Guangkuan Zhao

Mingxiang Zhu

Olivier Provot

Mouad Alami

Samir Messaoudi

Synthesis of 2,3-Substituted β-N -Glycosyl Indoles through C-H Activation/Annulation Process under Rh(III)-Catalysis

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Synthesis of 2,3-Substituted β-N -Glycosyl Indoles through C-H Activation/Annulation Process under Rh(III)-Catalysis.

Guangkuan Zhao, a Mingxiang Zhu, a Olivier Provot, a Mouad Alami, a and Samir Messaoudi a* a BioCIS, Univ. Paris-Sud, CNRS, University Paris-Saclay, Châtenay-Malabry, France ABSTRACT: An efficient and selective C-H activation/annulation of readily available β-N-aryl glycosides with various alkynes has been established. Using [Cp*RhCl 2 ] 2 as a catalyst and AgSbF 6 in DCE, this protocol proved to be general to prepare a variety of 2,3-substituted N-glycosyl indoles in good yields with exclusive β-selectivity. β-N-glycosyl indoles are of high importance in medicinal chemistry and commonly found in many compounds of practical importance, ranging from natural compounds to pharmaceutical agents 1 (Figure 1). While these derivatives clearly hold a great potential in medicinal chemistry, relatively little attention has been devoted to their syntheses (Figure 1-B), since the stereoselective induction of a nitrogen indole scaffold at the anomeric position remains as a particularly difficult task. From a synthetic point-of-view, 2,3-substituted β-N-glycosyl indoles 3 (Figure 1) were prepared through multi-steps syntheses by treating indoline derivatives with sugars lactols 1i,2 followed by (i) oxidation of the indoline into indole and (ii) functionalization of the C-2 and C-3 positions (Figure 2, Path A). The synthesis of 2,3-substituted β-N-glycosyl indoles is also possible via a β-glycosylation of indoles through a SN 2 Mitsunobu reaction 1k, 1l, 3 followed by the functionalization of the indole nucleus (Figure 2, Path B). However, the reaction necessitate the use of a well-defined α-sugar lactols which are difficult to synthesize through multisteps sequences. [START_REF] Bayle | A Stable Blocking Group Removable by Ozonolysis[END_REF] Moreover, a mixture of α-and β-anomers was obtained in most cases. Another way to prepare stereoselectively β-N-glycosyl indoles is the use of Danishefsky α-1,2-anhydro sugars (Figure 2, Path C). [START_REF] Gallant | A Stereoselective Synthesis of Indole-β-N-glycosides: An Application to the Synthesis of Rebeccamycin[END_REF] While this method is efficient, beside the necessary of C2-and C3 functionalization, Indeed, their synthesis from epoxidation of the corresponding glycals is inherently relies on the use of the Murray's reagent [START_REF] Murray | Chemistry of Dioxiranes. 12[END_REF] (DMDO) which is an instable volatile peroxide not easy to prepare and manipulate. Despite these advances, the absence of a general and predictable Figure 2 Strategies to access to 2,3-substituted β-N-glycosyl indoles method for the direct synthesis of 2,3-substituted β-N-glycosyl indoles with a minimum of steps remains a major gap in glycochemistry preventing greater investigation of the biology and applications of these compounds.

Recently, our group reported an efficient protocol for the synthesis of β-N-aryl glycosides via a copper-catalyzed Chan-Lam-Evans N-arylation of aryl boronic acids. [START_REF] Bruneau | Stereoselective Copper-catalyzed Chan-Lam-Evans N-arylation of Glucosamines with Arylboronic Acids at Room Temperature[END_REF] As part of our continued efforts to functionalize sugars under transition-metal catalysis to access complex glycosides, [START_REF] Brachet | Palladium-Catalyzed Cross-Coupling Reaction of Thioglycosides with (Hetero) aryl Halides[END_REF] we envisioned whether β-N-aryl glycosides could be utilized as building blocks in the synthesis of β-N-glycosyl indoles through a transition-metalcatalyzed activation/annulation reaction in the presence of various alkynes (Figure 1-C). This modular strategy is conceptually attractive in terms of diversifying the N-glycosyl indoles frameworks with the aim to identify novel scaffolds of biological interest. In this work, we showed for the first time, that β-Naryl glycosides and alkynes successfuly joined together through a C-H activation/annulation process 9 to afford in a single step stereoselectively, a variety of substituted β-Nglycosyl indoles (Figure 1-C). To achieve successfully our goal, initial investigations focused on identifying optimal conditions for coupling of β-Nphenyl glucopyranoside 1a with 1,2-diphenylethyne 2a as models study (Table 1). In preliminary experiments, the C-H activation/annulation reaction was examined under various conditions in the presence of different catalysts such as Ru, [START_REF] Ackermann | Cationic Ruthenium (II) Catalysts for Oxidative C-H/N-H Bond Functionalizations of Anilines with Removable Directing Group: Synthesis of Indoles in Water[END_REF] Rh, [START_REF] Stuart | Indole Synthesis via Rhodium Catalyzed Oxidative Coupling of Acetanilides and Internal Alkynes[END_REF] Ni, [START_REF] Song | Nickel-catalyzed Alkyne Annulation by Anilines: Versatile Indole Synthesis by C-H/N-H Functionalization[END_REF] and Pd, [START_REF] Dupont | Reactivity of Cyclopalladated Compounds. Part 18. Compared Reactivity of the Pd-C Bonds of Two Closely Related Six-membered Palladocyclic Rings with Substituted Alkynes. X-Ray and Molecular Structures of [Pd{C(Ph)=C(R)C(Ph)=C(R)(o-C6H4N=CMeNHPh)}Cl[END_REF] but unfortunately, β-N-glycosyl indole 3a was not detected under these conditions.

Further, inspired by the recent work described by Zhu 14a and Fan 14b whose reported the cyclization of N-nitrosoanilines under Rh(III)-catalysis, we evaluated the influence of a nitrososubstituent on the phenyl glucopyranoside nitrogen atom (compound 1b). When we used β-N-nitrosophenyl glucopyranoside 1b with [(Cp*RhCl 2 ) 2 ] (5 mol%), AgSbF 6 (20 mol%), and diphenylacetylene (2a, 2 equiv.) in 1,2-dichloroethane (DCE) at 100 °C for 3 h, we obtained β-N-glucosyl indole 3a (J 1,2 = 9.3 Hz) in 38% yield (entry 18). After screening several parameters, we finally found that the C-H annulation between 1b and 2a occured smoothly with 70% yield in the presence of [(Cp*RhCl2)2] (5 mol%), AgSbF 6 (20 mol%), in DCE at 90 °C for 5 h without adding any external oxidant (Table 1, entry 23). It should be noted that the Rh-catalyst and AgSbF 6 were necessary to achieve this transformation since no reaction occur when the coupling was conducted in the absence of [(Cp*RhCl 2 ) 2 ] or AgSbF 6 (entries 24 and 25).

Motivated by these results, we next explored the scope of the coupling reaction of 1b with a variety of internal alkynes and we are gratifyingly pleased with the generality of this method. Various internal alkynes reacted smoothly to afford the desired 2,3-disubstituted N-glucosyl indoles 3a-i in satisfactory yields. A variety of symmetric diaryl alkynes were efficiently converted into the corresponding products and electron-donating substituents on the aromatic rings had a positive electronic effect on yield than electron-withdrawing substituents (Table 2, 3a-d). The coupling reaction with 1,2-di(thiophen-2-yl)ethyne was also efficient to afford 3e in a moderate yield. Moreover, unsymmetrically substituted alkynes 2f-i were converted with variable yields and moderate regioselectivities (3f-i).

In a further set of experiments, we investigated the scope and generality of the method with respect to β-N-aryl glycosides 1jr. As depicted in Table 3, the protocol tolerated different β-Naryl glucosides and a multitude of 5-and

Scheme 1 Scope of alkynes coupling with tetraacetyl β-Nnitrosophenyl glucopyranoside 1b

Reaction conditions: reactions were performed in a flame dried resealable Schlenk tube using 1b (0.30 mmol), alkynes 2 (2 equiv), [RhCp*Cl2]2 (5 mol%), AgSbF6 (20 mol%), in 1,2-DCE (0.1 M) at 90 °C for 5 h. b Yield of isolated product 3.

6-substituted indoles 3j-o were readily prepared using this reaction. 5-Substituted β-N-aryl glycosides bearing elec trondonating or electron-withdrawing groups afforded the corresponding 5-substituted β-N glucosyl indoles 3j-m in acceptable yields. In addition, meta-substitution was tolerated furnishing 6-disubstituted β-N-glucosyl indoles 3n and 3o in 58% and 37% yields, respectively. Interestingly, this cross-coupling tolerated the presence of C-halogen bonds (e.g., F, Cl, Br) which offers a platform for further metal-catalyzed cross coupling reactions (compounds 3l, 3m and 3n). Moreover, the C-H activation/annulation process is not limited to β-N-aryl glucosides but also works successfully with β-N-aryl galactosides 1p,q and the peracetylated β-D-disaccharide 1r derived from D-β-cellobiose octaacetate, however only 25% isolated yield of the disaccharide 3r was obtained probably due to its intrinsic instability of 3r. Of note, the stereochemistry of the 1→4' glycosidic bond remained intact. It is noteworthy that the coupling of unprotected β-N-nitrosophenyl glucoside with 1,2diphenylethyne 2a under the above experimental conditions failed, as only starting materials were recovered unchanged. With substantial amounts of 3l in hand (Table 2), we focused our attention on demonstrating whether our method could be employed for molecular diversity. As shown in Scheme 1, β-Nglucosyl indole 4, which is an analogue of compound A, a highly promising cytotoxic and antitubulin agent developed in our group, [START_REF] Soussi | Discovery of Azaisoerianin Derivatives as Potential Antitumors Agents[END_REF] was easily Scheme 2 Scope of β-N-nitroso-aryl glycosides 1j-r coupling with 2a

Reaction conditions: reaction were performed in a flame dried resealable Schlenk tube using 1j-r (0.30 mmol), alkyne 2a (2 equiv), [RhCp*Cl2]2 (5 mol%), AgSbF6 (20 mol%), in 1,2-DCE (0.1 M) at 90 °C for 5 h. b Yield of isolated product 3. c The reaction was achieved at 80 °C prepared via a Pd-catalyzed coupling reaction of 3l with 3,4,5trimethoxy-N-methylaniline. Compound 5 was obtained from 4 by deprotection of the acetates groups under Zemplen's conditions using a catalytic amount of potassium carbonate in methanol. [START_REF] Zemplén | Studien über Amygdalin, IV: Synthese des Natürlichen l-Amygdalins[END_REF] Scheme 1 Application of this methodology the synthesis of bioactive compounds 4 and 5

The in vitro activity of derivatives 4 and 5 was evaluated by their growth-inhibitory potency against HCT-116 cancer cells (human colon carcinoma) at the concentration of 10 -6 M. The quantification of cells survival in this cell line was established by using MTT assays after 72 h of exposure. We found that analogues 4 and 5 displayed a moderate effect in the growth of HCT-116 (81% and 70% survival, respectively) compared to the reference compound A (IC 50 = 7 nM) (Scheme 1).

