PhD Dario Carradori

PhD Claudia Balducci

PhD Francesca Re
email: francesca.re1@unimib.it

PhD Davide Brambilla

PhD Benjamin Le Droumaguet

PhD Orfeu Flores

PhD Alice Gaudin¸

PhD Simona Mura

PhD Gianluigi Forloni

PhD Lara Ordoñez-Gutierrez

PhD Francisco Wandosell

PhD Massimo Masserini

PhD Patrick Couvreur

PhD Julien Nicolas
email: julien.nicolas@u-psud.fr.

PhD Karine Andrieux

Antibody-Functionalized Polymer Nanoparticle Leading to Memory Recovery in Alzheimer's Disease-like Transgenic Mouse Model

Keywords: Polymer Nanoparticles, Alzheimer's disease, antibody, blood-brain barrier, -amyloid peptide

Alzheimer's disease (AD) is a neurodegenerative disorder related, in part, to the accumulation of amyloid-β peptide (Aβ) and especially the Aβ peptide 1-42 (Aβ 1-42 ). The aim of this study was to design nanocarriers able to: (i) interact with the Aβ 1-42 in the blood and promote its elimination through the "sink effect" and (ii) correct the memory defect observed in AD-like transgenic mice. To do so, biodegradable, PEGylated nanoparticles were surfacefunctionalized with an antibody directed against Aβ 1-42 . Treatment of AD-like transgenic mice with anti-Aβ 1-42 -functionalized nanoparticles led to: (i) complete correction of the memory defect; (ii) significant reduction of the Aβ soluble peptide and its oligomer level in the brain and (iii) significant increase of the Aβ levels in plasma. This study represents the first example of Aβ 1-42 monoclonal antibody-decorated nanoparticle-based therapy against AD leading to complete correction of the memory defect in an experimental model of AD.

Background

Alzheimer's disease (AD) is a neurodegenerative disease with the highest incidence compared to other types of dementia. [START_REF] Anand | Therapeutics of Alzheimer's disease: Past, present and future[END_REF][START_REF] Kumar | A review on Alzheimer's disease pathophysiology and its management: an update[END_REF] Mid-2015, the World Health Organization (WHO) estimated that AD affected about 36 million people worldwide with an overall economic impact of nearly 818 billion USD per year. [START_REF] Prince | World Alzheimer Report 2015: The Global Impact of Dementia: an Analysis of Prevalence, Incidence, Cost and Trends[END_REF] Although AD is the focus of intensive research, its pathogenesis remains only partially understood and there is still no cure available.

