Biocompatible Soft Nanoparticles with Multiple Morphologies

Mar 1, 2017 - These original architectures of copolymers, when nanoprecipitated in water, the backbone-selective solvent, allowed us to obtain various...
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Biocompatible soft nanoparticles with multiple morphologies obtained from nanoprecipitation of amphiphilic graft copolymers in a backbone selective solvent. Gaëlle Le Fer, Clémence Le Coeur, Jean-Michel Guigner, Catherine Amiel, and Gisèle Volet Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b00471 • Publication Date (Web): 01 Mar 2017 Downloaded from http://pubs.acs.org on March 2, 2017

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Biocompatible soft nanoparticles with multiple morphologies obtained from nanoprecipitation of amphiphilic graft copolymers in a backbone selective solvent. Gaëlle Le Fer1, Clémence Le Cœur 1,3, Jean-Michel Guigner4, Catherine Amie11, Gisèle Volet1,2* 1 Université Paris Est, ICMPE (UMR7182), CNRS, UPEC, F 94320 Thiais, France 2 Université d’Evry Val d’Essonne, Rue du Père Jarlan, 91025 Evry cedex, France 3 Laboratoire Léon Brillouin, UMR 12 CEA – CNRS, CEA Saclay, 91191 Gif-sur-Yvette cedex, France 4 Institut de Minéralogie,de Physique des Matériaux et de Cosmochimie (IMPMC), Sorbonne Universités, UPMC univ Paris 6, IRD, CNRS UMR7590, MNHN; 4 place Jussieu, 75252 Paris Cedex 05, France

ABSTRACT Stealth nanocarriers are a promising technology for the treatment of diseases. However preparation and characterization of well-defined soft nanoparticulate systems remains challenging. Here is described a platform of amphiphilic graft copolymers leading to nanoparticles with multiple morphologies and the role of the hydrophilic backbone in their interaction with a model protein. The amphiphilic graft copolymers platform was composed of a hydrophilic backbone poly(2-methyl-2-oxazoline-co-2-pentyl-2-oxazoline) (P(MeOx-coPentOx)) prepared via cationic ring opening polymerization (CROP) and hydrophobic poly(D,L-lactide) (PLA) chains were grafted on the backbone via the Huisgen 1,3-dipolar cycloaddition. The “click” copper-catalyzed cycloaddition reaction of azides with alkynes (CuAAC) are successfully achieved and a series of amphiphilic copolymers were prepared containing a backbone with a number average molecular weight of 14.2×103 g.mol-1 and different hydrophobic PLA grafts with various molecular weight (2.8×103 to 12.4×103 g.mol1

). These original architectures of copolymers, when nanoprecipitated in water, the backbone

*

Correspondence to : G. Volet (E-mail: [email protected])

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selective solvent, allowed obtaining various structures of nanoparticles with a hydrodynamic diameter range of 65 nm to 99 nm. More interestingly, a plurality of morphologies going from unilamellar, multilamellar and large compound vesicles to core-shell nanoparticles and depending on the PLA molecular weights were evidenced by combining Cryo-Transmission Electron Microscopy (Cryo-TEM) and Small Angle Neutron Scattering (SANS) studies. A first evaluation of their stealthiness by the study of the stability and the interaction of these nano-objects with a model protein revealed the role played by the P(MeOx-co-PentOx) in these interactions, demonstrating the utility of this amphiphilic graft copolymers platform with well-defined architectures for design of nanocarriers in drug delivery applications.

INTRODUCTION Over the last few years, amphiphilic block copolymers self-assembly in solution has been extensively studied by groups led by Eisenberg, Bates, Discher, Kataoka and Lecommandoux.1-8 The various morphologies obtained depend mainly on the hydrophiliclypophilic balance.2, 9 Potential applications in the field of drug and gene delivery have been a particular recent focus.10-14 However, the major recurrent problem in the development of injectable polymeric delivery system is the initial burst release of drug which occurs during the first minutes after contact of the nano-carriers with external medium.15 This could constitute a serious limitation for in vivo application, particularly for the active drugs that need to be released over a long time and those that are toxic at high concentration.16 We can think that the elaboration of multilamellar nanoparticles or compound vesicles as nanocarriers can reduce this phenomenon due to the presence of various hydrophilic domains in the structure which can slow down the spread of a hydrophobic drug. Another advantage of these “reservoir systems” is the simultaneous encapsulation of the hydrophilic and 2 ACS Paragon Plus Environment

