 2D TROSY spectra of 15 N-labeled tRNA. 52 The new binders also showed a reinforced selectivity for the Darm of tRNA Lys3 , although interactions with the T-stem were observed at higher RNA:ligand ratios. The constant of derivative 77 was estimated tacking advantage of its florescence properties, a value in the micromolar range could be determined (Kd = 8 M). The considerably improved affinity and reinforced selectivity of ligand 77 in comparison with those of fragments 61 and 62 clearly demonstrated the validity of the fragment based approach for the design of new RNA binders. However, ligands 73-77 proved to have a relatively low stability as their linker arms could undergo retro-Michael reactions.

Scheme 28 Connection of kynuramine-like fragments and cyclic cis-1,3diamine compounds with short aminoester linkers.

The linkage strategy was reinvestigated with 3+2 cycloaddition reaction for connecting the fragments through a 1,4-triazole motif. A second series of RNA binders (81a-h) was synthesized in a straightforward manner using a practical one-pot procedure (Scheme 29). These compounds were isolated in 6% to 66% overall yield after catch-and release purification on sulfonic acid resin. 52 Binding of ligands 81a-h to tRNA Lys3 was monitored by using TROSY NMR spectroscopy. Improved affinity and reinforced selectivity for the D-arm was again observed. Fluorescence binding assays revealed that the use of a 1,4-triazole-based linker afforded better RNA binders, as compounds 81a, 81c and 81f showed enhanced affinity and selectivity when compared to derivative 77 (Table 1).

The stereospecificity of the binding was also evaluated by studying the interaction of tRNA Lys3 with the two enantiomers of compound 81f. Both racemic or enantiomerically pure compounds 81f binds to the same region of tRNA with similar strength (Table 1). To gain a deeper insight into the influence of the linker on the binding properties of our ligands, we envisioned the possibility to replace the 1,4-triazole moiety with its 1,5-isomer, and thus shortening the distance between the two fragments without modifying the chemical nature nor the size of the linking arm.

A ruthenium-catalyzed azide-alkyne cycloaddition reaction (RuAAC), 53 was successfully used to synthesize the desired compounds starting form fragments 78 and 80 (Scheme 30). 54 Fluorescence-quenching assays were employed to estimate the dissociation constants of ligands 82a-g to tRNA Lys3 . Altering the orientation of the two fragments was found to entail a modest increase of the Kd values. 52,54 However, selectivity was lost as NMR studies clearly showed that all ligands could bind to both the T and D stem of tRNA Lys3 . 54 Scheme 30 Connection of kynuramine-like fragments and cyclic cis-1,3diamine compounds with RuAAC reactions.

Interestingly, the binding interactions of ligand 82b with the T stem differed from those observed for compound 77. Since the interaction with the T stem was not detected in the presence of compound 81f, showing a 1,4 triazole moiety, 54 we concluded that rigid linkers capable of imposing a long distance between the two fragments should prove essential in the design of site selective binders.

To confirm this assumption we finally synthesized a ligand incorporating a triazine linker, a good mimic of the 1,4-triazole motif (Scheme 31). 54 Scheme 31 Synthesis of ligand 86 incorporating a triazine linker

Cyclic cis-1,3-diamine 26a and -hydroxy-naphtylacetophenone 85 were chosen as aliphatic and aromatic fragments, and could be connected to cyanuric chloride 83 through successive substitution reactions (Scheme 31). As depicted by NMR TROSY experiments, the resulting triazine-containing ligand 86 proved to selectively interact with the D stem of tRNA Lys3 (Kd = 1.4 M). 54 This last result allowed as to unequivocally confirm that, together with the nature of the connected fragments, the linker characteristics play a key role in controlling the binding selectivity of our ligands and have to be taken into account when designing new RNA binders.

