Emmanuel Anger

Hiroki Iida

Tomoko Yamaguchi

Koutarou Hayashi

Daisuke Kumano

Jeanne Crassous
email: jeanne.crassous@univ-rennes1.fr

Nicolas Vanthuyne

Christian Roussel

Eiji Yashima
email: yashima@apchem.nagoya-u.ac.jp

Kotaro Hayashi

Synthesis and chiral recognition ability of helical polyacetylenes bearing helicene pendants

Keywords: helicene, polyacetylene, helical structure, chirality, chiral recognition

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INTRODUCTION

Helicenes are unique polycyclic aromatic compounds and are inherently chiral because of their nonplanar screw-shaped skeletons. 1 Due to such intriguing features of the helicenes that involve helical chirality, conjugated π-systems, and fluorescent and chiroptical properties, 1,2 optically active helicene-containing polymers have the potential to be applied to a wide range of material sciences including optoelectric materials, asymmetric catalysts, and chiral separation and sensing materials.

To date, several groups have attempted to introduce helicene units into macromolecules, 3 but the applications of optically active helicene-containing polymers as chiral materials are quite limited. 1a,b,e In particular, the application of optically active helicene-containing polymers as enantioselective adsorbents and chiral stationary phases (CSPs) for high-performance liquid chromatography (HPLC) has not been reported previously, in spite of the recent significant advance in chromatographic enantioseparation that has become an essential technique for the development of chiral drugs in the pharmaceutical industry. 4 On the other hand, the development of artificial helical polymers with a controlled helical sense has advanced considerably during the past decades because of their possible applications to chiral materials. 5 Among the synthetic helical polymers prepared so far, polyacetylenes are one of the most extensively studied helical polymers, 5,6 and some of them have been successfully utilized as asymmetric catalysts 7 and chiral adsorbents 8 and CSPs, 9,[START_REF] Teraguchi | Solid membranes for enantioselective permeations have also been demonstrated[END_REF] where the preferred-handed helical structures appear to be dispensable for their enantioselectivities and resolving abilities, respectively.

We anticipated that an optically active [6]helicene derivative might be a suitable chiral component as the pendant group to develop a novel optically active helical polyacetylene, since the helicene residues, once introduced covalently to the polyacetylene backbone, could induce a preferred-handed helical structure in the polymer accompanied by a one-handed helical array of the helicene pendants, thus showing an intriguing chiral recognition ability originating from the helical chiralities of the helicene itself and polymer main-chain as well. To this end, we have designed and synthesized a series of novel optically inactive (poly-1-rac) and active polyacetylenes (poly-1-P and poly-1-M) (Scheme 1) bearing an racemic or optically active helicene unit as the pendant groups directly connected to the polymer backbone, and also investigated the chiroptical properties of poly-1-P and poly-1-M and their chiral recognition abilities evaluated on the basis of an enantioselective adsorption method toward several racemates. 11

RESULTS AND DISCUSSION

The helicene monomer 2-ethynyl-carbo[6]helicene (1-rac) was prepared and resolved into the enantiomers by chiral HPLC separation according to the previously reported method 2h (1-P and 1-M with right-and left-handed screw senses, respectively; enantiomeric excess (ee) > 99%) (Supporting Information).

Helically twisted helicene molecules are known to form self-associated supramolecular aggregates through noncovalent, face-to-face π-π interactions, which frequently triggers spontaneous resolution; P-and M-helicenes favorably form homochiral assemblies over heterochiral ones due to a greater π-π overlap between the helicenes with the same configuration. 1a,b,e,12 If this is the case, we anticipated that the enantiomerically pure 1-P and 1-M could polymerize faster than its racemic mixture (1-rac), giving higher molecular weight (MW) homochiral polymers (poly-1-P and poly-1-M, respectively) or the racemic 1-rac could polymerize in a stereoselective manner (poly-1-rac) via a growing chain-end control mechanism, thus producing a mixture of polymers rich in either 1-P or 1-M. 13 In order to investigate this possibility, we first polymerized 1s on a small scale under various conditions, namely in different solvents (tetrahydrofuran (THF) and chloroform) in the presence of NEt 3 , different concentrations of 1s (0.02 -0.2 M) and rhodium catalysts (1 -6 mol%) (neutral [Rh(nbd)Cl] 2 and cationic [Rh(nbd) 2 BF 4 ] (nbd: norbornadiene), respectively) based on the previously reported results. 14 However, the polymers precipitated during the polymerizations independent of the conditions, affording polymers with a relatively small MW in moderate yields.

Therefore, the difference in the polymerization reactivity between 1-rac and 1-P or 1-M was difficult to investigate.

Typical polymerization results in THF in the presence of NEt 3 using [Rh(nbd)Cl] 2 as a catalyst, yielding stereoregular (cis-transoidal) polyacetylenes, [START_REF]The stereoregularity of the polymers was investigated by laser Raman spectroscopy, which revealed that poly-1s are rich in cis-transoid (Figure S2)[END_REF] poly-1-rac, poly-1-P, and poly-1-M (Scheme 1), are shown in Table 1 (entries 1-3, 4, 9, and14). The number-average molecular weights (M n ) of poly-1-P and poly-1-M (M n = 2.6 x 10 3 ) were higher than that of poly-1-rac (M n = 1.6 x 10 3 ) (entries 1-3). However, this tendency was not reproducible (see entries 4, 9, and 14 in Table 1), because the polymers precipitated during the polymerization as mentioned above. The size exclusion chromatography (SEC) profiles of poly-1s showed multimodal peaks, thus we fractionated poly-1s (Table 1, entries 4, 9, and 14) into four fractions (indicated by a subscript notation using A-D) due to MW by SEC (Figure S4 and Table 1) [START_REF]The highest-MW part of poly-1-rac (poly-1-rac A ) (Table 1, entry 4) may be composed of aggregates of poly-1-rac, since the M n of poly-1-rac A became lower than that before the fractionation (Figure S4d)[END_REF] and their chiroptical properties were investigated. Figure 1A shows the circular dichroism (CD) and absorption spectra of optically active 1-P, 1-M, poly-1-P B , and poly-1-M B measured in THF at 25 ˚C. The monomers 1-P and 1-M gave rise to the typical absorption signals centered at around 270 and 320 nm assigned to the helicene moiety (g and h in Figure 1A) and characteristic intense Cotton effects in the regions (c and d). 2h On the other hand, poly-1-P B and poly-1-M B showed an apparent CD in the π-conjugated main-chain region (380-600 nm) induced by the chiral helicene pendants as well as CDs in the helicene chromophore region (235-380 nm) (a and b), whose CD spectral patterns were different from those of the monomers 1-P and 1-M. Figure 1B shows the differential CD spectra, where the contributions arising from the CD absorption due to the enantiomerically pure 1-P and 1-M themselves are subtracted from the observed CD spectra of the poly-1-P B and poly-1-M B , respectively (i and j in Figure 1B). The differential CD spectral patterns seem to be like those of helical polyacetylenes with optically active pendant groups, 18 suggesting that poly-1-P and poly-1-M most likely possess an excess of a one-handed helical conformation induced by the optically active helicene pendants, which may further result in a preferred-handed helical array of the helicene pendants along the single-handed helical polymer backbone.