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hydrophobic drugs.17 Indeed, the cavities can contain the hydrophilic compound, whereas the membrane traps the hydrophobic drug. This strategy was exploited to highlight the efficiency of a cocktail of water-insoluble paclitaxel with water-soluble doxorubicin and it led to better tumor regression compared to either drug alone.17 Recent theoretical works and simulations18, 19

showed the possibility to obtain these promising structures of nanoparticles (NPs) by self-

assembly of amphiphilic graft copolymers. However these copolymers architectures have been less extensively studied than their linear diblock counterparts and there is an open question about their morphological diversity although few experimental works report on it.2024

It should be outlined that the few examples reported here are related to graft copolymers

composed of a hydrophilic polymer backbone and hydrophobic grafts and the self-assembly of amphiphilic graft copolymers in a backbone selective solvent has not received much attention yet. In this work, amphiphilic graft copolymers made of poly(D,L-lactide) (PLA) grafts fixed on hydrophilic backbone based on poly(2-methyl-2-oxazoline) (PMeOx) were synthesized. Poly(D,L-lactide) (PLA) are synthetic polymers of special interest because of their biocompatibility and biodegradability. They have tremendous applications as engineering plastics and within the biomedical field including resorbable bone pins and screws, scaffolds for cells in tissue engineering, and drug-delivery systems.25, 26 An attractive feature of graft polyester copolymers is their potential to act as building blocks for elaboration of nanomaterials thanks to the hydrolytically degradable polyester grafts.27 For instance Leonard and coworkers28,

29

have prepared NPs from PLA graft dextrans which exhibited good

colloidal stability. Furthermore, poly(2-alkyl-2-oxazoline) are attracting growing interest nowadays because of their highly tunable structure, versatile properties and favorable biological safety profiles compared to the Poly(ethylene glycol) (PEG).30-32 The presence of PEG is of particular interest, as PEG-functionalized polymers can be used in aqueous media, 3 ACS Paragon Plus Environment

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are resistant to protein adsorption and exhibit enhanced residence time in delivery applications.33,

34

However, over the last few years, several studies showed important

disadvantages of PEG.35, 36 Because of the intensive use of PEG, the PEGylated systems have lost the aimed properties at once applied in vivo, due to the existence of specific and nonspecific recognition by the immune system.37-39 Thanks to its hydrophilicity, its biocompatibility, stealth behavior and biodistribution, PMeOx closely resemble the beneficial properties of PEG and is a potential alternative to PEG for use in biomedical applications.31, 32, 40

Poly(2-alkyl-2-oxazoline)s are accessible via living cationic ring opening polymerization

(CROP) of 2-alkyl-2-oxazoline which allows incorporation of various functionalities in the polymer. Recently, we reported on the synthesis of a thermoresponsive copolymer poly(2methyl-2-oxazoline-co-2-(5-azidopentyl)-2-oxazoline)

(P(MeOx-co-N3PentOx))

bearing

pendant azido groups on the macromolecular chain.41, 42 This azido copolymer can be coupled to alkyne end-functionalized poly(D,L-lactide) by the grafting-onto approach through the Huisgen 1,3-dipolar cycloaddition reaction to prepare graft copolymer composed of a hydrophilic backbone and hydrophobic grafts.43 The main objectives of this paper are to report on the synthesis, characterization and self-assembling properties in a selective solvent of a series of amphiphilic graft copolymer poly(2-methyl-2-oxazoline-co-2-pentyl-2-oxazoline)-g-poly(D,L-lactide)

(P(MeOx-co-

PentOx)-g-PLA). An earlier work by our group43 has shown that different self-assembling morphologies (polymersomes, core/shell or nanospheres self-assemblies) were obtained when the hydrophilic/lipophilic balance was varied for copolymers bearing 6 to 8 PLA graft per chains. It is obvious that these copolymers offer-up numerous self-assembling possibilities considering the large number of ways to alter the copolymer parameters, which can be done by varying independently the molecular weight of the main chain, of the PLA grafts and the number of PLA graft per chain. The present work will demonstrate that focusing on a 4 ACS Paragon Plus Environment