In conclusion, we successfully developed an efficient and practical method based on Rh(III)-catalyzed C-H activation/annulation process of various β-N-nitroso-aryl glycosides with alkynes. The protocol exhibited a broad substrate scope with respect to the coupling partners, thus providing an attractive access to a large molecular diversity of 2,3-disubstituted N-glycosyl indoles 3. This protocol developed is stereoretentive, functional-group tolerant, and proceeds in good yields. We believe that this methodology will find broad applications in organic synthetic chemistry as well as in combinatorial and pharmaceutical sciences.

ASSOCIATED CONTENT Supporting Information

The Supporting Information is available free of charge on the ACS Publications website. Experimental procedures, spectroscopic data and NMR spectra of new compounds.

Figure 1 .

 1 Figure 1. Heteroaryl N-glycoside-based bioactive molecules access to α-1,2-anhydrosugars (epoxides) remain no trivial.Indeed, their synthesis from epoxidation of the corresponding glycals is inherently relies on the use of the Murray's reagent[START_REF] Murray | Chemistry of Dioxiranes. 12[END_REF] (DMDO) which is an instable volatile peroxide not easy to prepare and manipulate. Despite these advances, the absence of a general and predictable

Table 1

 1 Survey of reaction conditions for the C-H activation/annulation of 1a,b with 2a a

	entry	R	Cat.	additive	solvent	Temp	Time	yield
						(°C)	(h)	3a
								(%) b
	1	H	Cat1	AgNO3	t-AmOH	120	12	0
	2	H	Cat1	Ag3PO4	t-AmOH	120	12	0
	3	H	Cat1	MeCO2Ag	t-AmOH	120	12	0
	4	H	Cat1	CF3CO2Ag	t-AmOH	120	12	0
	5	H	Cat1	AgSbF6	t-AmOH	120	12	0
	6	H	Cat1	AgSbF6	dioxane	120	12	0
	7	H	Cat1	AgSbF6	DCE	120	12	0
	8	H	Cat1	AgSbF6	Acetone	120	12	0
	9	H	Cat1	AgSbF6	DMF	120	12	0
	10	H	Cat1	AgSbF6	Toluene	120	12	0
	11	H	Cat2	AgSbF6	t-AmOH	120	12	0
	12	H	Cat3	AgSbF6	t-AmOH	120	12	0
	13	H	Cat4	AgSbF6	t-AmOH	120	12	0
	14	H	Cat5	AgSbF6	t-AmOH	120	12	0
	18	NO	Cat3	AgSbF6	DCE	100	3	38
	19	NO	Cat3	AgSbF6	DCE	120	3	25
	20	NO	Cat3	AgSbF6	DCE	rt	15	trace
	21	NO	Cat3	AgSbF6	DCE	60	15	31
	22	NO	Cat3	AgSbF6	DCE	100	5	55 c
	23	NO	Cat3	AgSbF6	DCE	90	5	70
	24	NO	Cat3	--	DCE	90	5	0
	25	NO	--	AgSbF6	DCE	90	5	0
	26	NO	Cat3	AgSbF6	DCE	90	5	72 d

a Reactions were conducted with substrate 1a or 1b (0.10 mmol), alkyne (0.2 mmol), catalyst, additive, and solvent (1.0 mL). b Yield of isolated product 3a. c 1b was completely consumed. d 0.4 mmol of 2a were used.

ACKNOWLEDGMENTS

Authors acknowledge support of this project by CNRS, University Paris Sud, ANR (ANR-15-CE29-0002) and by la Ligue Contre le Cancer through an Equipe Labellisée 2014 grant. We also thank the China Scholarship Council for a fellowship (CSC) to Guangkuan Zhao and Mingxiang Zhu. Our laboratory is a member of the Laboratory of Excellence LERMIT supported by a grant (ANR-10-LABX-33).

AUTHOR INFORMATION

Corresponding Author *E-mail: samir.messaoudi@u-psud.fr

Author Contributions

The manuscript was written through contributions of all authors. / All authors have given approval to the final version of the manuscript.

Notes

The authors declare no competing financial interest.


P Arguel

I Massiot
email: ines.massiot@laas.fr

Understanding the evolution of the geometry of silicon solar cells

Keywords: Silicon solar cells, Solar cell geometry, Photovoltaics engineering

Today, solar cells are very familiar to the broad public, and everyone has noticed their particular geometry resembling a square with rounded-off corners. However, the rationale behind this geometry is little known. This paper presents a precise study of the geometry of silicon solar cells and its evolution since their appearance in the early 1960s. Through a parametric approach taking into account different types of constraints, it is shown that the cell geometry is the result of an optimization process depending on both technical and economic aspects. The correlation between the parametric study and historical data shows that the photovoltaic panels that we see today on our roofs have been mainly shaped by the evolution of cost constraints over the past decades.

Introduction

The development of photovoltaic energy started at the end of the 1950s, motivated in particular by the crucial issue of powering up the first satellites [START_REF] Bailey | Solar Cells and Their Applications[END_REF]. A similar motivation arose for terrestrial applications in isolated areas opening up new prospects for research on solar cells at the beginning of the 1960s [START_REF] Starr | Applications of Photovoltaics[END_REF]. Most studies were focused on photovoltaic devices made of monocrystalline silicon leading to the best conversion efficiency at the time [START_REF] Green | Solar Cells: Materials, Manufacture and Operation[END_REF]. Later, the growth of photovoltaics and its various applications has relied on multiple technologies using different materials such as polycrystalline silicon, organic polymers or multijunction structures based on III-V semiconductors. Nevertheless, despite the diversity of architectures and materials, monocrystalline silicon remains currently the technology used in the fabrication of a third of commercial solar cells [4].

Solar cells are now very familiar to the broad public, as photovoltaic panels have started to cover more and more buildings. Everyone has noticed their particular geometry resembling, at first sight, a square with rounded-off corners as illustrated in figure 1(a). However, this is far from the original geometry of the disk directly obtained from the sawing of a cylindrical ingot of silicon in identical circular slices called wafers.

In this article, the evolution of the geometry of silicon solar cells is studied from its origins to the present day. It is shown that the current geometry results from an optimization process based on several parameters that are identified and described. The general framework of the study is the design of conventional photovoltaic panels (or modules) made of bidimensional arrays of identical elements, that are also called components, each containing one solar cell. Figure 1 illustrates such arrays based on circular (figure 1(b)) and square (figure 1(c)) cells. For the sake of simplicity, the process of optimizing the geometry of solar cells is carried out on a single component of the photovoltaic module. The photovoltaic module being an array of identical elements, the results of the optimization performed to design one component can then be extended to the whole module.

Description of the model to study the solar cell geometry

Definition of the optimization parameters

A single element of the module is defined as a square component with a side length L, enclosing a solar cell fabricated from a wafer of diameter D (figure 2(a)). The sides of the solar cell of surface area A are always considered to be tangent to the sides of the square component of surface area A = L 2 . A third useful parameter for this study is the length of the chord C as illustrated in figure 2(a).

Two extreme cases can then be distinguished: (i) the cell is directly made from the circular wafer (figure 2(b)) and (ii) the circular wafer is sawed to make a square cell of maximal area (figure 2(c)).

In the first case (circular cell with a diameter D), the side length of the component is equal to L = D and thus C = 0. The cell area is then given by A = A cir = πD 2 /4. In case (ii), the cell and the component overlap, leading to L = D/ √ 2 = C. The cell area can then be expressed as

A = A squ = D 2 /2.
In between those two cases, the cell is a regular truncated disk with an area A such as D 2 /2 ≤ A ≤ πD 2 /4. In order to characterize such a cell with respect to cases (i) and (ii), two parameters can be defined as follows:

• the usage rate of the silicon wafer U w that can be expressed as

U w = A A circ = 4A πD 2 ; (1)
• the surface coverage of the component C c that can be expressed as

C c = A A = A L 2 . ( 2
)
The case (i) of circular cells corresponds to A = A cir , leading to U w = 1 and C c = π/4 ≈ 0.79. Likewise, in the case (ii) of square cells with maximal area, C c = 1 and U w = 2/π ≈ 0.64. The parameters U w and C c thus vary inversely with respect to each other when the geometry of the solar cell goes from a circular wafer to a perfect square, or conversely. Designing the optimal geometry of the solar cell (and thus the module itself) requires to take into account a whole set of factors (material cost, fabrication cost, module footprint, power conversion efficiency, final cost of kWh, module weight, etc). The importance of these factors in the module design depends on the market module price and the application aimed for the module. For instance, the design process has to adapt to the specifications of building-integrated photovoltaics or embedded systems.

As an example, the case C c = 1 (square cells) could correspond to a module design combining a high conversion efficiency with a low footprint as sought after in the aerospace or space industry. On the other hand, for domestic photovoltaic applications where the solar panels are placed on a roof or a large terrace, the footprint is less restrictive and sawing a large part of the wafers can then be avoided. In this case, the U w parameter can get closer to 1 (circular cells).

The specifications dictated by the targeted application should be taken into account in the model for the geometrical design of the cell. To do so, a weighting coefficient γ can be defined to indicate the relative importance of U w with respect to C c :

γ = U w C c . (3)
Inserting the expressions for U w and C c given by ( 1) and ( 2) into (3) leads to the following relation between γ, L and D:

γ = 4 π L D 2 . ( 4
)
By definition, the range of values that γ can take is bounded by cases (i) and (ii):

D √ 2 ≤ L ≤ D =⇒ 2 π ≤ γ ≤ 4 π . (5)
The following study is thus restrained to γ ∈ [2/π; 4/π].