Fluorinated 1,3 diamino cyclopentanes as NMR probes

The use of cyclic 1,3 diamines as key components for the design of RNA binders being validated, we then explored the possibility to use fluorinated diaminocyclopentanes (Figure 9) as chiral sensitive NMR probes for studying RNA structures and dynamics as well as spy reporters in binding assays. F NMR is indeed known to be very sensitive, and the fluorine shielding parameters are also known to strongly rely on local environment. The interaction of compound 33 (racemic or enantioenriched) with tRNA Lys3 was investigated first by 19 F NMR. Because compound 33 is only a millimolar RNA binder, the exchange between the free and bound forms was confirmed to be in the fast regime on the NMR timescale. Thus, each enantiomer led to a single 19 F NMR signal. However, although the two enantiomers bind to tRNA Lys3 with the same dissociation constant (around 1 mM), and on the same binding site, 55 they form diastereomeric complexes with the chiral RNA. As a consequence, two different averaged resonance shifts of the 19 F NMR signal could be detected for each enantiomer of derivative 33 in the presence of tRNA Lys3 . The use of racemic derivative 33 led to a complete NMR resolution with a split of 0.5 ppm (Figure 9). The interaction of racemic compounds 45, 40 and 43 (Figure 8) with tRNA Lys3 was then investigated under similar conditions. Shifts and/or resolution of the fluorine resonances were observed for all compounds but with lower resolution. 55 Figure 9 19 F-NMR of fluorinated diaminocyclopentane 33 alone (black) and with tRNA Lys3 (tRNA/ ligand ratio = 16:1, blue). The samples were dissolved in a 10 mM KPO4 buffer (pH = 6.5) and the spectra were registered at 293K.

Interestingly, a smaller signal splitting was detected by fluorine NMR when the racemic probe 33 was mixed with a short oligonucleotide sequence mimicking the T-arm hairpin of tRNA Lys3 , and no splitting was observed in the presence of the tRNA Lys3 D-arm sequence, highlighting the sensitivity of fluorinated probes to the tertiary structure of RNA. Sensitivity to secondary structure was also confirmed by thermal denaturation experiments. 55 Ligand-based fluorine NMR screening is a powerful method for binding constant measurements, popularized by Dalvit and coworkers. 56 As 1,3 diamino cyclopentanes could mimic DOS, the central core of aminoglycosides, we assessed compound 33 as a fluorinated reporter against a 23-nucleotide hairpin mimicking the decoding A-site of 16S ribosomal RNA, the natural target of aminoglycosides. 57 Competition binding experiments were conducted by monitoring the 19 F chemical shift of the reporter. As exemplified in Figure 10 with neamine, chemical shift variation as well as splitting modification could be observed by increasing competitor concentration, leading to the dissociation constant measurement of the non-fluorinated ligand.

Interestingly, observation of a splitting signal variation ensures that chemical shift variations are the direct consequence of the displacement of the racemic probe bound to its chiral target.

Concluding remarks

In this account article, we have described our efforts in setting up a general access to cyclic cis-1,3-diamines. We have shown that bicyclic hydrazines, easily available at a multigram scale, can lead to a large variety of five-and six-membered rings incorporating the cis-1,3-diamine motif in a few synthetic steps.

All the transformations can be conducted on a large scale, and delivers intermediates and final substituted diamines in a regioand stereoselective fashion. This systematic chemical space exploration around the cyclic cis-1,3-diamine motif provides numerous valuable fragments and building blocks. We have illustrated the interest of this strategy in the field of RNA binders, with the design of ligands and probes. We have no doubt that potential applications of cyclic cis-1,3-diamines are not restricted to this field, and we hope that our work will contribute to further developments in other areas of research.

Funding Information

Part of the work conducted in our lab and reported in this account has been financed by MESR (ACI 2003), ANR (TrigerRNA), Université Paris Descartes and Satt IDFInnov.

Figure 1

 1 Figure 1 General structure of five-and six-membered cyclic cis-1,3-diamines

Figure 3 .

 3 Figure 3. Biologically active compounds incorporating six-membered cyclic cis-1,3-diamine moieties.

Figure 4

 4 Figure 4 Biologically active compounds incorporating six-membered cyclic cis-1,3-diamine moieties.

Figure 5

 5 Figure 5 Synthetic strategies to access cyclic cis-1,3-diamines

Scheme 8

 8 Scheme 8 Asymmetric hydroborations compound 21a with Rh-or Ir-based catalysts and subsequent reductive cleavage of the hydrazine bond Chiral diphosphines also play an essential role in controlling the seterochemical outcome of the asymmetric hydroboration (Scheme 9).

Scheme 14

 14 Scheme 14 Synthesis of enantioenriched fluorinated 1,3-cisdiaminocyclopentane ()-33.

Figure 6

 6 Figure 6 Mapping on tRNA Lys 3 structure (PDB code 1FIR) of the binding zones of compound 61 (Blue, T-arm) and compound 62 (green, D-arm). G15 and m7G46 are depicted in red.