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different range of copolymer parameters, particularly a larger number of PLA grafts per chain (16), more intriguing morphologies can be obtained. The strategy to prepare these graft copolymers,43 is to use a rapid and efficient method for CuAAC with copper nanoparticles easily removable by centrifugation.44 The coupling reaction was led under microwave irradiation to reduce the time of reaction. Graft copolymers were characterized by 1H NMR, COSY NMR, FTIR, Raman spectroscopy, thermogravimetric analysis (TGA), and Inductively Coupled Plasma–Optical Emission Spectroscopy (ICP-OES). Copolymers with various hydrophilic/lipophilic balance were synthesized by grafting PLA blocks of molecular weights varying from 2.8×103 to 12.4×103 g.mol-1 on a backbone of poly(2-methyl-2-oxazoline-co-2pentyl-2-oxazoline) of 14.2×103 g.mol-1. Nanoparticles from these graft copolymers were prepared by nanoprecipitation in water, the selective solvent of the backbone. The structure of the nanoparticles was explored by two independent and complementary experimental approaches. The direct imaging of the geometrical shape of the objects was explored by cryotransmission electron microscopy (cryo-TEM) and small angle neutron scattering (SANS) was used to determine averaged nanoscopic dimensions and structures of the objects through an appropriate model. In addition, Bovine serum albumin (BSA) was used as a model protein mimicking the behavior of the opsonins of the immune system45-47 for an interaction study with the NPs. Their non-bio-adhesive properties were evaluated to show the potential use of these NPs as stealth nanocarriers.

MATERIALS AND METHODS Materials Anhydrous acetonitrile (ACN), 1-iodobutane (purity 99%), anhydrous methanol were purchased from Aldrich and used as received. Dimethylsulfoxide (DMSO), diethyl ether, 5 ACS Paragon Plus Environment

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tetrahydrofuran (THF) and chloroform were purchased from Carlo Erba and used without further purification. The monomers 2-(5-azidopentyl)-2-oxazoline (prepared in the laboratory according the procedure described previously42) and 2-methyl-2-oxazoline (Sigma-Aldrich, Milwaukee WI, USA, purity 99 %) were dried overnight over calcium hydride and purified by distillation under an argon atmosphere. Copper commercially available nanopowder (Cu NPs) with nominal size from 20 nm to 40 nm was purchased from Alfa Aesar. The Bovine serum albumin (BSA) was supplied by Sigma-Aldrich.

Microwave assisted synthesis The synthesis under microwave irradiation was performed using a Monowave-300 reactor from Anton Paar. The microwave source was a magnetron with a 2.5 GHz frequency powered by a 900 W power generator, which could be operated at different power levels. The reaction mixtures were loaded inside quartz glass tubes with a magnetic stirrer; the tubes were closed with silicon caps using a pressure monitor unit, and placed into the reactor. The microwave reactor was programmed to maintain a constant temperature by adjusting the applied power. After the end of the reaction, solutions were cooled to room temperature with compressed air.

Synthesis of graft copolymers Synthesis

of

poly(2-methyl-2-oxazoline-co-2-(5-azidopentyl)-2-oxazoline)

(P(MeOx-co-

N3PentOx)) The synthesis of copolymers has been reported elsewhere.41,

42

Briefly, the cationic ring-

opening polymerization of 2-methyl-2-oxazoline with a home-made monomer 2-(5azidopentyl)-2-oxazoline (N3PentOx)42 was initiated by 1-iodobutane in acetonitrile. 6 ACS Paragon Plus Environment

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Copolymerization was stopped by adding an excess of basic solution of methanol (Scheme 1). The copolymer was recovered by precipitation into diethyl ether. In this study, a copolymer with a constant N3PentOx graft (11 mol% corresponding to 16 N3PentOx units determined by 1

H NMR) and a molecular weight (Mn = 14.2×103 g.mol-1 also determined by 1H NMR) was

synthesized (yield = 95 %). Unfortunately, the dispersity of this copolymer with 16 N3PentOx units cannot be determined by size exclusion chromatography because of the interaction between the azide groups and the column. However, when the number of azide groups is low enough to allow the analysis, dispersity determined in water is close to 1.1.41 Synthesis of alkynyl-terminated poly(D,L-lactide) Alkynyl-terminated

poly(D,L-lactide)