Expression of the solar cell area

Let us consider a solar cell as defined in figure 2(a). Its area A can be obtained from the initial area of the wafer A cir and the area A s of each of the four circular segments that have been sawed (depicted in shaded areas in figure 3):

A = A cir -4A s . ( 6
)
Such a circular segment can be described as the region of the wafer bounded by an arc of angle 2θ and by the chord of length C. The area A s can then be expressed as a function of A sec the area of the circular sector of angle 2θ and A tri the area of the triangular portion: As cos(θ) = L/D, the area of the circular sector is given by:

A s = A sec -A tri . ( 7
) D / 2 L/2 C/2 θ
A sec = θ π A cir = D 2 4 cos -1 L D . (8)
Furthermore, using D 2 = L 2 + C 2 , A tri can be expressed as:

A tri = CL 4 = L 4 √ D 2 -L 2 . ( 9
)
The expression for A s can then be obtained by combining [START_REF] Bp Solar | Data sheet of solar module BP[END_REF], ( 8) and (9):

A s = 1 4 D 2 cos -1 L D -L √ D 2 -L 2 . ( 10
)
Inserting [START_REF]Data sheet of solar cell ExcelTon II 4BB[END_REF] in (6) then gives an expression for A as a function of L and D:

A = πD 2 4 -D 2 cos -1 L D + L √ D 2 -L 2 . ( 11
)
The area A can subsequently be expressed as a function of the weighting coefficient γ by inserting (4) in [START_REF]Photowatt 2017[END_REF]:

A = πD 2 4 1 - 4 π cos -1 γπ 4 + 2 π γπ 1 - γπ 4 1/2 . ( 12
)
Finally, the parameters U w and C c can be obtained using ( 1) and ( 3):

U w = 1 - 4 π cos -1 γπ 4 + 2 π γπ 1 - γπ 4 1/2 . ( 13
)
C c = 1 γ 1 - 4 π cos -1 γπ 4 + 2 π γπ 1 - γπ 4 1/2 . ( 14
)

Results

Let us consider a silicon wafer with a given diameter value D. Figure 4(a) presents the variation of U w and C c as a function of the weighting coefficient γ (2/π ≤ γ ≤ 4/π). As expected, the two parameters vary inversely with respect to each other when γ changes.

Given the specifications of a targeted application for the solar module, the first step is to define a value for γ. Once the value of γ is known, the optimal values for U w and C c can be found using figure 4(a), thus leading to the best geometry for the solar cell.

Here, since the optimization process is carried out without specific constraints, the optimal value of γ is searched by maximizing the product U w * C c . According to figure 4(b) which displays the variation of U w * C c as a function of γ, the optimal value is γ = γ 0 ≈ 1.065. Using [START_REF] Neo | Data sheet of solar cell NS6WL[END_REF], the corresponding cell area is determined as:

A = A 0 ≈ 0.94 πD 2 4 . (15)
Likewise, the optimal values of the usage rate of the wafer U w and the surface coverage of the component C c can be obtained from ( 13) and ( 14) giving: U w 0 ≈ 0.94 and C c 0 ≈ 0.88. In practice, for manufacturing a photovoltaic module made of silicon wafers, the three parameters that are commonly used to define the geometry of the solar cell are D (diameter), L (full side length) and C (length of the chord). Figure 5 shows the variation of the normalized parameters L/D and C/D as a function of the coefficient γ. Table 1 gathers several values of L/D, C/D and the corresponding values of U w and C c .

In the case where γ = γ 0 , the corresponding optimal side length of the square component of the module can be calculated as a function of the wafer diameter D using (4), leading to: L = L 0 ≈ 0.915D. As for the length of the chord of angle 2θ, it can be determined using the following expression: The optimal value of C is then: C = C 0 ≈ 0.403D. Figure 6 shows the optimal geometry of the solar cell designed using L 0 and C 0 (figure 6(a)) and a module made of 3x3 cells with such a geometry (figure 6(b)).

C = √ D 2 -L 2 = D 1 - L D 2 1/2 = D 1 - γπ 4 1/2 . ( 16
Beyond the previous case where γ = γ 0 , figure 7 illustrates the evolution of the optimal geometry of the solar cell when the coefficient γ varies from 4/π to 2/π.

Discussion

In the 1960s and 1970s, the first solar panels made for terrestrial applications consisted of circular solar cells [START_REF] Salles | L'évolution industrielle des cellules solaires au silicium[END_REF]. The value of the weighting coefficient γ was then equal to 4/π. As mentioned earlier, current silicon solar cells have the geometry of a square with rounded-off corners (see figure 1(a)). In order to shed some light on this evolution of the solar cell geometry, technical data of solar cells over the last six decades were gathered using a review paper [START_REF] Salles | L'évolution industrielle des cellules solaires au silicium[END_REF] and commercial datasheets [6,[START_REF] Bp Solar | Data sheet of solar module BP[END_REF][START_REF] Sunpower | [END_REF][START_REF] Solar | Data sheet of solar cell[END_REF][START_REF]Data sheet of solar cell ExcelTon II 4BB[END_REF][START_REF]Photowatt 2017[END_REF][START_REF] Neo | Data sheet of solar cell NS6WL[END_REF]. For each solar cell, the values of the weighting coefficient γ and the diameter D were extracted. As a result, figure 8(a) shows the evolution of these two parameters (γ in full circles and D in open triangles) over time. In addition, figure 8(b) illustrates the evolution of the geometry with some examples of solar cells and the values of the corresponding geometrical parameters are given in table 2.

In order to illustrate the evolution of the solar module price over time, the standard metrics used is the cost per watt-peak expressed in $/Wp. The Watt-peak is the unit used to express the electrical power generated by a photovoltaic module under standard test conditions, i.e. a solar irradiance of 1000 W/m 2 with an air mass 1.5 (AM1.5) spectrum under normal incidence, and a temperature of 25 • C. Figure 9 displays the evolution of the cost per watt-peak for crystalline silicon solar cells over the last five Figure 7. Illustration of the evolution of the optimal cell geometry as a function of γ varying from 4/π (circular cell) to 2/π (square cell of maximal area). The component containing the cell is shown with a square for the following values of γ, from the outer edge to the center of the figure respectively: 4/π; 1.2; 1.1; γ = γ 0 ≈ 1.065 (dashed lines); 1; 0.9; 0.8; 0.7; 2/π. The dark grey area represents the initial circular wafer. decades, using data taken from [START_REF] Salles | L'évolution industrielle des cellules solaires au silicium[END_REF][START_REF] Swanson | A vision for crystalline silicon photovoltaics[END_REF][START_REF]IRENA 2012 Solar Photovoltaics Renewable Energy technologies: cost analysis series[END_REF][START_REF] Kavlak | Evaluating the causes of cost reduction in photovoltaic modules[END_REF]16]. The values given are the costs for the solar cell alone corrected to 2019 US dollars.

Comparing figures 8 and 9 provides an insight on the impact of the cost of the solar cell on its geometry. In the 1960s-1970s, the geometry of solar cells for terrestrial applications remained circular due to the high cost of silicon wafer fabrication. However, while keeping a circular geometry, the evolution and the growing control of silicon microelectronic technologies have led to an increase of the cell diameter from 20 mm in 1960 to 57 mm in 1975 (see figure 8(b)). Thereafter, silicon wafers have kept on growing larger and their production has accelerated to respond to the increasing demand of the booming microelectronic industry. This growth of silicon production has led to a significant drop in the fabrication cost of wafers translating into a drastic decrease in the cost per watt-peak as illustrated in figure 9, with a fortyfold cost reduction from 1970 to 1990. Releasing the cost constraint has induced a decrease of the usage rate of the silicon wafer, and thereby of the coefficient γ since the 1980s as can be clearly seen in figure 8a. Looking more closely at the evolution after 2000, one should note that the cost reduction is not monotonous as can be seen in the inset of figure 9. In particular, there was an increase of the cost per watt-peak in 2006 resulting from a shortage in the polysilicon material used to fabricate the monocrystalline silicon ingots [START_REF]IRENA 2012 Solar Photovoltaics Renewable Energy technologies: cost analysis series[END_REF]. The decrease in the cost slows down after 2010, with only a twofold cost reduction from 2010 to 2015. After 2015, the cost per watt-peak levels off around 0.4-0.8 $/Wp with slight variations following the silicon price. These low cost values translate into the fabrication of cells with γ ≈ 2/π corresponding to a square or quasi-square geometry (see figures 8a and 8b).

Along history, the geometry of silicon solar cells has gradually evolved from the initial circular wafers to regular truncated disks and, today, to square or quasi-square cells. This parametric study suggests that this evolution of the solar cell geometry can be mainly attributed to the economic constraint weighting on the usage rate of the silicon wafers used for the cell fabrication. [START_REF] Salles | L'évolution industrielle des cellules solaires au silicium[END_REF][START_REF] Swanson | A vision for crystalline silicon photovoltaics[END_REF][START_REF]IRENA 2012 Solar Photovoltaics Renewable Energy technologies: cost analysis series[END_REF][START_REF] Kavlak | Evaluating the causes of cost reduction in photovoltaic modules[END_REF]16]. All costs are corrected to 2019 US dollars.

Conclusion

The development of silicon solar cells for terrestrial applications began in the 1960s. Since that time, technologies have diversified and the geometry of monocrystalline silicon cells has gradually evolved from a disk to a square. This paper describes a simple parametric model to quantify the evolution of the solar cell geometry. The correlation with historical data suggests that the main parameter driving the solar cell geometry is the economic constraints weighting on solar cell fabrication. In the future, potential evolutions such as a drop in silicon market could add a new constraint and drive the optimization process towards another solar cell geometry. Interestingly, one can note that in the case of other technologies such as polycrystalline silicon or thin film materials relying on fabrication processes with a lower cost, the solar cells have kept a square geometry from the beginning.

Figure 1 .

 1 Figure 1. (a) Photograph of a monocrystalline silicon solar cell. (b,c) Schematics of a photovoltaic module based on a 3x4 array made with (b) circular and (c) square cells.

Figure 2 .

 2 Figure 2. Geometrical parameters of the model. (a) Schematics of a single square component of the module (side length L). The outline of the initial circular wafer of diameter D is shown with dashed lines. The grey area represents the area effectively covered by the cell, corresponding to a regular truncated disk of chord C. (b) Illustration of the case (i) where the cell is made of a circular wafer without sawing. (c) Illustration of the case (ii) where the circular wafer is sawed into a square cell of maximal area.