Figure 7

 7 Figure 7 Commercially available analogues of kynuramine 62 that do not bind to tRNA Lys3

Figure 8

 8 Figure 8 Fluorinated NMR probes

  19

Figure 10

 10 Figure10Competition experiments towards 16S23 RNA using compound 33 as the spy probe and neamine as a ligand : 19 F NMR spectra of compound 33 (0.7 mM) mixed to 16S23 RNA (0.3 mM) in KPO4 buffer pH 6.5 containing 50 mM KCl, at 293°K, with increasing concentration neamine.

Table 1

 1 Comparison of binding affinities of fragments 61, 62 and RNA ligands 77, 81a, 81c, 81f.

	Entry	Compound	Kd (M)	Method	Binding site
	1	61	1000	NMR	T arm, D arm
	2	62	5000	NMR	D arm
	3	77	8	Fluorescence	D arm, then T arm
	4	81a	6	Fluorescence	D arm
	5	81c	1.2	Fluorescence	D arm
	6	81f	1.8	Fluorescence	D arm
	7	(2R)-81f	1.2	Fluorescence	D arm
	8	(2S)-81f	1.1	Fluorescence	D arm

Connection of kynuramine-like fragments and cyclic cis-1,3diamine compounds with 3+2 cycloaddition reactions

Acknowledgment

CNRS , Université de Paris and Sanofi are acknowledged


Michel Abdalla
email: michel.abdalla@ens.fr

Security Analysis of Olvid's SAS-based Trust Establishment Protocol

In this report, we analyze the security of the trust establishment protocol used in the Olvid messaging protocol. The latter relies on the PV-SAS-MCA message cross-authentication protocol by Pasini an Vaudenay based on short authenticated strings (SAS). In order to make the implementation portable across different platforms, Olvid proposed particular instantiations of the underlying primitives used in PV-SAS-MCA in addition to some other minor modifications.

Here, we show that these changes have no impact on the security of the scheme. More precisely, we formally prove that the trust establishment protocol used in Olvid is a secure message crossauthentication protocol. The proof of security is in the random-oracle model and relies on the security of the underlying pseudorandom generator. It also assumes users know each other and have an authentic channel between them.

Introduction

Message cross-authentication (MCA) protocols allow users to identify each other and to simultaneously exchange messages in an authenticated manner. In [Vau05], Vaudenay proposed a variant of the above notion in which one assumes the existence of an extra channel which can authenticate short strings. The new notion allows for the construction of efficient authentication protocols based on short authenticated strings (SAS) and is known as SAS-based authentication.

Olvid3 is a secure instant messaging application which, contrary to other applications like What-sApp or Signal, does not impose its users to trust a central directory of users for the key distribution. Instead, Olvid offers multiple trust establishment protocols relying on the natural relationship between users so as to retain the same "security level" in the digital world as in real life. The absence of a central directory has two main advantages: (1) users do not need to disclose any personal information to the server, and (2) the key distribution is completely decentralized without any need for a trusted third party.

In this report, we analyze the security of the first trust establishment protocol used in Olvid, which assumes users know each other and have an authentic channel between them (e.g, phone calls, face-to-face meetings, voice chats). This protocol uses the SAS-based message cross-authentication (MCA) protocol PV-SAS-MCA by Pasini and Vaudenay [PV06,Pas09]. In Olvid, the SAS-based MCA protocol is used to allow users not only to identify each other but also to exchange their public keys in authenticated manner.

In the remainder of this report, we analyze the security of the particular instantiation of the PV-SAS-MCA protocol used in Olvid and show that it meets the security requirements listed in the security theorem for PV-SAS-MCA in [PV06,Pas09]. For completeness, we also provide a formal security proof that shows that the trust establishment protocol used by Olvid is a secure message cross-authentication protocol.

Organization. Section 2 recalls some basic notation used in this report and the cryptographic primitives used in the PV-SAS-MCA protocol. Section 3 recalls the description of the PV-SAS-MCA protocol by Pasini and Vaudenay [PV06,Pas09]. Section 4 describes the variant of the PV-SAS-MCA protocol used by Olvid together with a detailed security analysis. Finally, Section 5 concludes this report.