(PLA)

oligomers

were

synthesized

by

the

transesterification of high-molecular-weight PLA in presence of propargyl alcohol and dibutyltin dilaurate as reported elsewhere.42 The desired molar mass of the PLA obtained was dependent of the reaction time. The polymers were recovered by precipitation in petroleum ether. Four polymers of different number-average molecular weight determined by 1H NMR 2.8×103 ; 4.9×103 ; 8.0×103 ; 12.4×103 g.mol-1 were synthesized (yield = 90-95 %). The dispersities of this PLA determined by size exclusion chromatography in THF are from 1.3 to 1.5. Synthesis of graft copolymers P[(MeOx-co-PentOx)-g-PLA] A series of graft copolymers P[(MeOx-co-PentOx)-g-PLA] were prepared by coupling the alkynyl-PLA onto the copolymer-bound azides of the P(MeOx-co-N3PentOx). A previous test was carried out using CuI/PMDETA as catalytic entity and thermal heat.42 In this work, the strategy was to use commercially available copper nanoparticles in combination with microwave reaction conditions. A typical procedure was as follows : P(MeOx-co-N3PentOx) of 14.2×103 g.mol-1 (97.3 mg, 6.85×10-3 mmol corresponding to 1.15×10-1 mmol of azide 7 ACS Paragon Plus Environment

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units determined by 1H NMR), and alkynyl-terminated PLA of 4.9×103 g.mol-1 (564 mg, 1.15×10-1 mmol of alkylnyl function) were dissolved in DMSO (5 mL) and copper nanoparticles (10 mg) were added. The reaction mixture was heated at 80°C under microwave irradiation and stirred for 20 min. The copper nanoparticles were removed by centrifugation at 4000 rpm for 10 min. The supernatant was centrifuged a second time and the DMSO removed by dialysis against water. The graft copolymer was recovered by freeze-drying (yield = 65-86 %).

Preparation of nanoparticles Self-assembled P[(MeOx-co-PentOx)-g-PLA] nanoparticles were prepared using the nanoprecipitation method. Graft copolymer (10 mg) was dissolved in 5 mL of acetonitrile or tetrahydrofuran to have a concentration of 2 g.L-1 and the solution was stirred during 24 hours. 1 mL of solution of graft copolymer in organic solvent was precipitated dropwise in 2 mL of ultrapure water at 25 °C stirred at 220 rpm. After adding all the solution to the aqueous phase, a homogeneous nanodispersion was observed. The organic solvent was removed from the dispersion by evaporation under reduced pressure and the volume was adjusted by addition of ultrapure water to obtain a concentration of 1 g.L-1.

BSA adsorption study BSA adsorption study was carried out for NPs G16-2, G16-3 and G16-4 and for bare PLA NPs as control following the protocol described by Rouzes48 and El Fagui49. More precisely, a solution of BSA (1.4 g.L-1 was prepared in a buffer solution (2 × 10-2 mol L-1 H2PO4Na / HPO4Na2; pH = 7.4) and added to a NPs suspension (1v/1v) to get a concentration of BSA of

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0.7 g.L-1 (10-2 mol L-1 H2PO4Na / HPO4Na2; pH = 7.4) and a concentration of copolymers of 0.5 g.L-1.These mixtures were stored at 37 ° C in a thermostatic bath and a sample was taken after 15, 24 and about 48 hours. The samples were ultracentrifuged at 40 000 rpm for 20 minutes, allowing to get a pellet after recovery of the supernatant. This pellet was then redispersed in water. The hydrodynamic diameter of the re-dispersed nano-objects was determined by DLS and compared with the initial hydrodynamic diameter of NPs. A second ultracentrifugation of the supernatant at 100,000 rpm for 40 minutes enabled the elimination of residual NPs and to obtain a supernatant free of NPs. The concentration of free BSA in the supernatant was determined by UV-Visible spectrometry at 278 nm (with a spectrophotometer UV-Visible Agilent Cay 60).