Figure 3 .

 3 Figure 3. Illustration of the different areas used in the calculation of the solar cell area A (dark grey area): the area A s of each of the four circular segments (dashed area), the A sec of the circular sector of angle 2θ (area surrounded in a thick line) and A tri the area of the triangular portion (light grey area).

)Figure 4 .

 4 Figure 4. Variation of (a) U w and C c and (b) U w * C c as a function of the weighting coefficient γ (γ ∈ [2/π; 4/π]). The value γ = γ 0 ≈ 1.065 corresponding to the maximum of U w * C c is highlighted.

Figure 5 .

 5 Figure 5. Variation of L/D and C/D as a function of the weighting coefficient γ (2/π ≤ γ ≤ 4/π). The values of the side length L 0 and the chord length C 0 as a function of D corresponding to γ = γ 0 are highlighted.

Figure 6 .

 6 Figure 6. Optimal geometry of (a) a single cell and (b) a module of 3x4 cells, corresponding to γ = γ 0 .

Figure 8 .

 8 Figure 8. Historical evolution of the geometry of monocristalline silicon solar cells. (a) Evolution of the weighting coefficient γ (full circles) and the cell diameter D (open triangles). The dotted lines act as a guide for the reader's eye to highlight the evolution of the value of γ from 4/π in the 1960s-1970s to approximately 2/π today. The horizontal thin line shows the 2/π bound corresponding to square cells. (b) Comparison of some examples of solar cell geometries. The data used in both figures come from[START_REF] Salles | L'évolution industrielle des cellules solaires au silicium[END_REF]6,[START_REF] Bp Solar | Data sheet of solar module BP[END_REF][START_REF] Sunpower | [END_REF][START_REF] Solar | Data sheet of solar cell[END_REF][START_REF]Data sheet of solar cell ExcelTon II 4BB[END_REF][START_REF]Photowatt 2017[END_REF][START_REF] Neo | Data sheet of solar cell NS6WL[END_REF].

Figure 9 .

 9 Figure 9. Historical evolution of the cost per watt-peak of crystalline silicon solar cells. Data prior to 1970 are not shown, they have little significance as the photovoltaic industry really started in the 1970s. The inset displays a zoom on the time range from 1990 to 2019. The data used in the figure come from[START_REF] Salles | L'évolution industrielle des cellules solaires au silicium[END_REF][START_REF] Swanson | A vision for crystalline silicon photovoltaics[END_REF][START_REF]IRENA 2012 Solar Photovoltaics Renewable Energy technologies: cost analysis series[END_REF][START_REF] Kavlak | Evaluating the causes of cost reduction in photovoltaic modules[END_REF]16]. All costs are corrected to 2019 US dollars.

Table 1 .

 1 Values of the geometrical parameters of the solar cell for notable values of the weighting coefficient γ.

	γ	L/D	C/D U w (%) C c (%)
	2/π	0.707 0.707	63.7	100
	0.7	0.741	0.671	69.7	99.6
	0.8	0.793	0.610	78.0	97.6
	0.9	0.841	0.541	85.1	94.6
	1	0.886	0.463	90.9	90.9
	γ 0 ≈ 1.065 0.915 0.403	94.1	88.3
	1.1	0.929	0.369	95.6	86.9
	1.2	0.971	0.240	98.8	82.3
	4/π	1	0	100	78.5

Table 2 .

 2 Values of the geometrical parameters of the commercial solar cells depicted in figure 8(b).

	Year D (mm) L (mm)	γ	Reference
	1960	20	20	4/π	[5]
	1975	57	57	4/π	[5]
	1998	113	103	1.05	[6]
	2002	127	108	0.92	[7]
	2010	160	125	0.78	[8]
	2018	210	157	0.7	[12]

Acknowledgments

The authors gratefully acknowledge Guy Vialaret (former engineer at LAAS-CNRS), Etienne Rapp (CNES) and Lionel Séguier (LAAS-CNRS) for their help in gathering historical data on photovoltaic panels and solar cells.


Mingxiang Zhu

Mouad Alami

Samir Messaoudi

Electrochemical nickel-catalyzed Migita cross-coupling of 1-thiosugars with aryl, alkenyl and alkynyl bromides

Here we report a simple route towards the synthesis of thioglycosides, in which electrochemical cross-coupling is used to form the S-C glycosidic bond from protected and unprotected thiosugars with functionalized aryl bromides under base free conditions. The reaction manifold that we

report here demonstrates the power of electrochemistry to access highly complex glycosides under mild conditions.

1-Thioglycosides have emerged as a privileged class of glycosides with a diverse range of potential applications. 1 These glycomimetics are known to be more robust towards enzymatic cleavage under biological conditions. 2 Representative active glycosides with a thioether linkage are discolosed in Figure 1, including hSGLT1 inhibitor, ligand of lectine A, cytotoxic Hsp90 inhibitor, antimicrobial agent as well as galactin-3-inhibitors. Moreover, thioglycosides are well known to be useful and 20 versatile reactive intermediates in organic synthesis. 3 At present, for the functionalization of glycosyl thiols, various catalytic systems based on palladium, 4 nickel 5 and copper 6 have been reported (Figure 1,a-b). 7 Despite their high interest, these methods require the use of basic conditions with high catalyst loadings (30 mol% in the case of Ni-catalysis) or precious metals such as Pd-catalyst with specially designed ligands. 8 In addition, an elevated reaction temperature (up to 120 °C cases of Pd-or Cu-catalysis), and long reaction times often limit the practicability or the scope of substrates. Moreover, in all these cases the coupling is effective with only aryl iodides, however the cross-coupling with aryl bromides is rather unexplored. To address these issues, we have reported recently a Ni/Ru dual photocatalyzed arylation of thiosugars with aryl iodides under base-free conditions (Figure 1,c). 9 However, the more available and less expensive aryl bromides are still unreactive under these photoredox conditions. Only few examples were reported with yield never exceeded 43% yield. Moreover, this method was limited to only protected thiosugars. This set of limitations clearly indicates that glycosyl thiols often behave as a rather special and 40 challenging class of thiol nucleophiles for transition-metalcatalyzed thio-functionalization. Because of the high importance of thioglycosides, the development of a general method for the coupling of both aryl bromides and iodides to protected and unprotected thiosugars under room temperature and without an external base remains a challenging task.

In recent years, electrosynthesis 10 has emerged as a remarkably efficient tool for organic cross-coupling reactions through the use of less precious catalysts, waste-free and inexpensive electric current, thereby avoiding stoichiometric amounts of toxic and costly chemical reagents limiting our impact on the environment.

10

Usually, this method was used successfully in C-H activation process fort the formation of C-C and C-heteroatom bonds (C-O, C-P, and C-N) through the directing group strategy. [START_REF] Zhang | C-heteroatom bond through electrochemistry[END_REF] However, electrochemical C-heteroatom bond formation via cross coupling of nucleophiles with aryl (pseudo)halides was less explored. 12 In particular, methods which use electrosynthesis to form C-S bonds are rare. To the best of our knowledges, only one study from Mei et al. reported the cross-coupling reaction of thiophenols with aryl halides. 13 The authors showed elegantly that this approach allows thiolation of aryl halides. However, this 20 method exhibited several limitations with regards to potential applications: i) the use of a large excess of the thiophenol partner (3 equiv) which can be prohibitive for larger scale reactions; ii) the scope with respect to the thiols was limited to unfunctionalized thiophenols, thus precluding many synthetic applications. Moreover, the S-arylation of functionalized and complex alkyl thiols bearing free hydroxyl or amino groups such as thiosugars under electrochemistry has never been studied, probably because of the polyhydroxylated complexity of carbohydrates. Herein, we became interested in whether the S-(hetero)arylation of 1-thiosugars could be performed by the electrochemical process. The present study is presented (Figure 1).

To initiate this study, we conducted this electrochemical thioglycosylation by coupling of tetra-O-acetylated 1-thio-β-D-35 glucopyranose 1a with 4-bromobenzotrifluoride 2a as a model study under various reaction conditions. Representative results from this study are summarized in Table 1. The reaction of 1a (1 equiv) with 2a (3 equiv) was first investigated by using constantcurrent electrolysis at 4.0 mA in the presence of NiBr 2 .diglyme 40 (10 mol%), 4,4'-di-tert-butyl-2,2'-bipyridine (di-tBubpy) (10 mol%), LiBr (4 equiv) in DMF at room temperature for 6 h within an undivided cell (Table 1, entry 2). Pleasingly, these conditions afforded the 3a in a moderate 54% yield as a single βanomer (J 1, 2 = 9 Hz) (See ESI). Interestingly, decreasing the amount of the bromide partner 2a from 3 equiv to 1 equiv furnished the thioglucoside 3a in a good 74% yield (entry 1).

Using other anodes such as Zn was not effective (entry 9). In addition, reaction with DMA instead DMF was less efficient and gave product 3a in only 47% yield (entry 11). Interestingly, when 50 the current was increased up to 8mA, the reaction time was shortened to 3h and 3a was isolated in 74% yield (entry 5). Finally, performing the coupling reaction with 20 mol% of NiBr 2 .diglyme and 20 mol% of di-tBubpy furnished 3a in 80% yield (entry 10). Negative controlling experiments demonstrated 55 the catalyst, the LiBr and the current are necessary (entries 6-8).

Table 1. Optimization of the coupling reaction of 1a [a] Entry Deviation from the standard conditions Yield (%) With these encouraging results, we investigated next, the scope and limitations for this electrochemical process by systematically varying the nature of the bromide electrophile 2. As showed in 65 Table 2, all the S-arylations of tetra-O-acetylated 1-thio-β-Dglucopyranose 1a proceeded cleanly and selectively in good yields. Cross-coupling of 1a with aryl bromides bearing various functions (-CN, -Cl, -F, -CF3, -CHO and -OMe) have been successfully achieved under room temperature to afford the 70 corresponding thioglycosides (3a-m) in yields up to 87%. In addition, the presence of an ortho substitution at the aromatic ring of the coupling partner do not affect the coupling process as compounds 3h and 3k were obtained in 80% and 77% yields, respectively. It is noteworthy that a selectivity was observed 75 when a tribromobenzene was used, and only a di-glycosylated product 3m was obtained in 70% yield even after running the coupling with longer reaction time (6h) and using 6 equiv of 1a.