Preliminaries

Notation. We write x ← y for the action of assigning the value y to the variable x. We write x 1 , . . . , x n ←$ X for sampling x 1 , . . . , x n from a finite set X uniformly at random. If A is a probabilistic algorithm we write y 1 , . . . , y n ← $ A(x 1 , . . . , x n ) for the action of running A on inputs x 1 , . . . , x n with independently chosen coins, and assigning the result to y 1 , . . . , y n . PPT as usual abbreviates probabilistic polynomial-time.

Commitment Schemes. A keyed tag-based commitment scheme is defined by a triple of PPT algorithms (setup, commit, open). Via (K p , K s ) ←$ setup(1 λ ), the setup algorithm generates a publicsecret key pair (K p , K s ). For any message m, tag t, and public key K p , the commitment algorithm outputs a pair (c, d) corresponding to the commit and decommit values via (c, d) ←$ commit(K p , t, m). Given (c, d) and a tag t, the original message can be recovered using the opening algorithm open via m ← open(K p , t, c, d). As noted in [Pas09], K p may be empty.

For the security of the SAS-based message cross-authentication protocol used in Olvid and described in Section 3, the commitment scheme needs to satisfy the following properties:

-Completeness: For any key pair (K p , K s ) ← $ setup(1 λ ), any message m, any tag t, and any (c, d) ←$ commit(K p , t, m), we have m = open(K p , t, c, d). -Perfectly Hiding: For any key pair (K p , K s ) ←$ setup(1 λ ), any tag t, any values m 0 , m 1 , any random bit b, and any (c, d) ←$ commit(K p , t, m b ), c gives no information on b. -Computationally Binding: For any key pair -An honest run should lead to ( îd A , mA ) = (id A , m A ) and ( îd B , mB ) = (id B , m B ).

(K p , K s ) ←$ setup(1 λ ), it is hard to find (m, t, c, d 0 , d 1 ) such that m b ← open(K p , t, c, d b ) = ⊥ for b = 0,
-An adversary is successful if some instance of an uncorrupted node ended with a pair (m, id), but no instance of the node of identity id was launched with input m.

SAS-based MCA protocol

The Alice Bob

input : mA input : mB K ←$ {0, 1} κ R ←$ {0, 1} ρ (c, d) ← commit(idA, mA, K) K ← open( îdA, mA, ĉ, d) SAS ← R ⊕ h( mB, K) SAS ? = R ⊕ h(mB, K) Check SAS is the same Check idA = idB output : idB, mB output : idA, mA mA, c mB, R d auth id A (SAS) auth id B (SAS) Fig. 1. The SAS-based MCA protocol PV-SAS-MCA [PV06, Pas09].
As stated in [Pas09, Section 8.4], the SAS-based MCA protocol PV-SAS-MCA directly authenticates m B through the SAS and indirectly authenticates m A via the direct authentication of K and the tagged commitment scheme, which binds m A to K. Moreover, as discussed in [Pas09], h can be instantiated as h(x, K) = trunc(hash(K, x)) with a collision-resistant hash function hash such as SHA-256 [SHA04] and trunc is a function that truncates its output to the leading bits.

Olvid's SAS-based MCA protocol

The SAS-based MCA protocol used in Olvid is described in Fig. 2 and differs slightly from the PV-SAS-MCA protocol in Fig. 1. In particular, the values m A and m B being exchanged are public keys associated with Alice and Bob, respectively. In addition to this, there are a few other notable changes to the original protocol:

-Bob now sends m B to Alice at the beginning of the protocol; -Every message from Alice to Bob is encrypted under Bob's public key m B ; -With the exception of the first message, every message from Bob to Alice is encrypted under Alice's public key m A ; -The function h is instantiated with a hash function H, modeled as a random oracle in the security analysis; -The random value R is computed as H(k B , c) where H is the SHA-256 hash function [SHA04] and k B is a secret key held by Bob; and -The SAS value is computed as SAS = prng(R ⊕ H( mB , K)).

Alice Bob

input : mA input : mB K ←$ {0, 1} κ (c, d) ← commit(mA, K) R ← H(kB,m B , ĉ) K ← open( mA, ĉ, d) SAS ← prng( R ⊕ H( mB, K)) SAS ? = prng(R ⊕ H(mB, K)) Check SAS is the same Check idA = idB output : idB, mB output : idA, mA idB, mB {idA, mA, c}m B {R}m A {d}m B auth id A (SAS) auth id B (SAS)

Instantiations

Equivocable Commitments. In order to instantiate the commitment scheme in PV-SAS-MCA protocol in Fig. 1, Olvid uses the following random-oracle-based commitment scheme given in [Pas09, Section 3.5.9.2]. Let c , e , and h be three integers and let H be a hash function from {0, 1} h + e to {0, 1} c modeled as a random oracle. The equivocable commitment works as follows:

setup: There is no setup algorithm. Hence, As noted in [Pas09, Section 3.5.9.2], one can easily build the simcommit and equivocate algorithms using the fact that H is modeled as a random oracle.