The CD and absorption spectra of fractionated poly-1-P C,D and poly-1-M C,D were also measured in THF at 25 ˚C (Figure S5A, C, D), which, together with the differential CD spectra (Figure S5B), indicated that the CD intensities more or less depended on the MW of the polymers, but their difference was not significant, probably because poly-1-P and poly-1-M likely possess a rather stable helical conformation with an excess one-handedness stabilized by the attractive intramolecular π-π interaction between the helically arranged helicene pendants with the same configuration. This speculation was supported by the facts that the CD spectra of poly-1-P B,C and poly-1-M B,C hardly changed over the temperature range of -10 -55 ˚C in THF and remained unchanged after heating at 55 ˚C for 2 h (Figure S6). Such a thermal stability of the present helical poly-1-P and poly-1-M is noticeable, since most of the reported helical polyacetylenes are dynamic in nature and highly sensitive to temperature, which frequently exhibits an inversion of the macromolecular helicity with temperature. 5d,6a,c,f (see Supporting Information). 19 The calculated structure revealed that the helicene pendant units are closely packed to each other and the neighboring helicene pendant units are overlapped via π-π interactions, resulting in the formation of the one-handed helical array along the helical polymer backbone. 20 Next, we investigated if the racemic 1-rac could polymerize in a stereoselective fashion to produce a mixture of poly-1s rich in either 1-P or 1-M by an enantioselective adsorption technique 11 using optically active (-)-poly(triphenylmethyl methacrylate) ((-)-PTrMA) with a one-handed helical conformation. 5a-c,23 The optically active helical PTrMA is a practically useful chiral packing material for HPLC and has been used to resolve a variety of aromatic racemic compounds including [6]helicene as well as racemic helical polymers such as (±)-PTrMA. 5a-c,24 In fact, (-)-PTrMA selectively adsorbed poly-1-M from an equimolar mixture of poly-1-P and poly-1-M in hexane/THF (80:20, v/v), and its supernatant solution exhibited a CD whose spectral pattern and intensity were identical to that of poly-1-P (ee = 36.5 %) (for more detailed experimental procedures, see Supporting Information, Table S1, and Figure S7). In the same way, the enantioselective adsorption experiment was performed for poly-1-rac with (-)-PTrMA. However, the supernatant solution showed almost no CD due to poly-1-P or poly-1-M (Table S1 and Figure S7), suggesting that 1-rac

A B

polymerized in an almost random way, or a highly stereoselective polymerization may not occur during the present polymerization of 1-rac.

The chiral recognition ability of optically active poly-1-P and poly-1-M was then investigated for the aromatic racemic compounds (1-3) including 1-rac by enantioselective adsorption experiments (Table 2 and Supporting Information). 11, [START_REF]For representative examples of chiral recognition by helicene derivatives[END_REF]26 Table 2. Enantioselective Adsorption of Racemates on Poly-1-P and Poly-1-M a a Experimental conditions for entries 1 and 2: poly-1-P or poly-1-M 2.0 mg; analyte 0.04 mg (1 mL portion from a 0.04 mg/mL solution in methanol). Experimental conditions for entries 3-6: poly-1-P 2.0 mg or poly-1-M 3.5 mg; analyte 0.4 mg for poly-1-P or 0.7 mg for poly-1-M (1 mL or 1.75 mL portion from a 0.4 mg/mL solution in methanol). b Determined by chiral HPLC analysis of analytes adsorbed on poly-1-P or poly-1-M using a UV-visible detector. c Calculated according to the equation α = (F major (%)/F minor (%))/(A major (%)/A minor (%)), where F major and F minor are the percentages of major and minor enantiomers of free analyte in the supernatant solutions, respectively, and A major and A minor are those of major and minor enantiomers of adsorbed analyte, respectively. d Average values of two runs.

The poly-1-P and poly-1-M selectively adsorbed one of the enantiomers of 1 -3 in methanol. methanol and subsequently in a methanol/THF (50:50, v/v) mixture (Supporting Information), and their amounts and ee values were estimated by chiral HPLC analysis (Table 2).

As anticipated, 1a,b,e,12 poly-1-P and poly-1-M preferentially adsorbed 1-P and 1-M with the same configuration of the helicene residues, furnishing 3.6 and 3.7 % ee, respectively (entries 1 and 2). Interestingly, poly-1-P and poly-1-M showed a much better and remarkable chiral recognition toward the racemic 1,1'-binaphthyl analogues (2 and 3), and selectively adsorbed the (S)-and (R)enantiomers, respectively, with up to 37 % ee (entries 3-6). On the basis of the amounts and ee values of the analytes (1 -3) adsorbed on the polymers, the separation factor (α), which is a useful measure in chiral HPLC to evaluate the chiral resolving ability of optically active hosts toward racemic analytes 11 can be calculated (Table 2). 4 The calculated α values (1.71 ~ 2.18) are high enough for the complete separation of enantiomers when used as a CSP for HPLC, 4,5a,b indicating the helicene-bound helical polyacetylenes possesses a practically useful chiral recognition power for aromatic racemic compounds like a commercially available (+)-PTrMA column.