One can be noted that, 3-bromonaphtalene is less reactive at current conditions than 2-bromonaphatelene giving the desired S-80 naphtylglycoside 3j in only 22% yield (vs 77% with 2bromonaphatelene). However this yield could be increased significantly (73% yield) when 3-iodonaphtalene. The same low reactivity was observed with p-bromoanisole vs p-iodoanisole (compound 3l, 15% vs 83%). Interestingly, couplings with 85 heteroaryl bromides derived from quinoline and pyridine have also been successful, furnishing 3n,o in 46% and 86% yields, respectively. Next, we moved on to investigate the reactivity of a series of alkene and alkyne bromides. Pleasantly, both aryl-and alkyl-substituted alkene bromides ((E)-1-(2-bromovinyl)-4-90 methoxybenzene, E-β-styryliodide, (E)-1-bromooct-1-ene)) and 2-bromo-1H-indene were transformed efficiently into alkenyl thioglycosides 3p-s in excellent yields. Finally, alkynylthioglycosides 3t-u were also easily prepared in good yields from corresponding 1-(bromoethynyl)-4-methoxybenzene and 3- (bromoethynyl)thiophene by this electrochemical method.

The successful synthesis of thioglucosides 3a-u from tetra-O-10 acetylated 1-thio-β-D-glucopyranose 1a and various aryl bromides under our optimized electrochemical conditions encouraged us to investigate other glycosyl thiols. As depicted in Table 3, this coupling reaction tolerates different sugars such as O-benzoylated 1-thio-β-D-glucopyranose, thio-β-Dgalactopyranose and 1-thio-β-D-fucopyranose which furnished the thioglycosides 4a-c in yields up to 95%. In addition, this coupling could be applied successfully to the more complex and biologically relevant saccharide derivatives thio-β-D-cellobiose and thio-β-D-maltotriose delivering 4d and 4e in 90% and 88% 20 yields, respectively. One of the most important task in this area of organometallic chemistry is to discover mild and general methods for easy functionalization of polyhydroxylated compounds with high 30 selectivity without using any protection/deprotection sequence.

To further show the synthetic use of our protocol, we evaluated whether the S-arylation of unprotected thiosugars could be realized by this electrochemical method. To our delight, the reaction of unprotected thio-β-D-glucose and thio-β-D-cellobiose 35 yielded the respective products 5a and 5b in good yields, and no side products arising from the O-arylation were observed (Scheme 1). To this end, we tested the reaction of L-cysteine with pcyanophenyl bromide and we were pleased to observe the formation of the S-arylated cysteine 5c in 67% yield (Scheme 1). This result represent the first example of diversification of cysteine under electrochemical approach and may open new way to the bioconjugation of peptides and protein through this methodology.

In summary, we described here the first electrochemical method of coupling of various anomeric glycosyl thiols with aryl bromides. This method demonstrates unprecedented reactivity in regard to bromide partners (heteroaryl, alkenynyl and alkenyl bromides) and selectivity, including reactions with unprotected thiosugars, and cysteine aminoacid. Further studies on the applications of this method on the S-arylation of more complex polysaccharides and cysteine containing peptides are ongoing.

Table 2

 2 Scope of Coupling of Thioglucose 1a with Halo(hetero)arenes, Alkenyl Halides and Alkynyl Bromides 2

	3 mmol), 2

This journal is © The Royal Society of Chemistry [year] Journal Name, [year], [vol], 00-00 | 3 a Reactions conditions: Mg anode, Ni foam cathode, 1a (0.b Isolated yield.

Table 3 .

 3 Scope of 1 thio-β-glycosyls 1b-g a

		a Reactions conditions: Mg anode, Ni foam cathode, 1b-f (0.3 mmol), 2
	25	(0.3 mmol), NiBr2.diglyme (10 mol%), di-tBubpy (10 mol%), LiBr (1.2
		mmol), DMF (0.3 M), undivided cell, constant current = 8 mA, N2
		protection, room temperature, 3 h.

b Isolated yield.

Acknowledgements

Authors acknowledge support of this project by CNRS, University Paris-Saclay, ANR (CarNuCat, ANR-15-CE29-0002), and by la Ligue Contre le Cancer through an Equipe Labellisée 2014 grant. We also thank the China Scholarship Council for a 20 fellowship (CSC) to Mingxiang Zhu.


Nacim Meslem
email: nacim.meslem@grenoble-inp.fr

Thao Dang
email: thao.dang@imag.fr

On the Reachable Set of Uncertain Closed Loop Discrete-Time Linear Systems

Keywords: Uncertain linear systems, closed-loop systems, estimated state feedback control, reachability analysis, interval analysis, interval estimators

This work tackles the reachability problem of uncertain discrete-time linear systems controlled by estimated state feedback regulators. First an augmented system is considered in order to include the effect of the estimation error on the behavior of the closed loop system. Then, based on interval analysis, an interval predictor is proposed to compute tight trajectory tubes that contain in a guaranteed way the actual state trajectory of the augmented system. Moreover, under the standard controllability and observability assumptions of linear systems, the convergence of the width of these trajectory tubes is shown.

INTRODUCTION

Reachability problems have been received a great attention these last years, for example [START_REF] Chisci | Recursive state bounding by parallelotopes[END_REF][START_REF] Daryin | Reachability approaches and ellipsoidal techniques for closed-loop control of oscillating systems under uncertainty[END_REF][START_REF] Guernic | Reachability analysis of linear systems using support functions[END_REF][START_REF] Kühn | Rigorously computed orbits of dynamical systems without the wrapping effect[END_REF]Kurzhanski andVaraiya, 2002, 2007;Pico andAliprantis, 2016, 2018;[START_REF] Rakovic | Reachability analysis of discrete-time systems with disturbances[END_REF][START_REF] Ramdani | A hybrid bounding method for computing an overapproximation for the reachable set of uncertain nonlinear systems[END_REF][START_REF] Ramdani | Computing reachable sets for uncertain nonlinear monotone systems[END_REF]. In fact, reachability is a fundamental issue that appears in different engineering fields where the safety requirement is crucial, since safety properties specify the (unsafe/bad) state regions that a system should not enter. In the context of dynamical systems, the reachability problem consists in characterizing all the possible state trajectories generated from a given set of initial states and governed by a (possible uncertain) dynamics. In the literature, this problem has been tackled for continuous and hybrid systems1 , in both continuous-time and discrete-time domains, where various set representations (for example2 , polytopes in [START_REF] Bak | Numerical verification of affine systems with up to a billion dimensions[END_REF], parallelotopes in [START_REF] Dreossi | Reachability computation for polynomial dynamical systems[END_REF], ellipsoids in [START_REF] Kurzhanski | Ellipsoidal techniques for reachability analysis of discrete-time linear systems[END_REF], zonotopes in [START_REF] Pico | Reachability analysis of linear dynamic systems with constant, arbitrary, and lipschitz continuous inputs[END_REF], boxes in [START_REF] Ramdani | A hybrid bounding method for computing an overapproximation for the reachable set of uncertain nonlinear systems[END_REF][START_REF] Chen | Flow*: An analyzer for non-linear hybrid systems[END_REF], polynomial sub-level set in [START_REF] Rodríguez-Carbonell | Generating polynomial invariants for hybrid systems[END_REF], level sets in [START_REF] Mitchell | A toolbox of Hamilton-Jacobi solvers for analysis of nondeterministic continuous and hybrid systems[END_REF]) have been used in specific set-membership reachability algorithms. Notice that most of these approaches are based on numerical methods to compute outer approximations of reachable sets; however other methods using SMT and theorem proving have also been developed by [START_REF] Platzer | KeYmaera: A hybrid theorem prover for hybrid systems (system description)[END_REF][START_REF] Kong | dReach: δ-reachability analysis for hybrid systems[END_REF]. Among the numerical algorithms, we can further classify them into two categories: the algorithms that incrementally approximate the reachable set for each time step, and the algorithms that aim at approximating the whole reachable set using invariance conditions. While the second category enables us to prove unbounded-time safety properties, approximation accuracy may not be a stringent requirement since only an approximation sufficient to prove a given safety property is needed. Concerning the algorithms of the first category, a major problem they face is conservatism linked to the wrapping effect [START_REF] Alefeld | Interval analysis: theory and applications[END_REF][START_REF] Corliss | Survey of interval algorithms for ordinary differential equations[END_REF][START_REF] Moore | Interval analysis[END_REF]. More precisely, in order to reduce the computational effort, at every step of these methods, the exact reachable set of the system is overestimated by super-sets that are easy to design and computationally tractable. Repeating this procedure for a large number of steps may lead to significant error accumulation that returns overly conservative state enclosures.

Recently, a new approach to characterize tight enclosures of the reachable set of a class of discrete-time dynamical systems has been introduced in [START_REF] Meslem | Interval predictors for a class of uncertain discrete-time systems[END_REF]. To face the wrapping effect, their approach proposes to compute enclosures of the exact reachable set at every time instant directly from the initial set. The main idea brought by their approach is the use of a pre-processed interval version of the analytic expression of the state response of discrete-time linear systems, where interval assessments of the state enclosures are not carried out in a recursive way. The same idea has been also applied in [START_REF] Meslem | A luenberger-like interval observer for a class of uncertain discrete-time systems[END_REF][START_REF] Meslem | Using set invariance to design robust interval observers for discrete-time linear systems[END_REF] to characterize the reachable set of the estimation error of discrete-time linear setmembership state estimators. In this work, we extend the use of their approach to closed loop uncertain discrete-time linear systems, where the system outputs are not required to obtain guaranteed enclosures of the actual state vector. More precisely, we consider the case of uncertain discretetime linear systems that are controlled by estimated state feedback regulators. Thus, the effect of the estimation error on the behavior of the closed loop system has to be considered while computing outer approximations of its reachable set. To achieve that, we propose an augmented system that includes the dynamics of the estimation error and the bounds of the feasible domains of the measurement noise and the state disturbance. It is worth pointing out that, to the best of our knowledge, this reachability problem of this class of uncertain systems has not been yet treated in the literature. However, the error accumulation problem is addressed in [START_REF] Guernic | Reachability analysis of linear systems using support functions[END_REF] for uncertain affine systems by separating the deterministic linear part (for which the reachable set can be approximated without error accumulation) and the uncertain input part. However, their approach uses zonotopes of increasing geometric complexity of which (measured by the number of generators) grows step after step, and if additional over-approximation or reduction (as in [START_REF] Combastel | A state bounding observer based on zonotopes[END_REF][START_REF] Guernic | Reachability analysis of linear systems using support functions[END_REF]) is used to keep a reasonable number of generators, the resulting error accumulation is hard to control. Our proposed interval approach can reduce error accumulation and provides a good compromise between complexity and accuracy. Furthermore, the approach using interval approximation can be extended to nonlinear systems as future work pointed out in the conclusion.