The two most prominent histopathological hallmarks of AD are extracellular plaques, mainly composed of amyloid-β peptide (Aβ) species, and intracellular neurofibrillary tangles made of hyperphosphorylated τ protein. [START_REF] Stelzmann | An english translation of alzheimer's 1907 paper, "über eine eigenartige erkankung der hirnrinde[END_REF] Several lines of evidence converge towards what is known as the "amyloid hypothesis", according to which the neurofibrillary tangles and the associated neurodegeneration are believed to be secondary to the accumulation of Aβ species, which would result from the imbalance between the production of Aβ and its clearance from the brain. [START_REF] Karran | The amyloid cascade hypothesis for Alzheimer's disease: an appraisal for the development of therapeutics[END_REF] A considerable amount of knowledge has been accumulated since the first report of purification and characterization of Aβ species from the brain tissue of AD patients. [START_REF] Glenner | Alzheimer's disease: initial report of the purification and characterization of a novel cerebrovascular amyloid protein[END_REF] It is now well known that Aβ is produced by the proteolytic cleavage of a larger amyloid precursor protein (APP), a constitutively expressed transmembrane glycoprotein with still unclear functions. [START_REF] Goldgaber | Isolation, characterization, and chromosomal localization of human brain cDNA clones coding for the precursor of the amyloid of brain in Alzheimer's disease, Down's syndrome and aging[END_REF][START_REF] Van Der Kant | Cellular Functions of the Amyloid Precursor Protein from Development to Dementia[END_REF] Under normal conditions, the non-amyloidogenic cleavage of APP by αand γsecretases predominates and generates different non-aggregating fragments. [START_REF] Muresan | Amyloid-β precursor protein: Multiple fragments, numerous transport routes and mechanisms[END_REF] Alternatively, the proteolysis of the APP by the sequential enzymatic actions of β-site amyloid precursor protein-cleaving enzyme 1 (BACE1 or β-secretase) and γ-secretase leads to the production of Aβ of 37-42 amino acids found in the senile plaques of AD brains. [START_REF] Hardy | The amyloid hypothesis of Alzheimer's disease: progress and problems on the road to therapeutics[END_REF] Among the different species, the Aβ peptide 1-42 (Aβ 1-42 ) is believed to be the most representative and toxic species in AD physiopathology because of its high tendency to self-aggregate, [START_REF] Allaman | Amyloid-beta aggregates cause alterations of astrocytic metabolic phenotype: impact on neuronal viability[END_REF][START_REF] Garcia-Matas | In Vitro and In Vivo Activation of Astrocytes by Amyloid-beta is Potentiated by Pro-Oxidant Agents[END_REF] whereas the soluble Aβ 1-42 oligomer is toxic to neurons. [START_REF] Cizas | Size-dependent neurotoxicity of β-amyloid oligomers[END_REF][START_REF] Maezawa | Amyloid-beta protein oligomer at low nanomolar concentrations activates microglia and induces microglial neurotoxicity[END_REF] Aβ is indeed naturally prone to self-aggregation owing to its enriched sequence in hydrophobic residues. [START_REF] Pitschke | Detection of single amyloid [beta]protein aggregates in the cerebrospinal fluid of Alzheimer's patients by fluorescence correlation spectroscopy[END_REF][START_REF] Mastrangelo | Highresolution Atomic Force Microscopy of Soluble Aβ42 Oligomers[END_REF] It was recently shown that Aβ oligomers, rather than the monomeric form or misfolded Aβ fibrils, are mainly responsible for neurotoxicity. [START_REF] Hardy | The amyloid hypothesis of Alzheimer's disease: progress and problems on the road to therapeutics[END_REF] Alongside Aβ oligomerization and deposition, synaptic and neuritic injury occur, as well as alterations in neuronal ionic homeostasis, oxidative injury and altered kinase/phosphatase activities (responsible for the neurofibrillary tangles), which altogether ultimately lead to neuronal dysfunction and cellular death. [START_REF] Eckman | An update on the amyloid hypothesis[END_REF] In the past few years, a number of different therapeutic strategies against AD have been investigated, including: (i) inhibition/modulation of the secretase activity; (ii) enhancement of Aβ clearance from the brain with antibodies or nanoparticulate systems; (iii) decreasing of cholesterol levels (given the link between high cholesterol levels and AD incidence); (iv) chelation of metallic ions needed for Aβ aggregation and (v) protection of neurons against synaptic dysfunction and degeneration. [START_REF] Brambilla | Nanotechnologies for Alzheimer's disease: therapy, diagnosis and safety issues[END_REF][START_REF] Barone | Statins more than cholesterol lowering agents in Alzheimer disease: Their pleiotropic functions as potential therapeutic targets[END_REF][START_REF] Fernandez | Potential immunotargets for Alzheimer's disease treatment strategies[END_REF][START_REF] Eskici | Copper and Oxidative Stress in the Pathogenesis of Alzheimer's Disease[END_REF] However, current treatments against AD (e.g., NMDA receptor antagonist, [START_REF] Hellweg | Efficacy of memantine in delaying clinical worsening in Alzheimer's disease (AD): responder analyses of nine clinical trials with patients with moderate to severe AD[END_REF] AchE inhibitors [START_REF] Hansen | Efficacy and safety of donepezil, galantamine, and rivastigmine for the treatment of Alzheimer's disease: a systematic review and meta-analysis[END_REF] ) are solely palliative and limited to the alleviation of symptoms and behavioral disturbances. [START_REF] Querfurth | Alzheimer's Disease[END_REF][START_REF] Allgaier | An update on drug treatment options of Alzheimer's disease[END_REF] Lately, clearance of Aβ from the brain was identified as one of the most ambitious strategies for potential treatment of AD. [START_REF] Toyn | What lessons can be learned from failed Alzheimer's disease trials?[END_REF] The physiological elimination of Aβ 1-42 peptide from the brain towards the blood has been shown to be promoted by receptors like low-density lipoprotein receptor related protein-1 (LRP1) and P-glycoprotein (P-gp).