K p = K s = ε. -commit(t,
As shown in [Pas09], the commitment scheme above is perfectly hiding and statistically binding, In Olvid, the hash function H is instantiated with SHA-256 [SHA04].

SAS Computation. The implementation of the SAS differs from the original protocol in [Pas09] as follows:

-The SAS computation outputs a random 8-digit numbers, where the first and last 4 digits are displayed to Alice and Bob, respectively. To complete the authentication, both parties must type the 4 digits associated with the other party.

Pseudorandom Number Generator. Olvid instantiates the PRNG using the NIST SP 800-90A [BK15].

Changes to the Security Analysis of PV-SAS-MCA

Despite the numerous modifications, the proof of security for the PV-SAS-MCA protocol in [Pas09] also applies to the Olvid SAS-based MCA protocol described in Fig. 2. In the following, we describe the main changes to the proof.

Encryption of protocol messages. In Fig. 2, with the exception of the very first message from Bob to Alice, all messages are encrypted using the ECIES encryption scheme [ABR01] under the public-key of the recipient. Since the security of the PV-SAS-MCA protocol in [Pas09] does not require privacy for the messages being exchanged and given that the public-key encryption scheme is perfectly binding, this change has no impact on the security of Olvid's protocol. In particular, the security analysis of Olvid's protocol makes no assumption about the security of the underlying public-key encryption scheme and simply assumes that these messages are sent in the clear.

Computation of R. Unlike in the PV-SAS-MCA protocol where R is chosen uniformly at random in each session, the random value R in Olvid's version of the protocol is computed as

H(k B,m B , c)
where H is the SHA256 hash function [SHA04], k B,m B is a secret key held by Bob which is chosen uniformly at random for each different message m B , and c is Alice's commitment. Since the commitment scheme being used is statistically binding and H is a random oracle, this has no impact on the security of the protocol given that each execution of protocol between Alice and Bob yields a random and uniformly distributed R value.

PRNG used in the SAS computation. In the PV-SAS-MCA protocol, the short authenticated string used for the SAS authentication is equal to R ⊕ H(m B , K). In Olvid's protocol, the value R ⊕ H(m B , K) is used to derive the short authenticated string via a PRNG function prng. Since the security analysis of the PV-SAS-MCA protocol shows that the string R ⊕ H(m B , K) is random and uniformly distributed, the same is true for prng(R ⊕ H(m B , K)) due to the security of the PRNG function prng. As a result, the probability that an adversary can impersonate either Alice or Bob on a given interaction is 1/10 4 .

Bob's first message. Unlike the PV-SAS-MCA protocol, Bob now sends message m B to Alice at the beginning of the protocol. Since m B is directly authenticated through the SAS authentication and given that the value R computed by Bob remains random and uniformly distributed even when Alice chooses its first flow message as a function of m B , this change has no impact on the security of the protocol.

Universal hash function. In the PV-SAS-MCA protocol, the function h is assumed to be an almost strongly universal hash function. In Olvid's protocol, the function h is is simply instantiated with the SHA-256 hash function, which we assume behaves as a random oracle.

Security Proof

As indicated in Section 4.2, the modifications made to the PV-SAS-MCA protocol do not affect the security of the scheme. To prove this formally, we prove the security of the Olvid SAS-based MCA protocol described in Fig. 2 against one-shot adversaries, in which adversaries can launch at most one instance of each party. As shown by Vaudenay [Vau05], this result already implies security with respect to adversaries that can launch polynomially many instances of each party.

Security model. As indicated in [Vau05], adversaries are assumed to have full control of the communication network, except for the authenticated channel. in particular, adversaries are allowed to launch new instances of a protocol party (via queries to a launch oracle) or to send messages to protocol instances (via queries to a send oracle). The adversary is also allowed to specify the input of a party when querying the launch oracle.

launch(n, r, x) denotes a query to the launch oracle where n is a node of the network, r is the role of this node, and x is the input. It outputs a unique instance tag π i n .

send(π n , m) denotes a query to the send oracle through which the adversary can send a message m to an instance π n . It outputs a message m that is meant to be sent to the other protocol party.