The remainder of this paper is structured as follows. Section 2 is devoted to the problem statement, where the considered reachability problem is formulated. A brief introduction on interval analysis is presented in Section 3. The main novelty of this work is stated in Section 4, where an interval predictor for the augmented closed loop system is introduced, and the convergence of the width of the computed state enclosures is examined. Section 5 illustrates the performance of the proposed reachability approach.

PROBLEM STATEMENT

Consider a discrete-time linear system subject to additive state disturbances and measurement noises that is described by

x k+1 = Ax k + Bu k + w k y k = Cx k + v k (1)
where x k ∈ R n stands for the state vector, u k ∈ R nu is the input vector, w k ∈ R n stands for an unknown but bounded state disturbance vector, y k ∈ R ny is the output vector and v k ∈ R ny stands for the unknown but bounded measurement noise vector. The matrices A ∈ R n×n , B ∈ R n×nu and C ∈ R ny×n are known and constant. Throughout this paper, system (1) is assumed to be controllable and observable.

Thus, based on this standard assumption, estimated statefeedback controllers can be designed to drive system (1). These controllers are based on the Luenberger observer and have the following dynamical structure:

xk+1 = (A -LC)x k + Bu k + Ly k u k = -Kx k + Gr k (2)
where xk ∈ R n is the estimated state vector and r k ∈ R ny stands for the desired setpoint. The gain matrices L ∈ R n×ny , K ∈ R nu×n and G ∈ R ny×ny stand respectively for the observer gain, the state-feedback gain and the feedforward gain. Notice, in the ideal case where system

(1) is free from the state disturbances and measurement noises (w k = 0, v k = 0), the linear controller (2) succeeds to impose the desired performance (settling time, tracking error,. . . ) to the controlled system. Now, a legitimate question that arises is about the assessment of the robustness of the proposed controller against state disturbances and measurements noises. That is, does the controller keep its theoretical performance in the presence of state disturbances and measurement noises that could affect both system dynamics and state estimation algorithm?

Based on the reachability analysis, this issue could be tackled in the bounded error context. That is, the state disturbance and the measurement noises are considered unknown but bounded with known bounds.

Hence, relying on this assumption, a reliable decision about the expected performance of the linear controller (2) could be taken by the use of the reachability analysis.

In this work, we introduce a set-membership algorithm that yields an outer approximation of the reachable set of the closed-loop system described by ( 1) and ( 2). First, we define by e k = x k -xk the estimation error. Then, under the control law (2), the dynamics in (1) can be rewritten as:

x k+1 = (A -BK)x k + BKe k + BGr k + w k (3)
On the other hand, the dynamics of the estimation error is described as follows:

e k+1 = (A -LC)e k + (I n -L) w k v k (4)
where I n stands for an identity matrix of dimension n. Thus, from (3) and (4) the closed-loop system (1)-( 2) is equivalent to the following coupled (augmented) system:

x k+1 e k+1 = A -BK BK 0 n,n A -LC x k e k + I n 0 n,ny I n -L w k v k + BG 0 n,ny r k (5)
where by 0 l,s one denotes a zero matrix of dimension l × s.

Thereby, the reachable set of the closed-loop system can be defined as follows.

Definition 1. Let x 0 ∈ X 0 ⊂ R n and e 0 ∈ E 0 ⊂ R n . The reachable set of the system (5) at the time instant k is the set Ω k = {(x k , e k ) | e 0 ∈ E 0 , x 0 ∈ X 0 }
of all the solutions to (5) that are:

• generated from the initial sets X 0 and E 0 ,

• driven by the desired set-point input vector r k and • affected by the exogenous bounded signals w k , v k .

Moreover, the reachable set of the system (5) over a finite period k ∈ {k 0 , . . . k m } is the trajectory set that can be defined by: Ω {k0,...,km} = {Ω k0 , . . . , Ω km }

In what follows, interval analysis [START_REF] Alefeld | Interval analysis: theory and applications[END_REF][START_REF] Jaulin | Applied interval analysis: with examples in parameter and state estimation, robust control and robotics[END_REF][START_REF] Moore | Interval analysis[END_REF] will be used to compute an outer approximation of the trajectory tube defined in (6).

INTERVAL ANALYSIS

One way to represent and manipulate sets of real values is interval arithmetic. By definition, a real interval denoted by [x] = [x, x] is a connected and closed subset of R, [x] = {a | x ≤ a ≤ x, x, x ∈ R} where x and x stand respectively for its lower and upper endpoints. The set of all real intervals of R is denoted by IR. Moreover, the basic arithmetic operations • ∈ {+, -, ×, ÷} on real numbers are extended to the interval case, over IR, according to the following set-membership rule 7) More formally, the set-membership definition (7) can be detailed as follows for • ∈ {+, -}:

∀[x], [y] ∈ IR [x] • [y] = {a • b | a ∈ [x], b ∈ [y]} (
[x] + [y] = [x + y, x + y], [x] -[y] = [x -y, x -y] (8)
On the other hand, the width of an interval is defined by

w([x]) = x -x.
Note that, interval analysis are also extended to vector and matrix expressions in [START_REF] Alefeld | Interval analysis: theory and applications[END_REF][START_REF] Moore | Interval analysis[END_REF]. In this work, only addition and subtraction of two interval vectors of the same dimension and the multiplication between a real matrix and an interval vector of appropriate dimensions are needed. Thus, the arithmetic operation • ∈ {+, -} between two interval vectors [x], [y] ∈ IR n provides an interval vector defined as follows:

[z] = [x] • [y] = {a • b | a ∈ [x], b ∈ [y]} (
p = M + x -M -x and p = M + x -M -x (12)
where the element-wise non-negative matrices M + and M -are determined by M + = max{M, 0 m,n } and M -= M + -M (13) In ( 13), the max operator is applied component-wise. Finally, the maximal diameter of an interval vector [x] ∈ IR n is given by,

W ([x]) = max 1≤i≤n w([x i ]) (14)

MAIN RESULTS

Before introducing the main contribution of this work, let us state our assumptions. Assumption 1. The initial state x 0 of the system (1) belongs to a bounded box:

x 0 ∈ [x 0 ] = [x 0 x 0 ],
where the lower and upper bounds respectively x 0 , x 0 are assumed to be known. Remark 1. To reduce the impact of the estimation error at the transient regime, the initial state x0 of the Luenberger observer in (2) has to be picked from the box [x 0 ].

Assumption 2. The state disturbances w k and the measurement noises v k are assumed to be unknown signals but bounded with known bounds:

∀k ≥ 0, w k ∈ [w w] and v k ∈ [v v] (15
) Assumption 3. The pairs (A, B) and (A, C) of system (1) are respectively controllable and observable. That is, there exist an observer gain and a state feedback gain with appropriate dimensions such that the matrices A -LC and A -BK are Schur stable.

For the sake of simplicity, let us introduce the following matrices and vectors before presenting the main contribution of this work:

N = A -BK BK 0 n,n A -LC , F = I n 0 n,ny I n -L H = BG 0 n,ny , g k = x k e k (16)
Notice, based on the definition of g k in ( 16), in what follows the state vector x k and the estimation error vector e k will be re-denoted by 1 g k and 2 g k , respectively. That is,

g k = 1 g k 2 g k (17)
and

g k+1 = Ng k + Fd k + Gr k (18) where d k = (w k , v k ) T ∈ ([w], [v]) T = [d].
Proposition 1. Let Assumptions 1, 2 and 3 be satisfied and define:

[f 0 ] = F[d], [g 0 ] = [x 0 ] [e 0 ] and b 0 = Hr 0 (19)
Then, there exist a state feedback gain K and an observer gain L such that, for all k ≥ 1, the following interval predictor:

[g k ] = N k [g 0 ] + [f k-1 ] + b k-1 [f k ] = N k [f 0 ] + [f k-1 ] b k = Nb k-1 + Hr k , (20)
yields a converging outer approximation of the reachable set of the closed-loop system (5). That is,

∀ x 0 ∈ [x 0 ] and ∀ e 0 ∈ [e 0 ]: Ω {0,...,s} ⊆ {[g 0 ], . . . , [g s ]} (21) and lim k→∞ W ([g k ]) = c ( 22
)
where c is a positive constant.

Proof. Note that ( 20) is obtained by iterating successively the interval extension of the dynamic equation ( 18). That is, (20) can be rewritten as follows,

[g k ] = N k [g 0 ] + k-1 i=0 (N i [f 0 ]) + k-1 i=0 N k-1-i Hr i (23)
On the other hand, based on Assumption 2 and the definition of [f 0 ], we have ∀k ≥ 1:

k-1 i=0 N k-1-i F w k v k ∈ k-1 i=0 (N i [f 0 ]) (24)
Therefore, we can affirm that the exact solution of the augmented system (5) given by,

g k = N k g 0 + k-1 i=0 N k-1-i F w i v i + k-1 i=0 N k-1-i Hr i (25) at each time instant k, is included inside the box [g k ].
Moreover, the first part of the box [g k ] represents an outer approximation of the reachable set of the augmented system. That is, ∀k ≥ 1

x k ∈ [ 1 g k ] and {x 0 , x 1 , . . . , x s } ⊆ {[ 1 g 0 ], [ 1 g 1 ], . . . , [ 1 g s ]}
(26) This completes the framing proof of this proposition. Now, let us consider the convergence proof. Since system (1) is assumed controllable and observable, there exist a state feedback gain K and an observer gain L such that the matrix N is Schur stable. This implies that, lim

k→∞ N k = 0 (27)
Therefore, we can claim that the second interval equation in ( 20) has a fixed point. That means, lim

k→∞ [f k ] = lim k→∞ [f k-1 ] = [f * ] (28)
Thus, according to the first equation in ( 20), we can affirm that lim

k→∞ [g k ] = lim k→∞ W N k [g 0 ] + [f k-1 ] = W ([f * ]) = c (29)
This ends the convergence proof.