Nanotechnology and its application to the biomedical field (usually termed nanomedicine) hold great hope for the treatment of severe diseases. [START_REF] Sanhai | Seven challenges for nanomedicine[END_REF][START_REF] Farokhzad | Impact of Nanotechnology on Drug Delivery[END_REF] So far, only three nanoparticulate systems with enhanced affinity for Aβ have been designed to counterbalance the deficient Aβ clearance mechanism in vivo: (i) PEGylated liposomes bifunctionalized with a peptide deriving from the Apo E receptor-binding domain for blood-brain barrier (BBB) targeting and phosphatidic acid for Aβ binding, 29,[START_REF] Balducci | Multifunctional Liposomes Reduce Brain β-Amyloid Burden and Ameliorate Memory Impairment in Alzheimer's Disease Mouse Models[END_REF] (ii) Apo E3-reconstituted high density lipoprotein (Apo E3-rHDL) [START_REF] Song | Lipoprotein-Based Nanoparticles Rescue the Memory Loss of Mice with Alzheimer's Disease by Accelerating the Clearance of Amyloid-Beta[END_REF] and (iii) liposomes surface-functionalized by a monoclonal anti-A antibody. [START_REF] Ordóñez-Gutiérrez | ImmunoPEGliposome-mediated reduction of blood and brain amyloid levels in a mouse model of Alzheimer's disease is restricted to aged animals[END_REF] This latter study is very interesting owing to the high affinity and specificity of antibodies towards their targets (antigens), reducing off-site binding and preventing substantial decrease of the therapeutic efficiency. [START_REF] Fay | Antibody-targeted nanoparticles for cancer therapy[END_REF] However, a proof of efficacy; that is the correction of the memory defect, was not investigated and thus, to date, no successful in vivo example of antibody-decorated nanoparticle-based therapy against AD leading to complete memory recovery has ever been reported, which would represent a crucial step forward.

Among the different classes of materials suitable for drug delivery purposes, biodegradable polymer nanoparticles have attracted much attention due to their safety, the flexibility offered by macromolecular synthesis methods, the wide range of (co)polymer compositions and the possibility to functionalize them for targeting and/or imaging purposes. [START_REF] Nicolas | Design and Functionalization Strategies for Biodegradable/Biocompatible Polymer-Based Nanoparticles Applied in Targeted Drug Delivery[END_REF] In this context, poly(alkyl cyanoacrylate) (PACA) nanoparticles represent a wellestablished drug delivery platform leading to promising results against different pathologies such as cancer or infections [START_REF] De Verdiere | Reversion of multidrug resistance with polyalkylcyanoacrylate nanoparticles: towards a mechanism of action[END_REF][START_REF] Balland | The uptake of ampicillin-loaded nanoparticles by murine macrophages infected with Salmonella typhimurium[END_REF][START_REF] Nicolas | Synthesis of poly(alkyl cyanoacrylate)-based colloidal nanomedicines[END_REF][START_REF] Vauthier | Poly(alkylcyanoacrylates) as biodegradable materials for biomedical applications[END_REF] and whose degradation is catalyzed in vivo by esterases. [START_REF] Lenaerts | Degradation of poly(isobutyl cyanoacrylate) nanoparticles[END_REF] Importantly, doxorubicin-loaded poly(isohexyl cyanoacrylate) nanoparticles 40 are currently in phase III clinical trial for the treatment of MDR resistant hepatocarcinoma. Moreover, it was evidenced that PEGylated nanoparticles of poly[hexadecyl cyanoacrylate-comethoxypoly(ethylene glycol) cyanoacrylate] (P(HDCA-co-MePEGCA)) (denoted as NPs in this paper) exhibited affinity towards the Aβ 1-42 peptide, as observed by surface plasmon resonance (SPR), where preferential interaction between PEG chains and Aβ have been established. [START_REF] Brambilla | PEGylated Nanoparticles Bind to and Alter Amyloid-Beta Peptide Conformation: Toward Engineering of Functional Nanomedicines for Alzheimer's Disease[END_REF] This affinity was further strongly increased via 'click chemistry'-based surfacefunctionalization by different A targeting ligands such as curcumin derivatives (Curcumin-NPs) and anti-Aβ 1-42 monoclonal antibody (anti-Aβ 1-42 -NPs, see Figure 1). [START_REF] Droumaguet | Versatile and Efficient Targeting from a Single Nanoparticulate Platform: Application to Cancer and Alzheimer's Disease[END_REF] SPR experiments evidenced an enhanced affinity towards Aβ 1-42 monomers (and fibrils) thanks to this surface modification, with K D values of 0.8 µM (and 0.3 µM) and ~700 pM (and ~300 pM) for Curcumin-NPs and anti-Aβ 1-42 -NPs, respectively. [START_REF] Droumaguet | Versatile and Efficient Targeting from a Single Nanoparticulate Platform: Application to Cancer and Alzheimer's Disease[END_REF] Thus, anti-Aβ 1-42 -NPs appeared as the most promising system to target Aβ peptide to inhibit its aggregation, to favor its clearance from the brain and to finally decrease its neurotoxicity. However, the robustness of this strategy and therefore its potential as therapy against AD mainly depend on whether these nanocarriers can demonstrate improved biological performances in vivo in AD-like animal models. In this study, we investigated the in vivo pharmacological efficacy of anti-Aβ 1-42 -NPs in ADlike transgenic mice and demonstrated that they were able: (i) to bind blood-circulating Aβ 1- 42 , (ii) to eliminate Aβ 1-42 species through nanoparticle clearance pathways and (iii) to decrease Aβ 1-42 brain levels, likely via the sink effect. [START_REF] Toyn | What lessons can be learned from failed Alzheimer's disease trials?[END_REF] Importantly, AD-like transgenic mice treated with anti-Aβ 1-42 -NPs showed a complete correction of the memory defect, together with a significant reduction of the Aβ soluble peptide and oligomer brain levels, resulting in a significant increase of the Aβ levels in plasma. This study represents the first example of Aβ 1- 42 monoclonal antibody-decorated nanoparticle-based therapy against AD leading to complete correction of the memory defect in an experimental model of AD.