In a SAS-based MCA protocol, there are two parties, called Alice and Bob. On Alice's side, the protocol input is (m A , id A ) and its output is ( mB , îd B ). On Bob's side, the protocol input is (m B , id B ) and its output is ( mA , îd A ). Finally, the adversary is considered successful in attacking the authentication protocol if there exists at least one instance of Alice (resp. Bob) that terminates and outputs ( mB , îd B ) (resp. ( mA , îd A )) such that no instance of Bob (resp. Alice) exists with input

(m B , id B ) = ( mB , îd B ) (resp. (m A , id A ) = ( mA , îd A )).
Security theorem. As the following theorem shows, Olvid's SAS-based MCA protocol is secure against one-shot adversaries, if H is modeled as a random oracle, prng is a secure pseudorandom generator, and the underlying commitment scheme is the one described in Section 4.1. The proof is based on the proof of security for the original SAS-based MCA protocol PV-SAS-MCA [PV06, Pas09].

Theorem 4.1. Let H be a random oracle, let commit be the random-oracle-based commitment scheme in Section 4.1, and let prng be a secure pseudorandom generator. Then any one-shot adversary against the Olvid's SAS-based MCA protocol is successful with probability negligibly close to 1/10 4 . Proof. We prove the security of Olvid's SAS-based MCA protocol using a sequence of games, starting with the original protocol and ending with a protocol in which the success probability of the adversary is 0.

Game G 1 . This is the initial security game, so we have:

Adv Olvid-SAS A ( ) = Pr[ G 1 ⇒ T ]
Game G 2 . In this game, we change the simulation of the protocol so that Bob chooses the value R uniformly at random in the range of H.

Since H is a random oracle. the advantage of the adversary remains the same unless it finds a collision on the output of the random oracle, which is negligible for any polynomial number of queries to the latter. As a result,

| Pr[ G 2 ⇒ T ] -Pr[ G 1 ⇒ T ] | is negligible.
Game G 3 . In this game, we extract the value K committed in ĉ and abort if extraction fails.

Since the random-oracle-based commitment scheme described in Section 4.1 is statistically binding, the probability of aborting is negligible, This is due to the fact that an adversary can only break the binding property if it finds a collision in the random oracle used by the commitment scheme, which is negligible for any polynomial number of queries to the latter. Hence, | Pr[

G 3 ⇒ T ] -Pr[ G 2 ⇒ T ] | is negligible.
Game G 4 . If the two instances created by the one-shot adversary are Alice's instances, then the seed used by the pseudorandom generator prng of Alice's instance that terminates last is chosen at random.

Let K (i) , m(i) B , R(i) be the values seen or chosen by Alice's instance i ∈ {1, 2}, where the index 2 indicates the instance that terminates last. Since the commitment scheme is perfectly hiding, the value K (2) is uniformly distributed and independent from the rest. As a result, the adversary can

only detect this change if R(1) ⊕ H( m(1) B , K (1) ) = R(2) ⊕ H( m(2) B , K (2)
), which happens with negligible probability. Hence, it follows that | Pr

[ G 4 ⇒ T ] -Pr[ G 3 ⇒ T ] | is negligible.
Game G 5 . If the two instances created by the one-shot adversary are Bob's instances, then the seed used by the pseudorandom generator prng of the instance of Bob that sends its R value last is chosen at random.

Let K(i) , m

B , R (i) be the values seen or chosen by instance i ∈ {1, 2} of Bob, where the index 2 indicates the instance that sends its R value last. Since the value R (2) is uniformly distributed and independent from the rest, the adversary can only detect this change if R (1) ⊕ H(m

(1) B , K(1) ) = R (2) ⊕ H(m (2) B , K(2) ), which happens with negligible probability. Thus, | Pr[ G 5 ⇒ T ] -Pr[ G 4 ⇒ T ] | is negligible.
In Games G 6 -G 8 , we consider the cases in which the one-shot adversary creates one instance of Alice and one of Bob. Without loss of generality, we will assume that Bob's instance terminates last since the other cases can be dealt with in a similar way.

Game G 6 . If the one-shot adversary creates one instance of Alice and one of Bob and Alice's instance's terminates before Bob's instance sends its R value, then the seed used by the pseudorandom generator prng of Bob's instance is chosen at random.