The enantioselective adsorption results on the poly-1-P and poly-1-M (Table 2) revealed insight into the chiral recognition mechanism, in other words, the stereochemistry of poly-1-analyte interaction; helically arranged helicene-pendant residues along the poly-1 helical backbone (Figure 2) appear to be responsible for the observed efficient separation of aromatic analytes (1 -3). The two naphthyl rings of the more retained enantiomers of (R)-2 and (R)-3 on poly-1-M may be favorably twisted into a left-handed helix so as to interact with the adjacent M-helicene groups arranged in a left-handed helical array through the aromatic interactions, whereas due to a considerable π-π overlap between the helicene pendants may interfere with favorable close helicene-helicene interactions between poly-1 and 1-rac, resulting in a relatively lower separation factor than we anticipated.

CONCLUSIONS

In summary, we have, for the first time, prepared stereoregular helical polyacetylenes bearing optically active and racemic [6]helicene groups as the pendants. Although the molecular weights of the polymers were rather low because of the low solubility of the helicenes, the optically active helicene-bound polymers exhibited an apparent CD in the polymer backbone chromophore region due to a preferred-handed helix formation biased by the chiral helicene residues. Moreover, we found that the optically active helicene-bound polyacetylenes showed a high chiral recognition ability toward racemic 1,1'-binaphthyl derivatives largely arising from a one-handed helical array of the pendant helicene residues.

The present results suggest that a more practically useful chiral material for separation of enantiomers may be developed based on analogous helicene-and organometallic helicene 2f -bound helical polyacetylenes by introducing a long alkyl chain on the helicene residue, which will overcome the solubility problem of polymers, thereby providing a unique chiral stationary phase with high enantioselectivity composed of one-handed helical polyacetylenes with screw-shaped helicene and organometallic helicene pendants, and further studies along this line are now in progress.

Supporting Information. Experimental details in the synthesis and characterization of poly-1s and the enantioselective adsorption on poly-1-M and poly-1-P, and the molecular modeling and calculations of poly-1-M. This material is available free of charge via the Internet at http://pubs.acs.org.

ACKNOWLEDGEMENT

Instruments

The NMR spectra were measured using a Varian AS500 spectrometer (Agilent, Santa Clara, CA) operating at 500 MHz for 1 H using tetramethylsilane (TMS) as the internal standard. The absorption and CD spectra were obtained in a 0.1, 0.5, or 5 cm quartz cell using a JASCO V570 spectrophotometer and a JASCO J-820 spectropolarimeter, respectively (JASCO, Tokyo, Japan).

The temperature was controlled with a JASCO PTC-423L apparatus. The concentrations of the polymers were calculated based on the monomer units. The size exclusion chromatography (SEC) measurements and fractionations were performed with a JASCO PU-980 liquid chromatograph equipped with a UV-visible detector (300 nm, JASCO UV-1570) and a column oven (JASCO CO-1565). The number-average molecular weight (M n ) and its distribution (M w /M n ) were determined at 40 °C using a Tosoh TSKgel Multipore H XL -M (0.78 (i.d.) x 30 cm) SEC column (Tosoh, Tokyo, Japan), and tetrahydrofuran (THF) or chloroform (CHCl 3 ) was used as the eluent at a flow rate of 0.5 or 1.0 mL/min, respectively. The molecular weight calibration curve was obtained with polystyrene standards (Tosoh). The chiral HPLC analyses and separations were performed on a JASCO PU-2080 liquid chromatograph equipped with a UV-visible detector (JASCO UV-2075 or MD-2010) and an optical rotation (JASCO OR-2090) or a CD detector (JASCO CD-2095) using Chiracel OD-H, Chiralpak IA, and Chiralpak IB columns (0.46 (i.d.) x 25 cm, or 2 (i.d.) x 25 cm, Daicel, Osaka, Japan). The laser Raman spectra were taken on a JASCO RMP-200 spectrophotometer.

Materials

All starting materials and anhydrous solvents were purchased from Aldrich (Milwaukee, WI), Wako Pure Chemical Industries (Osaka, Japan), or Tokyo Kasei (TCI, Tokyo, Japan), and were used as received, except for triethylamine (Et 3 N) and THF. Et 3 N was dried over KOH pellets and distilled onto KOH under nitrogen. THF was dried and deoxygenized by passing through purification columns (Glass Contour Solvent System, Nikko Hansen, Osaka, Japan), distilled onto LiAlH 4 under nitrogen, and redistilled under high vacuum just before polymerization. 2-Ethynyl-carbo[6]helicene (1-rac) was synthesized according to the previously reported method. S1 The helicene 1-rac was separated into the enantiomers (1-P and 1-M) using a chiral HPLC column (Chiralpak IB, 2 (i.d.) x 25 cm, n-hexane/EtOH = 99:1 (v/v), 11 mL/min), and the enantiomeric excess (ee) values of the 1-P and 1-M separated were estimated to be >99% using a chiral HPLC column (Chiralpak IB, 0.46 (i.d.) x 25 cm, n-hexane/EtOH = 99:1 (v/v), 1 mL/min) (Figure S1). (-)-Poly(triphenylmethyl methacrylate) (PTrMA, [α] D [START_REF]For representative examples of chiral recognition by helicene derivatives[END_REF] -336 (c 2.0, THF))

was synthesized according to the previously reported method. S2 The M n and M w /M n of (-)-PTrMA estimated as its methyl ester (poly(methyl methacrylate) were 7.0 x 10 3 and 1.19, respectively, as determined by SEC using poly(styrene) standards in CHCl 3 as the eluent.

Synthesis of Polymers

Polymerization. Polymerizations of 1-rac, 1-P, and 1-M were carried out according to Scheme 1 in a dry glass ampule under a dry nitrogen atmosphere using [Rh(nbd)Cl] 2 as a catalyst in a similar way as reported previously. S3 The polymerization results are summarized in Table 1. A typical polymerization procedure is described below.