Methods

Materials

Poly[(hexadecyl cyanoacrylate-co-rhodamine B cyanoacrylate-co-methoxypoly(ethylene glycol cyanoacrylate)] (P(HDCA-co-RCA-co-MePEGCA)) and poly[methoxypoly(ethylene glycol) cyanoacrylate-co-Biotin-poly(ethylene glycol) cyanoacrylate-co-hexadecyl cyanoacrylate] (MePEGCA-co-Bio-PEGCA-co-HDCA) copolymers were obtained following previously reported procedures. [START_REF] Droumaguet | Versatile and Efficient Targeting from a Single Nanoparticulate Platform: Application to Cancer and Alzheimer's Disease[END_REF][START_REF] Brambilla | Design of fluorescently tagged poly(alkyl cyanoacrylate) nanoparticles for human brain endothelial cell imaging[END_REF] Radio-labeled [START_REF] Maezawa | Amyloid-beta protein oligomer at low nanomolar concentrations activates microglia and induces microglial neurotoxicity[END_REF] C-P(HDCA-co-MePEGCA) copolymer was prepared at the Commissariat à l'Energie Atomique (Saclay, France) according to previously described protocols [START_REF] Peracchia | Stealth (R) PEGylated polycyanoacrylate nanoparticles for intravenous administration and splenic targeting[END_REF] and had a specific activity of 5.8 mCi/mg. The Anti-Aβ 1-42 monoclonal Ab was prepared by StabVida (Costa da Caparica, Portugal). Dulbecco's phosphate buffer saline (DPBS) without CaCl 2 and MgCl 2 and EMB-2 medium were purchased from Lonza. Penicillin 10,000 units-streptomycin 10,000 μg.mL -1 , trypsin/EDTA and chemically defined lipid concentrate were obtained from Invitrogen, Gibco. Pluronic F-68, hydrocortisone, trypsine/EDTA 1X, human basic fibroblast growth factor (bFGF) cell culture tested, glycerol, Tris buffer, sodium dodecyl sulfate (SDS) (≥ 98.5%), Triton X100, 3',3",5',5"-tetrabromophenolsulfonphthalein (bromophenol blue), 1,4-dithioerythritol (DTE), acrylamide (40%), tetramethylethylenediamine (TEMED) (~99%), paraformaldehyde (PFA),

CHAPS (3-[(3-cholamidopropyl) dimethylammonio]-1-propanesulfonate), thiourea, urea, dithiothreitol (
DTT), iodocetamide, glycine, Sypro ruby, agarose, Bradford reagent ammoniun persulfate (APS) (98%), iminothiolane hydrochloride (≥ 98%), streptavidin FITC ( ≥ 5 units/mg protein), N-hydroxy-succimide (98%), Supedex Prep Grade 200 gel and ascorbic acid were purchased from Sigma-Aldrich. Acetone (> 99.95%) was purchased from Carlo Erba. Soluene-350 and Hionic-Fluor were purchased from Perkin Elmer.