Since the value R is uniformly distributed and independent from the rest, the adversary can only detect this change if R ⊕H( mB , K) = R ⊕H(m B , K), which happens with negligible probability. Thus,

| Pr[ G 6 ⇒ T ] -Pr[ G 5 ⇒ T ] | is negligible.
Game G 7 . If the one-shot adversary creates one instance of Alice and one of Bob and Alice's instance does not terminate before Bob's instance sends its R value and ĉ = c, then the seed used by the pseudorandom generator prng of Alice's instance is chosen at random.

Since the value K remains hidden until Alice's instance sends the decommitment value d, its value is uniformly distributed and independent from the other values used in the seed computation of both instances. As a result, the probability that the adversary can only detect the change in the game is negligible. Hence

, | Pr[ G 7 ⇒ T ] -Pr[ G 6 ⇒ T ] | is negligible.
Game G 8 . If the one-shot adversary creates one instance of Alice and one of Bob and Alice's instance does not terminate before Bob's instance sends its R value and ĉ = c, then the seed used by the pseudorandom generator prng of Alice's instance is chosen at random.

Since the commitment scheme is statistically binding, it follows that mA = m A . Hence, we can assume that mB = m B as otherwise the adversary would not be considered successful. Moreover, as in Game G 7 , the value K remains hidden until Alice's instance sends the decommitment value d and its value is uniformly distributed and independent from the other values used in the seed computation of both instances. As a result, the probability that the adversary can only detect the change in the game is negligible. Hence, | Pr[ G 8 ⇒ T ] -Pr[ G 7 ⇒ T ] | is negligible.

Game G 9 . In this game, the output of the pseudorandom generator prng is replaced with a random string whenever its seed is chosen uniformly at random.

Since it is straightforward to convert any adversary that detects this change into one that breaks the security of the pseudorandom generator prng, it follows that | Pr[

G 9 ⇒ T ] -Pr[ G 8 ⇒ T ] | is negligible.
Finally, the claim in Theorem 4.1 follows by noticing that, for every instance in Game G 9 , at least one of the values used for the SAS comparison is chosen uniformly at random.

Conclusion

As the analysis in Section 4 formally shows, the trust establishment protocol used by Olvid is a secure message cross-authentication protocol. The analysis assumes users know each other and have an authentic channel between them (e.g, phone calls, face-to-face meetings, voice chats). The security proof is in the random-oracle model and relies on the security of the underlying pseudorandom generator, which has been instantiated according to NIST SP 800-90A [BK15].

  PV-SAS-MCA protocol byPasini and Vaudenay [PV06, Pas09] is recalled in Fig.1and works as follows. First, users Alice and Bob initiate the protocol with their respective inputs m A and m B and then each picks a random key, denoted by K and R. Next, Alice computes a commitment c of her key K and sends both m A and c to Bob, who replies with m B and R. After receiving m B and R, Alice reveals her hidden key K by sending the decommitment value d to Bob. Finally, both Alice and Bob authenticate the string SAS = R ⊕ h(m B , K) using the extra authenticated channel, where h is an almost strongly universal hash function.

Fig. 2 .

 2 Fig. 2. Olvid's SAS-based MCA protocol. The notation {a}m indicates the public-key encryption of the plaintext a using m as the public key.

  m): Sample e ←$ {0, 1} e , set c = H(t, m, e) and d = (m, e), and output (c, d). -open(t, c, d): Parse d as (m, e), check that c = H(t, m, e). If true, return m, else return ⊥.

  1 and m 0 = m 1 .-Equivocable: There exist algorithms simcommit and equivocate capable of generating fake commitments that can opened to arbitrary values. That is, given the secret key K s and tag t, (c, ξ) ←$ simcommit(K s , t) yields a fake commit value c and an additional information value ξ. Alice and Bob starts with inputs m A and m B and ends with outputs ( îd B , mB ) and ( îd A , mA ), respectively.

	Then, given t, K s , and the output (c, ξ) of simcommit(K s , t), equivocate(K s , c, ξ, m) can output a
	decommit value d such that m = open(K p , c, d) for any message m.
	Message Cross-Authentication. Message Cross-Authentication (MCA) protocols allow users to
	exchange messages in an authenticated manner and are often used to exchange public keys.
	Definition 2.1 (Message Cross-Authentication [Pas09]). A Message Cross-Authentication (MCA)
	protocol between

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