The monomer 1-M (50.0 mg, 0.142 mmol) was placed in a dry ampule, which was then evacuated on a vacuum line and flushed with dry nitrogen. After this evacuation-flush procedure was repeated three times, a three-way stopcock was attached to the ampule, and a mixture of

Characterization of Monomers and Polymers

The 1 H NMR spectra of poly-1s were too broad to determine the stereoregularity. Therefore, their stereoregularities were investigated by laser Raman spectroscopy. Figure S2 shows the Raman spectra of poly-1-rac, poly-1-P, and poly-1-M (Table 1, entries 4, 9, and14

Enantioselective Adsorption

Enantioselective Adsorption of Poly-1s by (-)-PTrMA. A typical experimental procedure for the enantioselective adsorption of an equimolar mixture of the enantiomeric poly-1-P and poly-1-M by (-)-PTrMA is described below. A mixture of (-)-PTrMA (0.4 mg), poly-1-P (0.04 mg), and poly-1-M (0.04 mg) was dispersed in THF (2 mL) in a screw-capped sample bottle, and to this was added n-hexane (8 mL) at ca. 25 ˚C, affording a white precipitate of (-)-PTrMA on which poly-1 was adsorbed. After centrifugation, the absorption and CD spectra of the supernatant solution were measured to determine the amount and ee of poly-1 (Figure S7). In the same way, the enantioselective adsorption experiment was performed for poly-1-rac and the results are summarized in Table S1. S1). quickly washed with methanol (1 mL) to remove the racemic solution of 2 attached on the surface of the polymer. The recovered poly-1-P was placed into a screw-capped sample bottle containing methanol (5 mL), and the mixture was permitted to stand at ca. 25 ˚C for 3 h to desorb 2 adsorbed on the poly-1-P. This desorption procedure was repeated with methanol (5 mL) and then with a methanol/THF (50:50, v/v) mixture (5 mL). After evaporation of the solvents, the residue was subjected to SEC fractionation using a Tosoh TSKgel Multipore H XL -M (30 cm) SEC column (THF, 0.5 mL/min) to recover 2, and the amount and ee of 2 adsorbed on the poly-1-P were estimated by chiral HPLC using a Daicel Chiralpak IA column (0.46 (i. In the same way, the enantioselective adsorption experiment was performed for racemic 3 and the results are summarized in Table 2.

Scheme 1 . 1 [

 11 Scheme 1. Rhodium-Catalyzed Polymerization of 1

c

  [Rh]/[monomer] = 17. d Determined by SEC (polystyrene standards) with THF as the eluent. e Not measured because the amounts of poly-1-P A and poly-1-M A obtained by SEC fractionation were too small.

Figure

  Figure 2A,B displays a possible right-handed helical structure of poly-1-M (20 mer) calculated

Figure 1 .Figure 2 .

 12 Figure 1. (A) CD (a-d) and absorption (e-h) spectra of poly-1-P B (a,e), poly-1-M B (b,f), 1-P (c,g),

Figure S1 .

 S1 Figure S1. HPLC traces of 1-rac (a), 1-M (b), and 1-P (c) using a Daicel Chiralpak IB column [n-hexane/EtOH = 99:1 (v/v), 1 mL/min].

Et 3 N

 3 (19.8 μL, 0.142 mmol) and anhydrous THF (0.64 mL) was added with a syringe. To this was added a solution of [Rh(nbd)Cl] 2 (0.06 M) in THF (0.07 mL) at 30 °C. The concentrations of the monomer and the rhodium catalyst were 0.2 and 0.006 M, respectively. After 13 h, a solution of [Rh(nbd)Cl] 2 (0.06 M) in THF (0.07 mL) was added again to complete the polymerization. After additional 5 h, the resulting polymer (poly-1-M) was precipitated into a large amount of Et 2 O, washed with Et 2 O, and collected by centrifugation. The product was purified by reprecipitation from CHCl 3 to Et 2 O, and the precipitated poly-1-M was washed with

  ). All the polymers showed intense peaks at 1505, 1311, 873 cm -1(a), 1509, 1310, 882 cm -1 (b), and 1508,1311, 874 cm -1 (c), which can be assigned to the C=C, C-C, and C-H bond vibrations in the cis polyacetylenes, respectively, S4 indicating that these polymers possess a highly cis-transoidal structure.

Figure S2 .

 S2 Figure S2. Raman spectra of poly-1-rac (a), poly-1-P (b), and poly-1-M (c).

Figure S3. 1 HFigure S4 .Figure S5 .Figure S6 .

 1S4S5S6 Figure S3. 1 H NMR spectra of poly-1-rac (a), poly-1-P (b), and poly-1-M (c) in CDCl 3 at 50 ˚C. X denotes the proton from CHCl 3 .

Figure S7 .

 S7 Figure S7. (A) Experimental procedures for the enantioselective adsorption of (I) poly-1-rac and (II) an equimolar mixture of poly-1-P and poly-1-M by (-)-PTrMA. (B) CD and absorption spectra of poly-1-rac (a,b, red lines) and an equimolar mixture of poly-1-P and poly-1-M (c,d, blue lines) in THF (0.04 mg/mL, 0.1 cm cell) and those of the supernatant solutions after enantioselective adsorption of poly-1-rac (e,f, red lines) and the equimolar mixture of poly-1-P and poly-1-M (g,h, blue lines) by (-)-PTrMA in THF/n-hexane (2:8, v/v, 5 cm cell) at 25 ˚C (see TableS1).

  d.) x 25 cm, CH 2 Cl 2 , 0.5 mL/min) equipped with UV and CD detectors. The total amount of the analyte 2 obtained by this procedure was almost coincident with that estimated by the absorption spectral change of the supernatant solution during the enantioselective adsorption experiments.