Nanoparticle preparation and characterization

Fluorescently-and radio-labelled nanoparticles (NPs). The different non-functionalized NPs were prepared following previously described protocols from P(HDCA-co-RCA-co-MePEGCA) or [START_REF] Maezawa | Amyloid-beta protein oligomer at low nanomolar concentrations activates microglia and induces microglial neurotoxicity[END_REF] C-P(HDCA-co-MePEGCA) copolymers. [START_REF] Brambilla | Design of fluorescently tagged poly(alkyl cyanoacrylate) nanoparticles for human brain endothelial cell imaging[END_REF][START_REF] Peracchia | Stealth (R) PEGylated polycyanoacrylate nanoparticles for intravenous administration and splenic targeting[END_REF] For biotinylated NPs (Bio-NPs), 9.5 mg of polymer (either P(HDCA-co-RCA-co-MePEGCA) or [START_REF] Maezawa | Amyloid-beta protein oligomer at low nanomolar concentrations activates microglia and induces microglial neurotoxicity[END_REF] C-P(HDCA-co-MePEGCA)) and 0.5 mg of P(MePEGCA-co-Bio-PEGCA-co-HDCA) were dissolved in acetone (2 mL) and added dropwise to an aqueous solution 0.5% (w/v) of Pluronic F-68 (4 mL) under vigorous stirring. Acetone was then evaporated under reduced pressure and the NPs purified by ultracentrifugation (150,000 g, 1 h, 4 °C, Beckman Coulter). The supernatant was discarded and the pellet was resuspended in the appropriate volume of nanopure water to reach a final concentration of 2.5 mg.mL -1 . All nanoparticle formulations were stored at 4 °C in the dark.

Conjugation of streptavidin FITC with anti-Aβ

In vivo pharmacokinetic and biodistribution of the nanoparticles in normal mice

Animals. 32 Animal treatment. Mice were intravenously (IV) injected with radio-labelled anti-Aβ 1-42 -NPs (40 mg/kg). The dose was selected to allow comparisons with previously published results [START_REF] Balducci | Multifunctional Liposomes Reduce Brain β-Amyloid Burden and Ameliorate Memory Impairment in Alzheimer's Disease Mouse Models[END_REF] and according to the Standard Operating Procedures and Guidelines in animal experimentation. The animals were sacrificed after different times (5 min, 1 h, 4 h and 8 h) by euthanasia with pentobarbital. The blood was withdrawn and further centrifuged (4,500 g, 15 min, room temperature, 5810 R centrifuge) to collect the plasma while liver, spleen, kidneys and brain were collected, weighed, and stored at -80 °C.

Organ analysis. To quantify the amount of NPs, the different tissues were incubated overnight with H 2 O 2 at 50 °C. Then, Soluene-350 was added and the samples were stored for 1 h at 50 °C. Hionic-Fluor was added and the samples were vortexed for 30 s. After 1 h at room temperature, the samples were finally analyzed by scintigraphy (Beckman Coulter L5 6500

Multi-Purpose Scintillation Center).

In vivo evaluation of the nanoparticles in AD-like transgenic mice

Animals. 17 stretching the head toward the object at a distance of not more than 2 cm were scored as object investigation. Twenty-four hours later (test phase) mice were again placed in the arena containing two objects: one of the objects presented during the familiarization phase (familiar object, FAM), and a new, different one (novel object, NOV), and the time spent exploring the two objects was recorded for 10 min. Mice were tested following a pre-defined scheme (5 mice for each treatment group and the remaining mice by following the same scheme) so to precisely maintain the 24 h of re-test for each mouse. Results were expressed as % time of investigation on objects/10 min or as discrimination index (DI), that is: (seconds spent on novelseconds spent on familiar)/(total time spent on objects). Animals with no memory impairment spent longer time investigating the novel object, giving a higher discrimination index.