Characterizations and analytical techniques Analytical methods FTIR spectra were recorded on a Bruker Tensor 27 instrument equipped with a Digi Tect DLaTGS detector; 32 scans were collected at a resolution of 1 cm−1 using an ATR accessory (single-reflection horizontal ATR, Miracle, Pike Technologies). Chemical structure of the polymers was investigated using a Raman apparatus Xplora from Horiba Jobin Yvon (Longjumeau, France) equipped with a laser emitting at 638 nm. The acquisition time was fixed at 1 min. 1H NMR and COSY NMR spectra were recorded in CDCl3 or deuterated DMSO using a Bruker Advance 400 MHz spectrometer. Size exclusion chromatography (SEC) was performed on a chromatograph equipped with a pump P 100 (Spectra-Physics, Fremont, CA, USA), a Rheodyne injector and a set of two columns PL-gel Mixed C for the analysis of polymers in tetrahydrofuran. Two detectors were connected in series at the end of the columns: a differential refractometer RI 71 (Shodex, Japan) and a MiniDawn light 9 ACS Paragon Plus Environment

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scattering detector (Wyatt Technology, Santa Barbara, CA, USA). The chromatographic analysis of polymers was done with polymer solutions at a concentration of 10 mg.mL-1. Inductively Coupled Plasma–Optical Emission Spectroscopy (ICP-OES) measurements were conducted with a Varian Vista-PRO CCD Simultaneous ICP-OES for the determination of trace of copper in copolymers. The mineralization of the samples is done by solubilizing 20 mg in 1 mL of superpure HNO3, then a dry residue is obtained after 3 h at 150 °C. This residue was redissolved in 15 mL of a solution of HNO3 0.1 mol.L-1 and the solution was injected into the plasma in the form of aerosol. Thermogravimetric analysis (TGA) was performed on a Setaram Setsys Evolution 16 apparatus by heating the samples at a rate of 20 °C.min-1 from 20 to 800°C under argon atmosphere. Size measurements The nanoparticles were characterized by dynamic light scattering (DLS). All measurements were performed at 25 °C with a Zetasizer Nano ZS from Malvern Instruments operating with a He–Ne laser source (wavelength 633 nm, scattering angle 173°). The correlation functions were analyzed using the cumulants method. The mean diameters and polydispersity index (PDI) of the nanoparticles were derived on the basis of volume distribution. Cryo-Transmission Electron Microscopy The morphology and the size of nanoparticles (NPs) were determined from CryoTransmission Electron Microscopy (cryo-TEM) micrographs. The nanoparticles solutions were prepared at the concentration of 1 g.L-1 and the solutions were concentrated under vacuum to obtain a concentration of 3 g.L-1. A drop of NPs dispersion was deposited on “quantifoil”® (Quantifoil Micro Tools GmbH, Germany) carbon membrane. The excess of liquid on the membrane was absorbed with a filter paper and the membrane was quenchfrozen quickly in liquid ethane to form a thin vitreous ice film including NPs in the holes of 10 ACS Paragon Plus Environment

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the grid. Once placed in a Gatan 626 cryo-holder cooled with liquid nitrogen, the samples were transferred in the microscope and observed at low temperature (-180 °C). Cryo-TEM micrographs were recorded on ultrascan 2k CCD camera (Gatan, USA), using a LaB6 JEOL 2100 (JEOL, Japan) cryo microscope operating at 200kV with a JEOL low dose system (Minimum Dose System, MDS) to protect the thin ice film from any irradiation before imaging and reduce the irradiation during the image capture. The average diameters were calculated on at least 100 nanoparticles. Neutron scattering measurements Small-angle neutron scattering (SANS) measurements were performed at the Laboratoire Léon Brillouin (“ ORPHEE ” reactor, CEA Saclay) on the “ PACE ” spectrometer. The experimental scattering vector q ( q = ( 4π / λ ) sin( θ / 2 ) ) range was 0.0032 < q (Å-1) < 0.44, and were covered by three sample-to-detector distances and two different wavelengths (1 m or 3 m at the neutron wavelength of 5 Å and 4.5 m at 13 Å). The samples were loaded into Hellma quartz cells with a 2 mm optical path length. The cells were placed in a sample changer and the scattering for each sample was measured for about 1.5 hours at room temperature. Scattering intensities from solutions were corrected for empty cell scattering, and sample transmission. I(q) is in absolute scale (cm-1). For all samples, the solvent scattering intensity was substracted but we have not succeeded to completely remove the sample background (which is inferior to 5×10-2 cm-1). All the polymer solutions were prepared at a concentration of 10 g.L-1 in deuterated acetonitrile. These solutions were kept under stirring for 24 hours before measurement by SANS. All nanoprecipitated samples were prepared as previously explained and then dialyzed against D2O.