Table 1 . Polymerization of 1-rac, 1-P, and 1-M with [Rh(nbd)Cl] 2 and SEC Fractionation Results of Poly-1-rac, Poly-1-P, and Poly-1-M a

 1

	polymer

Table of Contents Use Only Synthesis and Chiral Recognition Ability of Helical Polyacetylenes Bearing Helicene Pendants Emmanuel

  This work was supported in part by Grant-in-Aid for Scientific Research (S) from the Japan Society for the Promotion of Science (JSPS) and Research Grants in the Natural Sciences from the Mitsubishi Foundation. E.A. expresses his thanks for JSPS Summer Program.

	H For (M)	n	favor	(M)	disfavor	(P)

Anger, Hiroki Iida, Tomoko Yamaguchi, Kotaro Hayashi, Daisuke Kumano, Jeanne Crassous, * Nicolas Vanthuyne, Christian Roussel, and Eiji Yashima * enantioselective adsorption

Supporting Information for:

Synthesis and Chiral Recognition Ability of Helical Polyacetylenes Bearing

Helicene Pendants

Emmanuel Anger, †, ‡ Hiroki Iida, † Tomoko Yamaguchi, † Kotaro Hayashi, † Daisuke Kumano, † Jeanne Crassous, ‡, * Nicolas Vanthuyne, § Christian Roussel, § and Eiji Yashima 13397 Marseille Cedex 20, France E-mail: yashima@apchem.nagoya-u.ac.jp; jeanne.crassous@univ-rennes1.fr

Molecular Modeling and Calculations

The molecular modeling and molecular mechanics (MM) calculations of poly-1-M were conducted with the Compass force field, S5 as contained in the MS Modeling software (version 4.4, Accelrys, San Diego, CA) operated using a PC running under Windows XP.

The initial monomer unit structure was constructed using the crystal structure of 2-methyl-carbo[6]helicene. S6 The main-chain helix sense of poly-1-M in THF where the negative first Cotton effect was observed at the main-chain region (380-600 nm) is assumed to be right-handed on the basis of the CD spectral pattern of the analogous L-alanine-bound helical poly(phenylacetylene) whose helix sense was by the direct observation of the helical structure obtained the high-resolution atomic force microscopy (AFM). S7 When the main-chain of poly-1-M is right-handed, it possesses an opposite, left-handed helical array of the pendants. The polymer model (40 repeating monomer units) of poly-1-M was constructed using the Polymer Builder module in the MS Modeling software. The starting main-chain geometrical parameters such as the bond lengths were defined as a 23 unit/10 turn (23/10) helix on the basis of the helical structure of a cis-transoidal poly(phenylacetylene) bearing N,N-diisopropylaminomethyl pendants determined by X-ray analysis. S8 The dihedral angle between the main chain and the aromatic residue of the pendant (θ) was varied from 0° to 300° at 60˚ intervals, and the dihedral angle of the single bond (C=C-C=C) from planarity (φ) was set to be 140.3°, S8 screw-capped sample bottle containing methanol (5 mL), and the mixture was permitted to stand at ca. 25 ˚C for 3 h to desorb 1 adsorbed on the poly-1-M. This desorption procedure was repeated with methanol (5 mL) and then with a methanol/THF (50:50, v/v) mixture (5 mL),

which almost completely desorbed 1 on the poly-1-M; the remaining 1 on the poly-1-M after the present desorption procedure was found to be negligibly small (< 0.05%) on the basis of SEC fractionation using a Tosoh TSKgel Multipore H XL -M (30 cm) SEC column (THF, 0.5 mL/min) followed by HPLC analysis of 1 adsorbed on the poly-1-M after dissolving in THF. The amount and ee of 1 adsorbed on the poly-1-M were estimated by chiral HPLC using a Daicel Chiralpak IB column (0.46 (i.d.) x 25 cm, n-hexane/EtOH = 99:1 (v/v), 1 mL/min) equipped with UV and CD detectors. The results are summarized in Table 2.

Enantioselective Adsorption of


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Stéphanie Prigent

MODELLING AND ANALYSIS OF PROTEIN AGGREGATION -COMPETING PATHWAYS IN PRION (PRP) POLYMERISATION *

Protein aggregation leading to the formation of amyloid fibrils is involved in several neurodegenerative diseases such as prion diseases. To clarify how these fibrils are able to incorporate additional units, prion fibril aggregation and disaggregation kinetics were experimentally studied using Static Light Scattering (SLS). Values that are functions of i≥1 i 2 ci, with ci being the concentration of fibrils of size i, were then measured as a function of time. An initial model, adapted from the Becker-Döring system that considers all fibrils to react similarly is not able to reproduce the observed in vitro behaviour. Our second model involves an additional compartment of fibrils unable to incorporate more prion units. This model leads to kinetic coefficients which are biologically plausible and correctly simulates the first experimental steps for prion aggregation.

Résumé. L'agrégation des protéines conduisant à la formation de fibres amyloïdes est impliquée dans plusieurs maladies neurodégénératives telles que les maladies à prion. Pour clarifier la manière dont les fibres de prion incorporent des unités supplémentaires, les cinétiques d'agrégation et de désagrégation des fibres ont été étudiées expérimentalement (par SLS, "Static Light Diffusion"). Ainsi des valeurs fonction de i≥1 i 2 ci avec ci la concentration en fibres de taille i ont été obtenues en fonction du temps.

Un premier modèle, adapté du système de Becker-Döring, qui considère que la totalité des fibres réagit de manière similaire ne permet pas de reproduire le comportement observé in vitro. Notre deuxième modèle met en jeu un compartiment additionnel de fibres incapables d'incorporer davantage d'unités de PrP. Celui-ci aboutit à des coefficients cinétiques biologiquement plausibles et simule correctement les premières étapes expérimentales de l'agrégation de prions.