Brain immunohistochemistry. Tg2576 plaque deposition was examined using the 6E10 monoclonal anti-Aβ antibody (Covance, CA) following a previously reported procedure. [START_REF] Balducci | Multifunctional Liposomes Reduce Brain β-Amyloid Burden and Ameliorate Memory Impairment in Alzheimer's Disease Mouse Models[END_REF] Brain coronal cryostat sections (30 μm; 3 slice/mouse) were incubated for 1 h at RT with blocking solutions (10% normal goat serum in PBS) then overnight at 4°C with the primary antibodies (1:500). After incubation with the anti-mouse biotinylated secondary antibody (1:200; 1h RT, Vector Laboratories) immunostaining was developed using the avidin-biotin kit (Vector Laboratories) and diaminobenzidine (Sigma, Italy). Tissue analysis and image acquisition were done using an Olympus image analyzer and the Cell-R software.

Statistical analysis. Data were expressed as mean ± standard error of the mean (SEM). For the NOR test, data were analysed by a one-way analysis of variance (ANOVA). In the presence of a significant effect of treatment the Tukey's post-hoc test was applied. A p-value < 0.05 was considered significant.

Ex vivo experiments after treatment in AD-like transgenic mice

Brain homogenate preparation and quantification of Aβ 1-42 and Aβ 1-40 . Brain homogenates were prepared as reported elsewhere [START_REF] Balducci | Multifunctional Liposomes Reduce Brain β-Amyloid Burden and Ameliorate Memory Impairment in Alzheimer's Disease Mouse Models[END_REF][START_REF] Mcdonald | The levels of water-soluble and triton-soluble Aβ are increased in Alzheimer's disease brain[END_REF] with some modifications. Frozen mouse brain hemispheres were thawed for ~30 s, weighted and homogenized Western blot analysis of brain Aβ assemblies. Western blot analysis of A oligomers was performed as reported elsewhere [START_REF] Balducci | Multifunctional Liposomes Reduce Brain β-Amyloid Burden and Ameliorate Memory Impairment in Alzheimer's Disease Mouse Models[END_REF][START_REF] Robb | High order W02-reactive stable oligomers of amyloid-beta are produced in vivo and in vitro via dialysis and filtration of synthetic amyloid-beta monomer[END_REF] with some modifications. Aliquots of TX-soluble A following the manufacture's instruction. All samples were analyzed at least in triplicate.

Results

Preparation of anti-Aβ 1-42 -NPs

Nanoparticles with high affinity for the Aβ 1-42 peptide were produced by coupling an anti-Aβ 1-42 monoclonal antibody at the PEG chain ends of P(HDCA-co-MePEGCA) nanoparticles via Bio-SAv ligation. [START_REF] Droumaguet | Versatile and Efficient Targeting from a Single Nanoparticulate Platform: Application to Cancer and Alzheimer's Disease[END_REF] In particular, the P(MePEGCA-co-Bio-PEGCA-co-HDCA) copolymer was nanoprecipitated in water and the resulting Bio-NPs were incubated with a streptavidin FITC-anti-Aβ 1-42 mAb conjugate to yield anti-Aβ 1-42 -NPs (see Methods section).

The nanoparticles presented an average diameter of 125 nm with rather narrow particle size distributions (PDI = 0.1-0.2) and negative ζ-potentials (-20 to -30 mV). These characteristics did not significantly change during the duration of the experiments, thus showing great colloidal stability. To enable accurate tracing during biological evaluations, radio-labelled anti-Aβ 1-42 -NPs were also prepared, simply by blending a small amount of [START_REF] Maezawa | Amyloid-beta protein oligomer at low nanomolar concentrations activates microglia and induces microglial neurotoxicity[END_REF] C-P(HDCA-co-MePEGCA) copolymer with P(MePEGCA-co-Bio-PEGCA-co-HDCA) prior their conanoprecipitation to give 14 C anti-Aβ 1-42 -NPs (average diameter = 182 nm, PDI = 0.22).