Hereafter some useful remarks about Proposition 1. Remark 2.

• Note that the convergence of the interval predictor ( 20) is ensured by the stability of the augmented matrix N, while iterative interval methods required the stability of the element-wise non-negative matrix |N| to provide a non-diverging outer approximation of the reachable set of the augmented system (5). This highlight the efficiency of the proposed method against the wrapping effect. • The contribution of this work is the new numerical scheme (20) that allows avoiding the wrapping effect. Moreover, to obtain tighter outer approximation of the reachable set of the augmented system (5), we suggest to use zonotopic sets in (20) instead of interval vectors.

• The proposed reachability method can be extended at a moderate effort to other types of controllers described by state space models. • Algorithm (20) can be also applied to detectable and stabilizable systems.

ILLUSTRATIVE EXAMPLE

Consider the following discrete-time uncertain system,

x k+1 = 1 0.63 0 0.37 x k + 0.37 0.63 u k + w k y k = ( 1 0 ) x k + v k (30)
where, the state disturbances w k and the measurement noise v k are assumed unknown but bounded with well known bounds: (-0.02, -0.02

) T ≤ w k ≤ (0.02, 0.02) T -0.01 ≤ v k ≤ 0.01 (31)
In addition, the initial state of ( 30) is poorly-known but belongs into a bounded box:

x 0 ∈ [0.8 1.5] × [1.2 3] (32)
Since this system is observable and controllable, an estimated state-feedback controller can be designed to drive its output to a desired set-point trajectory, r k = 1.5 sin(0.2k)

To do that, first the uncertainties of the system are neglected and LQR (Linear-Quadratic Regulator) method is applied to design both state feedback and observer gains: K = (0.8743 0.7237)

L = (1.0841 0.1278) T (34)
and then the feed-forward gain G is obtained by

G = (C(I 2 -A + BK) -1 B) -1 = 0.8743 (35)
The considered initial state for the Luenberger observer is x0 = (0, 0) T . Thus, the initial box of the estimation error is:

[e 0 ] = [x 0 ] -x0 = [x 0 ] (36)
Now, it is of great interest to be able to evaluate a priori the impact of the neglected uncertainties on the behavior of the closed-loop system. To do that, the result of Proposition 1 is useful. In fact, based on the outer approximation of the reachable set of the closed loopsystem all possible behaviors of the system can be observed and analyzed a priori.

Simulation results

In this subsection we have applied the interval predictor (20) to compute an outer approximation of the reachable set of the uncertain system (30) driven by the proposed estimated state feedback controller (2),( 34),(35).

Figure 1 shows the time evolution of both state variables x 1 (k) and x 2 (k). The solid lines in the top pictures stand for the outer enclosure of each state variable while the dashed lines represent the actual state variables of the system generated from the initial point x 0 = (1, 2) T . On the other hand, the bottom pictures show an enclosure of the reachable set of the estimation error generated by the augmented system.

-

1 0 1 2 3 4 [x 1 (k)] -1 -0.5 0 0.5 1 1.5 2 2.5 3 [x 2 (k)]
Fig. 2. The outer approximation of the reachable set of the closed-loop system plotted in the state plan (x 1 , x 2 ).

Figure 2 shows the same outer approximation of the reachable set of the controlled system (30) in the plan phase (x 1 , x 2 ). The rectangles represent the boxes [x k ] of the state vector x k at each time instant k and the solid line stands for the actual state trajectory of the controlled system.

Comparison with other methods

To show the interest of the proposed algorithm (20), the results of four reachability methods are considered in this subsection. The efficiency of each method is measured by the width of its computed outer approximation of the reachable set of the augmented system. The first method is a direct application of algorithm (20) (with interval computation). The second one is a zonotopic version of this algorithm, where zonotope sets are used instead of interval vectors. In this case the maximal number of the segments of each zonotope is d = 5. The third and the fourth methods are respectively iterative interval and zonotopic methods applied directly to the dynamics of the augmented system. Note that, for the zonotopic iterative method the maximal number of the segments of each zonotope is d = 5. As shown in Figure 3, algorithm (20) provides more accurate outer approximations than that computed by the iterative methods. Indeed, for this example, the smallest width of computed outer approximations is given by the zonotopic version of algorithm (20). Note that, with a high maximal number of segments d = 90, the zonotopic iterative method provides similar results than that obtained by the zonotopic version of algorithm (20). However, using a high value of d leads to a huge oneline computational effort.

CONCLUSION

In this paper, an interval predictor has been proposed to compute tight outer approximations of the reachable set of a class of uncertain discrete-time linear systems driven by linear estimated state feedback controllers. The considered closed loop systems are rewritten as augmented systems that encompass the dynamics of the estimation error. This result is mainly based on a recent non-conservative interval scheme developed to reconstruct all the possible behaviors of a class of uncertain dynamical systems. Moreover, under the standard observability and controllability assumptions of linear systems, the convergence of the size of the computed state trajectory tubes is proven.

In forthcoming work, we will investigate two directions.

One is combining this interval predictor with the consistency techniques in order to certificate numerically the performance of linear controllers. The other is to extend our method to some class of nonlinear systems.

  9)where the two first interval equations in (8) are applied element-wise, in both cases, to compute the lower z ∈ R n and upper z ∈ R n bounds of[z].The parallelotope P defined byP = {Mx | M ∈ R m×n , x ∈ [x] ⊂ IR n }(10) can be outer approximated by the following interval vector[p] = M[x] (11) Notice, the upper and lower bounds of the box [p] can be computed directly by:

For

  this example, the design matrices N, H and F of the interval predictor (20) are: box [f 0 ] is given by the following interval Cartesian product: [f 0 ] = [-0.02 -0.02] × [-0.02 -0.02] . . . ×[-0.0308 0.0308] × [-0.0213 0.0213] (38)

Fig. 1 .

 1 Fig. 1. Top figures show guaranteed enclosures for each state variables x 1 and x 2 . Bottom figures represent all possible estimation errors considered in the augmented system.

Fig. 3 .

 3 Fig. 3. The width of each component of the outer approximations of the reachable set computed by different methods.

This problem for discrete systems is traditionally treated in model checking originated in computer science.

The reader is referred to a survey in the chapter "Verification of Hybrid Systems" of the book entitled "Handbook of Model Checking"[START_REF] Doyen | Verification of hybrid systems[END_REF].


Mingxiang Zhu

Mouad Alami

Samir Messaoudi

Room Temperature Pd-Catalyzed Synthesis of 1-(Hetero)aryl Selenoglycosides

A general protocol for functionalization of anomeric selonate anion at room temperature has been reported. By using PdG3 XantPhos catalyst, the cross-coupling between the in situ generated glycosyl selenolate and a broad range of (hetero)aryl and alkenyl iodides furnished a series of functionalized selenoglycosides in excellent yields with a perfect control of the anomeric configuration.

1-selenoglycosides constitute a privileged class of glycosides and are considered as a one of the promising family of glycomimetics with some significant enhanced in vivo stabilities. Although it has been less explored in carbohydrate chemistry, they received during the last years an increasing attention due to their versatile biologically significant applications [1] including ligand based-therapeutics to carbohydrate-binding proteins, [START_REF] Mcdonagh | Lewis Acid Induced Anomerization of Se-Glycosides. Application to Synthesis of α-Se-GalCer[END_REF] X-ray crystallographic studying of carbohydrate-protein complexes, [3] and 77 Se-NMR spectroscopy handles for carbohydrate-proteine interactions (compound 1, Figure 1). [4] Since the discovery of the first selenoglycoside (2) (Figure 1) in rat and human urine as a hepatic Se-metabolite, [START_REF] Kobayashi | Selenosugars are key and urinary metabolites for selenium excretion within the required to low-toxic range[END_REF] the design and synthesis of bioactive seleglycosides have gained more attention within the chemistry community. Selected biologically active 1-selenoglycosides are represented in Figure 1, including Se-KRN7000 (3) [START_REF] Mcdonagh | Lewis Acid Induced Anomerization of Se-Glycosides. pplication to Synthesis of α-Se-GalCer[END_REF] as analogue of the potent immunostimmulant α-GalCer (KRN7000), the antiinvasive and anti-metastatic organoselenium glycoconjugate (4) [START_REF] Bijian | Synthesis and biological activity of novel organoselenium derivatives targeting multiple kinases and capable of inhibiting cancer progression to metastases[END_REF] which targets multiple kinases in preclinical models, the di-β-D-galactopyranosyl-diselenide (5) [START_REF] André | Thio-and selenoglycosides as ligands for biomedically relevant lectins: Valency-activity correlations for benzene-based dithiogalactoside clusters and first assessment for (di)selenodigalactosides[END_REF] with an antitumoral activity and ligand for lectin and the selenoglycoside AcSG (6) [START_REF] Kónya | Aralkyl selenoglycosides and related selenosugars in acetylated form activate protein phosphatase-1 and -2A[END_REF] as activator of phosphatases 1 and 2-A, two proteins involved in cancers. From the organic synthesis point of view, selenoglycosides have been investigated as unique glycosyl donors in glycosylation reactions. [START_REF] Witczak | Synthetic Applications of Selenium-Containing Sugars[END_REF] The usual approach employed for the synthesis of selenoglycosides relies on the use of selenium sources in the nucleophlic substitution of 1-halosugars [11] (Figure 2A) or a nucleophilic opening of the Danishefsky 1, 2-anhydro sugars (Figure 2B). [12] Azidophenylselenation of glycals, [13] was also as an efficient method to prepare selenoglycosides bearing an azido group at the C2-position. However, selenium nucleophiles such as arylselenols are not commercially available and must be prepared usually from potassium selenocyanate (KSeCN) and aryliodonium salts followed by the reduction of the Se-CN bond in the presence of reducing agent such as NaBH 4 . [11e] Another major drawback of this approach is the use of toxic, unstable and unpleasantly smelling arylselenols and diselenides. Ando, Ishihira and co-workers [14] developed an elegant method to prepare various alkyl Se-glycosides by an in situ generating of an anomeric selenolate anion from β-pmethylbenzoylselenoglycoside, which reacts with alkyl halides (Cl, Br and I) to yielde selenoglycosides with retention of the anomeric stereochemistry (Figure 2C). However, this method is limited to only alkyl halides. Only one example for the synthesis of aryl-selenoglycoside was reported by this approach using the highly activated 4-nitro fluorobenzene substrate in a nucleophilic aromatic substitution reaction (SN Ar ). Recently, Walczak and co-workers [1a] described an original approach (Path D) to the preparation of selenoglycosides based on the coupling of glycosyl stannane with symmetrical diselenide in the presence of 1.5 equiv of CuCl. This method is clearly attractive. However, it necessitates high reaction temperature (110 °C) and requires the use of the toxic alkyltin species in a multi-steps approach to prepare the starting 1stanylated sugars.