Introduction

More than twenty diseases, known as amyloid diseases and which include Alzheimer's, Hungtington's and prion diseases are due to the conversion of a protein structure into a misfolded conformation that induces protein aggregation. Recent studies have shown that injections of aggregated proteins involved in Alzheimer's and Parkinson's diseases follow a prion-like mechanism behaviour by a cell-to-cell transmission [START_REF] Polymenidou | Prion-like spread of protein aggregates in neurodegeneration[END_REF]. However, the exact mechanisms for incorporating prion (PrP) protein to a prion aggregate remain to be clarified. To better understand how PrP fibrils can recruit PrP molecules that were not misfolded, we have studied the kinetics of PrP fibril elongation, focusing on polymerisation through the addition of monomer(s) on fibrils. A loss of fibril ability for further polymerisation was experimentally observed after a given number of additions of monomers. Among several hypothetical mechanisms and according to different experiments [START_REF] Prigent | Molecular basis of structural information transference during prion conversion[END_REF], a hypothetical occurrence of a structural defect on fibrils could explain this loss of the ability to polymerise. Relying on the iterative modeling process, we have written two models for the simulation of prion polymerisation/depolymerisation. The first model, the basic one, led us to the Becker-Döring system for phase transition phenomena. Nevertheless, this model was unable to reproduce the loss of fibril ability for polymerisation. The second model involves the formation of fibrils with a structural defect preventing them from any further polymerisation. This model, with suitable parameter estimations, numerically reproduces part of the empirically observed curves of prion fibril kinetics.

Biological experiments

To clarify prion aggregation mechanisms, polymerisation and depolymerisation of prion fibrils were studied in vitro as a function of time, starting from preformed fibrils. These experiments were set up and performed by the team of Dr. H. Rezaei, Dr. D. Martin and Dr. J. Torrent Y Mas from INRA (Jouy-en-Josas, France). The kinetics were followed by SLS (Static Light Scattering) performed on a cuvette containing PrP fibrils in an aqueous buffer solution. SLS measures α( i≥1 i 2 c i ) + β, with c i being the concentration of polymers of size i, and α and β being parameters that are constant during each experiment. Because the experimental sample is a heterogeneous mixture of polymers of various sizes, this technique does not give access to the concentration of polymers of a precise size. To clarify the mechanisms involved in fibril growth once fibrils have formed, i.e. how additional monomers are recruited by fibrils, the experimental approach consisted in successive additions of monomers on fibrils (Figure 1). At time 0 a first addition of monomers was performed on the fibrils. The following additions of monomers were made once the SLS measurement had reached a quasi plateau. After a certain number of additions of human PrP monomers on human PrP fibrils, the SLS signal did not increase anyfurther. However, the SLS signal was increased by a further addition of mutant PrP monomers that differ from human PrP monomers in their structure. This observation and other experiments indicate that fibrils growth probably does not stop simply because the fibril has reached a certain size [START_REF] Prigent | Molecular basis of structural information transference during prion conversion[END_REF]. Complementary experiments using another technique (spectrophotometry) were also performed to get a crude estimate of the concentration of free monomers at the end of the first two plateaux; these indicated that only a small percentage of initial monomers was incorporated onto fibrils.

Basic model for fibril-monomer reaction

We start this modeling process by considering only classic prion fibrils with their basic reactions, which results in the following model.

Basic model

An i-sized polymer can gain a free monomer to become an (i + 1)-sized polymer. This reaction is called polymerisation, it occurs with a non-negative size-dependant rate k oni . An i-sized polymer can also release a monomer, giving rise to a smaller polymer, of size i -1, and a free monomer. This reaction is named depolymerisation. It occurs with a non-negative size-dependant rate, k dep i . Denoting by c i the concentration in polymers of size i, with i = 1, 2, ..., we get the following scheme of reactions:

c i + c 1 kon i -→ c i+1 , i ≥ 1, c i -→ k dep i c i-1 + c 1 , i ≥ 2.
In terms of equations, it gives the well-known Becker-Döring system [2] :

     dc i dt = -k oni c 1 c i + k oni-1 c 1 c i-1 -k dep i c i + k dep i+1 c i+1 i ≥ 2, dc 1 dt = - ∞ i=2 k oni c 1 c i -k dep i+1 c i+1 -2 k on1 c 2 1 -k dep 2 c 2 . (1)
This is obtained directly from the schemes reactions, using the law of mass action several times. One can notice that a particular equation is needed for the variation of concentration of 1-sized particules that are called monomers. This is due to their interaction in all the processes. A detailed qualitative study of this system was carried out in several articles, such as [START_REF] Ball | The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions[END_REF], [START_REF] Da | Asymptotic behaviour of low density solutions to the generalized Becker-Döring equations[END_REF], and [START_REF] Niethammer | On the Evolution of Large Clusters in the Becker-Döring Model[END_REF]. In [START_REF] Ball | The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions[END_REF], the authors give theorems of existence, uniqueness, continuous dependence of initial data and the fundamental mass conservation property:

c 1 (t) = ∞ i=1 (ic i (0)) - ∞ i=2 (ic i (t)) .
For new fibrils to be created through the reaction

c 1 + c 1 kon 1
-→ c 2 biogically requires a greater time than the time-scale of the experiments reported here. This was deduced through a control experiment where a constant signal was observed when using only monomers in the same duration (300min). Therefore, the coefficient k on1 is assumed to be null (k on1 = 0).

Simplified system

To gain further qualitative insight into the dynamics of these equations and a concrete idea about the order of magnitude of the kinetics coefficients k on and k dep , we initially consider a simplified system (which we will make more complex afterwards). This involves a summation of all the equations of the infinite Becker-Döring system (1) over all sizes. The simplification is the assumption we make on the coefficients k on and k dep by considering them to be constant. As all the fibrils are almost the same size, this assumption is justified. Moreover, we consider a time-scale where the concentration in dimers, c 2 , is negligible. This results in the following simplified system :

             dP dt = 0 dM dt = k on c 1 P -k dep P dc 1 dt = -k on c 1 P + k dep P (2)
where P = i≥2 c i (t) represents the total concentration of fibrils summed over all sizes i ≥ 2, M = i≥2 ic i (t) represents the total mass of fibrils summed over all sizes i ≥ 2, c 1 the concentration of monomers. The total mass conservation is then obvious.