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Synthesis and characterization of graft copolymers A series of graft copolymers P[(MeOx-co-PentOx)-g-PLA] was synthesized as represented in Scheme 1. First, poly(2-methyl-2-oxazoline-co-2-(5-azidopentyl)-2-oxazoline) (P(MeOx-coN3PentOx)) was prepared by cationic ring opening polymerization according to a procedure described previously by our group41, 42 and fully characterized using 1H NMR and SANS.43 A copolymer with  = 14.2×103 g.mol-1 was prepared bearing 11 mol% N3PentOx grafts  corresponding to 16 N3PentOx units. In a second step, an alkyne group was introduced at the end of the PLA oligomers via a transesterification of high-molecular-weight PLA according a procedure described previously.42 Four polymers of different number-average molecular weights (2.8×103 to 12.4×103 g.mol-1) were characterized by SEC and by 1H NMR (Table S1). Alkyne-functionalized PLA were coupled with azide moiety belonging to P(MeOx-coPentOx) through 1,2,3 triazole ring using Huisgen [3+2] cycloaddition “click” chemistry between azide and alkyne.50,

51

Usually, this reaction is catalyzed using a non-negligible

amount of copper in the form of salts, but its residual quantity after purification may pose problem with regard to biological applications. In the present work, the “click” reaction was catalyzed by copper nanoparticles (Cu NPs) in dimethylsulfoxide at 80°C. The [N3]/[C≡C] feed ratio was maintained at 1/1, and the reaction was conducted for 20 min under microwave irradiation. The purification of the graft copolymers was simplified by the use of copper nanoparticles as the Cu NPs could be easily removed by centrifugation. The level of residual copper in the purified copolymers was analyzed by Inductively Coupled Plasma–Optical Emission Spectroscopy (ICP-OES) measurements. The obtained value of 0.09 ppm is an inferior value to the reference value of 2 ppm for serum copper in humans.52 The efficiency of the coupling reaction between the azido pendant groups and acetylene chain ends under microwave irradiation was followed by FTIR, Raman spectroscopy, 1H and COSY NMR. The success of the click reaction was confirmed by FTIR with the total disappearance 12 ACS Paragon Plus Environment

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of the azide signal at 2097 cm-1 (Figure 1). In the Raman spectra (Figure S1), the peaks of the alkyne function of PLA at 2131 cm-1 and of the azide group at 2097 cm-1 disappeared after the cycloaddition which showed a complete coupling. The chemical structure of P(MeOx-co-PentOx)-g-PLA was investigated by 1H NMR in CDCl3 (Figure 2). Coupling was attested by the appearance of the triazole proton t at 7.59 ppm and by the downfield shift of the signal 1 (from 3.26 to 4.32 ppm) observed for the methylene protons in α of triazole compared to the one in α of the azide group. Futhermore, the signal 1’ corresponding to methylene proton of PLA in α of the triazole ring was totally shifted (from 4.69 in native PLA to 5.13 ppm in graft PLA). In COSY NMR (Figure S2), the absence of correlation signal corresponding to the methylene proton adjacent to terminal alkyne group before the cycloaddition confirmed the success of the click reaction and the absence of free alkynyl-terminated PLA. The molecular weight of PLA grafts was calculated from 1H NMR in deuterated DMSO (data not shown) using the integration of the signal at 1.45 ppm, which corresponds to the lateral methyl of the repetitive unit of PLA, and the integration of the signal 11 at 4.19 ppm, corresponding to the proton in α of the terminal alcohol of the PLA chain or the integration of the signal 1 at 4.30 ppm corresponding to the protons of the pendant chain in α of triazole. The values obtained were very similar to the one calculated before the click reaction (Table S1). Knowing the  of the P(MeOx-co-N3PentOx) backbone, of PLA grafts and their  number,  of P(MeOx-co-PentOx)-g-PLA was estimated and reported in Table 1. The  values are close to the one expected (Table 1). The copolymers were also characterized by SANS in deuterated acetonitrile which is a solvent of both macromolecular chains POXZ and PLA. At small angles, the scattering intensity can be approximated by the Zimm equation (1), yielding a radius of gyration (Rg).

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2 1 1  Rg 2  = 1+ q  I I 0  3 

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(1)

2

In the plots 1/I versus q (not shown), only the linear part of the scattering data (such as I.q²