To fully harness the potential of this class of carbohydrates there is a strong impetus to develop mild and general reactions for the synthesis of complex selenoglycosides. As part of our interest to explore the reactivity of sugars under transitionmetal catalysis, [START_REF]Recent Advances in Transition Metal-Catalyzed Functionalization of 1-Thiosugars. Recent Advances in Transition-Metal-Catalyzed Functionalization of 1-Thiosugars[END_REF] combined with the lack of general method to Se-glycosides, we became interested in developing an efficient and simple Pd-catalyzed cross-coupling protocol for the preparation of (hetero)aryl-selenoglycosides which may occur under mild reaction conditions. Catalytic methods generally avoid the use of harsh reaction conditions and maintain the compatibility with most organic functions. However, the use of selenium nucleophilic partners in Pd(0)-cross couplings is highly challenging as they are well known to be poison of the catalyst through the formation of off-cycle complexes. [16] Herein we report a direct and programmable synthesis of selenoglycosides via Pd-G3-XantPhos catalyzed coupling of an in situ generated anomeric selenolate-anion with various (hetero)aryl and alkenyl iodides at room temperature.

In the first part of this study, the coupling of tetra-O-acetylated β-p-methylbenzoylselenoglycoside 1a (1 equiv) with piodoanisole 2a (2 equiv) was used as a model study, and various reaction conditions were examined. Representative results are summarized in Table 1 and the full optimization is reported in SI. The coupling of 1a with 2a was first investigated under our previously reported protocol for the coupling of thiosugars [START_REF] Bruneau | Stereoretentive Palladium-Catalyzed Arylation, Alkenylation, and Alkynylation of 1-Thiosugars and Thiols Using Aminobiphenyl Palladacycle Precatalyst at Room Temperature[END_REF] using PdG3-XantPhos precatalyst (10 mol%).

Cs 2 CO 3 (3 equiv) and piperazine (1.2 equiv) were added and the reaction was performed in THF at room temperature (Table 1, entry 1). By the action of piperazine and Cs 2 CO 3 , the βselenolate anomer anion was readily generated in situ (according to the Ando and Ishihira methods, [14] ) and fully converted under our conditions to the β-selenoglycoside 3a in 95% yield as a single β-anomer (J 1, 2 = 9 Hz). Decreasing the catalyst loading from 10 mol% to 5 mol% led to 3a in a slightly lower yield (81% yield, entry 2). With these encouraging results, we continued the optimization of the reaction conditions with respect to other parameters (base, solvent, reaction time, amount of the reagents). We confirmed that Cs 2 CO 3 is the base of choice since the use of CsF or NaOAc produced the desired selenoglycoside 3a in 83% and 65% yields, respectively. Moreover, performing the cross-coupling in other polar solvents such as dioxane, DCE or DMF produced the desired product 3a in yields ranging from 83% to 95% (entries 5-7). Finally, performing the reaction for 1 h furnished 3a in only 80% yield (entry 8), while 95 % yield was obtained when the reaction time adjusted to 2 h (entry 9). A control experiment showed that the PdG3-XantPhos precatalyst was essential for this coupling since no reaction occurred when Pd(OAc) 2 (10 mol%) and XantPhos (10 mol%) were used (see SI). With these results in hand, we investigated the scope for this coupling by varying the nature of the aryl iodide partner 2 and the p-methylbenzoylselenyl glycosides 1a-g (Scheme 1). Pleasantly, this coupling proceeded cleanly in excellent yields. Various electron-rich and electron-deficient aryl iodides having para-and meta-substitution effectively underwent reaction with β-p-methylbenzoylselenoglycoside 1a in yields up to 99% (products 3a-q). Interestingly, this coupling tolerates various reactive functional groups such as isopropyl (3f), nitrile (3h), halogens (3j), nitro (3k), aldehyde (3l), ketone (3o) and ester (3m, n and 3p, q). Moreover, the coupling is not limited to aryl halides, but works also with halogenated alkenes such as E-βstyryl iodide (compound 3g). In addition, the reaction of 1a with heteroaryl halides such as 4-iodopyridine and 5-iodoindole have been performed successfully, furnishing 3r-s in 87% and 85% yield, respectively. Ortho-substituted aryl iodides such as ortho-methoxyl iodobenzene, ortho-cyano iodobenzene and ortho-CF3 iodobenzene were not effective coupling partners Reaction conditions: A sealable tube was charged with β-pmethylbenzoylselenoglycoside 1a (1 equiv, 0.1 mmol), 2 (2 equiv), PdG3-XantPhos precatalyst (10 mol %), piperazine (1.2 equiv), Cs2CO3 (3 equiv) in dry and degassed THF (0.1 M) under Ar for 2 h. a Yield of isolated product.

probably due to the hindrance effect, implying that the oxidative addition of aryl iodides to palladacycle is not facile. Finally, this method tolerates other selenosugars such selenogalactose, selenofucose and selenocellobiose or OBz-protected selenoglucose which were successfully coupled with 2a or 2d to give β-Se-glycosides 3t-v in good yields. Besides p-Figure 3. Strategy for the trapping of the p-methylbenzoyl group of 1a by using a double action (electrophilic-nucleophilic) of the partner methylbenzoylselenyl β-glycosides α-glycoside is also an effective substrate for this methodology giving the desired Seglycosides 3x in excellent yield and with exclusive αselectivity. Of note attempts to react the unprotectd selenoglycoside failed: 3y was never detected and only the degradation of starting materials was observed.

In a next part of this study, we focused our attention on demonstrating whether our cross-coupling could be employed with functionalized aryl iodides bearing a secondary amino group (Figure 3). We may assume that these bis-electrophilicnucleophilic partners could be involved in two concomitant reactions: at first by a nucleophilic action, the secondary amine such as piperazine, traps the p-methylbenzoyl group of the selenoglycoside 1a and generates the anomeric selenolate anion. Then, this later will react with the electrophilic part (C-I bond) via the Pd-catalzyed Se-C(sp 2 ) bond forming reaction. By this way, all the atoms of the reactants will be incorporated into the final desired products, resulting in high atom economy and environmental process which avoid the generation of wastes. As shown in Scheme 2, the proof-of-concept of this green process was demonstrated by coupling of bis-electrophilicnucleophilic partners 5a-c with pmethylbenzoylselenoglycoside 1a in the presence of PdG3-XantPhos (10 mol%), Cs 2 CO 3 (3 equiv) in THF at room temperature. Under these conditions, the synthesis of phenylpiperazine functionnalized selenoglycosides 6a and 6c were achieved with 90% and 82% yields, respectively. This finding is of high interest since piperazine is known as one of the most promising ring displaying various biological activities for the treatmentof Parkinson's disease, Alzheimer's disease and depression. [18] In addition, this methodology led to the synthesis of the N-benzyl selenoglycoside 6b in good 95% yield (Scheme 2).

In summary, we have shown that anomeric selenolate anions are competent nucleophilic partners in the Pd-catalyzed crosscoupling reactions. To our best of our knowledge the C(sp 2 )Se bond of (hetero)aryl selenoglycosides was formed, for the first time, directly by a Pd-catalyzed cross-coupling of selenosugar with aryl halides at room temperature. Because of the mildness of the reaction conditions, this protocol tolerates a wide range of functional groups and proceeds in good to excellent yields. In addition, a variety of selenosugars could be used. This method opens news opportunities for using glycosyl selenolate in synthetic methodology and medicinal chemistry.

ASSOCIATED CONTENT Supporting Information

The Supporting Information is available free of charge on the ACS Publications website. Experimental procedures, spectroscopic data and NMR spectra of new compounds.

Figure 1 .

 1 Figure 1. Example of Biologically Active Selenoglycosides

Figure 2 .

 2 Figure 2. Strategies for the synthesis of selenoglycosides.

  Scheme 1 Scope of Coupling of β-pmethylbenzoylselenoglycoside 1a with Halo(hetero)arenes and Alkenyl Halides 2

Scheme 2

 2 Scheme 2 Scope of Coupling of 1a with Functionalized Aryl iodides Bearing a Secondary Amino Group 5a-c

Table 1

 1 Optimization of the Coupling Reaction of 1a with 2a a

	entry	base	solvant	Time (h)	3a (%) b
	1	Cs2CO3	THF	12	95
	2	Cs2CO3	THF	12	81 c
	3	CsF	THF	2	83
	4	NaOAc	THF	2	65
	5	Cs2CO3	dioxane	2	95
	6	Cs2CO3	DCE	2	83
	7	Cs2CO3	DMF	2	89
	8	Cs2CO3	THF	1	80
	9	Cs2CO3	THF	2	95

a A sealable tube was charged with β-p-methylbenzoylselenoglycoside 1a (1 equiv, 0.1 mmol), iodoanisole 2a (2 equiv), [PdG3-XantPhos precatalyst (10 mol %), piperazine (1.2 equiv), base (3 equiv) in dry and degassed solvent (0.1 M). b Yield of isolated product. c 5 mol% of PdG3-XantPhos were used.

ACKNOWLEDGMENTS

Authors acknowledge support of this project by CNRS, University Paris-Sud, ANR (CarNuCat, ANR-15-CE29-0002), and by la Ligue Contre le Cancer through an Equipe Labellisée 2014 grant. We also thank the China Scholarship Council for a fellowship (CSC) to Mingxiang Zhu.

AUTHOR INFORMATION

Corresponding Author *E-mail: samir.messaoudi@u-psud.fr

Author Contributions

The manuscript was written through contributions of all authors. / All authors have given approval to the final version of the manuscript.

Notes

The authors declare no competing financial interest.


F Zhang

Haizhou Xue

J Keum

Alexandre Boulle

Yanwen Zhang

W Weber

teaching and research institutions in France or abroad, or from public or private research centers.