The equation of the second order moment (measured data) variation is also simplified into :

dM measured 2 dt = 2M (k on c 1 -k dep ) + 2k dep P
and once the closed simplified system (2) has been solved, an analytical expression for M measured 2 can be easily deduced. Indeed, after integrating (2) we get :

       P = P in M (t) = M in -k on c in 1 -c eq 1 e -konP in t P in + c in 1 -c eq 1 c 1 (t) = c in 1 -c eq 1 e -konP in t + c eq 1 (3)
where P in = P (0) represents the initial total concentration of polymers and M in = M (0) their initial total mass, c in 1 = c 1 (t = 0) the initial concentration of monomers and c eq 1 = k dep kon the concentration of monomers at equilibrium state (when

dc 1 dt = 0, see (2)). Thus, M measured 2 = A + B(e -konP in t -1) + C(e -2konP in t -1) + 2k dep P in t, with A = M in 2 , B = -2 P in (M in + c in 1 -c eq 1 )(c in 1 -c eq 1 ) and C = (c eq 1 -c in 1 ) 2 P in .
One can notice from the mass expression M (t) above that we need to assume c eq 1 = 1) is not valid and M (t) defined by (3) would become negative for large time. The next step is then to estimate the parameters k on , k dep , α and β using the experimental data of SLS with SLS = αM measured 2 + β. We reformulate this inverse problem into the minimisation of the corresponding least squares criterion

k dep kon ≤ c in 1 + M in , else neglecting c 2 in (
J (k on , k dep , α, β) = n i=1 αM measured 2 (t i ; k on , k dep ) + β -SLS(t i ) 2
The uniqueness of the solution is, as for most non-linear inverse problems, non-trivial. This is essentially due to the non-convexity of the cost functional J. This minimisation allows us to get a satisfactory estimation for the first step of the experiments (Figure 2), but only for the first addition of monomers. At each new addition of monomers to the solute, fibrils once again start polymerising, consuming monomers. Despite the fact that the consumption of monomers is lower than in the first step ( due to volume dilution in the cuvette), the gap between two successive additions is higher and higher (Figure 3). This is due to the i 2 term in the simulated quantity α( 3). We conclude here that a stop pathway is needed to slow down the polymerisation of fibrils and the growth of their average size i and, hence, i 2 .

n i≥1 i 2 c i ) + β (Figure

Two-compartment model

Discrete model

To force the polymerisation to stop, we set the hypothesis of creating a different fibril that will no longer be able to polymerise due to a defect in its structure. This defective fibril would be the result of the polymerisation of a classic fibril with a polymerisation rate k m on . We denote by c m i the concentration of the defective fibrils of size i. The reactions are now:

c i + c 1 kon i -→ c i+1 , i ≥ 1, c i -→ k dep i c i-1 + c 1 , i ≥ 2, c i + c 1 k m on i -→ c m i+1 , i ≥ 2, c m i -→ k m dep i c i-1 + c 1 , i ≥ 3.
Which can be translated into the following ordinary differential equations system :

                           dc i dt = -k oni c 1 c i + k oni-1 c 1 c i-1 -k m oni c 1 c i -k dep i c i + k dep i+1 c i+1 + k m dep i+1 c m i+1 , i ≥ 2 dc m i dt = k m oni-1 c 1 c i-1 -k m dep i c m i , i ≥ 3 dc 1 dt = - ∞ i=2 k oni c 1 c i - ∞ i=2 k m oni c 1 c i + ∞ i=2 k dep i c i + ∞ i=3 k m dep i c m i -2 k on1 c 2 1 -k dep 2 c 2

The two-compartment simplified system

We denote by P m = i≥3 c m i (t) the total concentration of defective polymers. M m = i≥3 ic m i (t) the total mass of defective polymers. Summing once again over all sizes, we obtain the following system :

                               dP dt = -k m on c 1 P + k m dep P m dP m dt = k m on c 1 P -k m dep P m dM dt = k on c 1 P -k m on c 1 M -k dep P + k m dep (M m -P m ) dM m dt = k m on c 1 (P + M) -k m dep M m dc 1 dt = -k on c 1 P -k m on c 1 P + k dep P + k m dep P m (4)
and the measured data :

dM measured 2 dt = 2c 1 M (k on + k m on ) + 2k dep (P -M) + 2k m dep (P m -M m ) (5)
In order to fit the experimental data for the first addition of monomers, kinetic parameters for the twocompartment model were obtained as follows. We suppose the fibrils to be exclusively composed of classic fibrils at time 0, and that defective fibrils are absent. We start with a simplified view of the system to obtain certain values of kinetic constants able to fit the data. For this reason, we consider that at the beginning of the experiment the system is nearly a model with only one compartment. For the basic model, we can deduce from (3) that after a certain time denoted t eq (for equilibrium time), the quantity c in 1 -c eq 1 e -konP in t eq is negligible when compared to c eq 1 . We consider e -konP in t eq in the 10 -1 range, i.e. -k on P in t eq ≃ -2. Therefore, for the basic model, we have approximately :

k on P in t eq = 2 ⇒ k on = 2 P in t eq (6)
By analogy with the basic model, after replacing P in (P initial) by P eq (P of classic fibrils at equilibrium state) in the k on formula ( 6) from the basic model, we can obtain certain values of k on and k dep , for the two-compartment model, able to fit the first minutes. To calculate k on , we use a fixed value for t eq = 8 min (chosen from the minimal time where the experimental SLS slope reaches nearly 0). For k dep , with k dep = k on c eq 1 , we use 75 100 c in 1 ≤ c eq 1 ≤ 96 100 c in 1 (from experimentally estimated values of c 1 ) where c in 1 = 5.10 -6 mol.L -1 . In this way, during the first minutes, the concentration of consumed c 1 is modulated by the classic fibrils (more exactly by the ratio k dep /k on ) whereas any effect of the few defective fibrils on the concentration of monomers can be ignored. This analogy with the 'basic model' offers access to a part of the sets of plausible kinetic constants able to fit the first addition of monomers.

Once k on and k dep were calculated as functions of P m eq (the concentration of defective fibrils at equilibrium state, with a value from 0 to 100 % of P in ), the kinetic constants k m dep and then k m on for defective fibrils were deduced. k m dep was got by a small inverse problem (using the Matlab routine fminsearch) from a broad range of potential k m dep values (0 ≤ k m dep ≤ 10 4 ). k m on was deduced from the equilibrium state between classic and defective fibrils:

dP dt = dP m dt = 0 ⇒ -k m on c 1 P + k m dep P m = 0 ⇒ k m on = k m dep P m c 1 P
Table 1 presents calculated values for the kinetic parameters allowing us to reproduce the SLS signal for the first addition of monomers. Each percentage of defective fibrils, from 0 up to 100 % of the initial concentration of total fibrils, P in , allows the SLS signal to be mimicked for the first addition of monomers; this was accompanied by a broad range of k m on (0 to 3.10 12 mol.L -1 .min -1 ) and of k m dep (0 to 1.10 4 min -1 ).

The next step is to fit the effect of a second addition of monomers on the SLS signal once this signal has reached a plateau. To reach this goal, let us focus on and try to simplify the term representative of the SLS signal, α M measured Table 1. Kinetic constants (k on , k m on , k dep and k m dep ) able to fit the first addition of monomers for the two-compartment model. k on and k dep are obtained by analogy with the basic model, k m on and k m dep , as a function of P m eq (total concentration of defective fibrils at equilibrium state), through an inverse problem. Various values of P m eq were used in order to test a wide range of initializations of the minimization algorithm. The lack of identifiability of the inverse problem leads to several sets of kinetic constants which are reported above. The last line summarizes the values for the whole range of P m eq . whose variation is given in Equation ( 5):

dM measured 2 dt = 2c 1 M (k on + k m on ) + 2k dep (P -M) + 2k m dep (P m -M m )
From this equation, as logically expected, the positive part depends on the polymerisation constants (k on and k m on ) and the negative parts on the depolymerisation constants (k dep and k m dep ), the subtraction between fibril concentration and fibril mass (P-M) rendering these parts negative. It was experimentally observed that with a given number of successive additions, the SLS signal almost stops increasing (Figure 1). We thus can postulate that nearly 100 % of fibrils became defective at that moment and it might indicate a high stability of defective fibrils i.e. k m dep ≈ 0. However, even if such an assumption k m dep ≈ 0 could be verified, neglecting the term 2k m dep (P m -M m ) in Equation ( 5) is not possible: indeed the very low values of c 1 (5.10 -6 ), P (0 ≤ P ≤ 6.67.10 -10 ) and M (0 ≤ M ≤ 2.10 -5 roughly) and the potential numerical values for k on , k m on and k dep easily turn the three terms of Equation ( 5) into values of similar or almost similar numerical ranges.

To fit the second addition of monomers, we tested all the previously found sets of kinetic constants (k on , k m on , k dep and k m dep ) that closely mimicked the first addition of monomers. After adjusting these values for a satisfactory curve-fitting of the height of the second addition of monomers, one set of values was able to reproduce the first addition and the height of the second one (Figure 4, left panel). It is therefore confirmed that, in contrast to the basic model, the two-compartment model makes it possible to mimick the increase in the SLS signal by a second addition of monomers. However, the beginning of the slope of the second addition was not satisfactory, nor was the height for the third and subsequent additions of monomers (Figure 4, right panel). The fact that the third addition did not make it possible to increase sufficiently the simulated signal can be explained by the low k m dep and the high percentage of modified fibrils (Figure 5). Regarding the slopes, the contribution of the different biological parameters will be mathematically studied in a future sensitivity problem. This study will enable us, for instance, to confirm (or infirm) that a high ratio k m on k m dep compared to the ratio kon k dep could decrease the slopes (as the formation of defective fibrils prevents a further increase in its size).

This work has shown that the basic model with only one type of fibril could not explain the observed kinetic behavior of prion fibrils. Therefore, a model with at least two compartments is necessary. Biologically speaking, this could be the result of the occurrence of defective fibrils in addition to classic fibrils. This twocompartment model leads to polymerising constant values, k on , from 4.10 8 up to 4.10 12 mol.L -1 .min -1 . Such a range of values for polymerising constants (also named in literature as association rate constants, k on , k a or k + ) is commonly encountered in other examples of protein-protein associations (aggregation) when fast polymerising constants are involved [START_REF] Zhou | Modeling protein association mechanisms and kinetics[END_REF]. On the other hand, the depolymerising constant values (k dep from 2.10 3 up to 2.10 7 min -1 ) are representative for a very high dissociation rate constant which, although rare, as they are usually lower than 1min -1 , can be encountered for another aggregating protein, tubulin [START_REF] Gardner | Rapid Microtubule Self-Assembly Kinetics[END_REF]. For a better modulation of slopes and heights, an additional assumption could be added to the two-compartment model such as a progressive formation of a defective fibril by more than only one monomer. Recent experimental observations from our collaborating team are indeed in favor of such a complementary hypothesis for explaining the kinetics of prion fibrils.

Figure 1 .

 1 Figure 1. SLS experimental data of successive additions of monomers on PrP human fibrils

Figure 2 .

 2 Figure 2. Simulation of the first addition of monomers. (Quantities are in mol.L -1 .)

Figure 3 .

 3 Figure 3. Simulation of the first three additions of monomers with solute-dilution in the second and third ones.

Figure 4 .Figure 5 .

 45 Figure 4. Curve fitting for two (left panel) and four (right panel) successive additions of monomers, with the "two-compartment model" (using k on = 3.5.10 8 mol.L -1 .min -1 ; k dep = 1.6.10 3 min -1 ; k m on = 1.7.10 4 mol.L -1 .min -1 and k m dep = 1.10 -6 min -1 )

+ β (with α and β being constant as previously mentioned)

The authors would like to warmly thank the head of this project, Marie Doumic-Jauffret (INRIA), for her major contribution to this work and Human Rezaei, Davy Martin (INRA) and Joan Torrent Y Mas (INSERM) for initiating this topic, the experimental part and discussions. This research was supported by the ERC Starting Grant SKIPPER