Pharmacokinetics and biodistribution of anti-Aβ 1-42 -NPs

To evaluate their biodistribution profile, wild type (WT) mice were intravenously injected with [START_REF] Maezawa | Amyloid-beta protein oligomer at low nanomolar concentrations activates microglia and induces microglial neurotoxicity[END_REF] C anti-Aβ 1-42 -NPs (40 mg/kg) and the radioactivity was quantified at increasing time point in blood, liver, spleen, kidneys and brain (Figure 2). The nanoparticles demonstrated long blood circulation half-life because of their PEG coating (more than 19% of the injected dose/g of organ after 8 h) with preferential accumulation in liver (plateauing at 13% after 1 h), spleen (constantly more than 50% after 1 h) and, to a lesser extent, in kidneys (4-7% at 8 h).

Importantly, although liver accumulation was previously observed for non-functionalized P(HDCA-co-PEGCA) NPs, [START_REF] Vauthier | Poly(alkylcyanoacrylates) as biodegradable materials for biomedical applications[END_REF] the high amount accumulating in the spleen seems peculiar for 14 C anti-Aβ 1-42 -NPs. The spleen has a blood filtration system presenting tight reticular mesh of 200 nm in width. [START_REF] Yoo | Factors that Control the Circulation Time of Nanoparticles in Blood: Challenges, Solutions and Future Prospects[END_REF] Consequently, [START_REF] Maezawa | Amyloid-beta protein oligomer at low nanomolar concentrations activates microglia and induces microglial neurotoxicity[END_REF] C anti-Aβ 1-42 -NPs could be preferentially taken up by the spleen due to their average bigger size compared to the non-functionalized NPs, as previously described for other NPs. [START_REF] Lundy | Distribution of Systemically Administered Nanoparticles Reveals a Size-Dependent Effect Immediately following Cardiac Ischaemia-Reperfusion Injury[END_REF] A low, but detectable amount of 14 C anti-Aβ 1-42 -NPs was also found in the brain (~0.5%), confirming the ability of these NPs to diffuse into the brain tissue to a certain extent. [START_REF] Kim | Translocation of poly(ethylene glycol-co-hexadecyl)cyanoacrylate nanoparticles into rat brain endothelial cells: Role of apolipoproteins in receptor-mediated endocytosis[END_REF][START_REF] Kim | Low-density lipoprotein receptor-mediated endocytosis of PEGylated nanoparticles in rat brain endothelial cells[END_REF]

Brain and plasma levels of Aβ

To elucidate the mechanisms governing the observed in vivo efficacy of anti-Aβ 1-42 -NPs, dosage of Aβ species in brain and plasma of mice was performed. After treatment with anti-Aβ 1-42 -NPs, mice were sacrificed and brain hemispheres were analyzed for the soluble and insoluble Aβ content by ELISA. The results showed that the treatment did not induce any significant reduction of insoluble brain Aβ burden (Figure 4a), whereas a significant reduction (-20 %) of the Triton-soluble Aβ 4b).

Discussion

Altogether, these data support the hypothesis of a mechanism explaining the therapeutic efficacy of anti-Aβ 1-42 -NPs to reduce soluble forms of A and rescue memory in AD mice.

The therapeutic effect observed in vivo could be mainly driven by the higher concentration of anti-Aβ 1-42 -NPs detected in the blood than in the brain, likely through the sink effect. [START_REF] Toyn | What lessons can be learned from failed Alzheimer's disease trials?[END_REF] Nevertheless, we do not exclude the contribution of the NPs found in the brain since low brain concentrations of nanomedicines (e.g., 0.1-0.2 %) have shown significant improvement of neurological and histological phenotype in neurodegenerative disease animal models. [START_REF] Gartziandia | Intranasal Administration of Chitosan-Coated Nanostructured Lipid Carriers Loaded with GDNF Improves Behavioral and Histological Recovery in a Partial Lesion Model of Parkinson's Disease[END_REF] After intravenous injection of anti-Aβ 1-42 -NPs, their surface would be covered by blood-circulating Aβ, followed by clearance of the resulting Aβ-coated NPs by classic nanoparticle elimination pathways such as phagocytic uptake and hepatic filtration or through kidney excretion. [START_REF] Alexis | Factors Affecting the Clearance and Biodistribution of Polymeric Nanoparticles[END_REF] This would result in a global reduction of blood-circulating Aβ, leading to a drastic decrease of its aggregation kinetics. Moreover, the peripheral elimination of Aβ peptides would shift the blood-BBB equilibrium of the peptide leading to Aβ translocation from the brain to the peripheral circulation mediated by receptors like LRP1 or P-gp, in agreement with the socalled "sink effect".

These results demonstrated the beneficial effect of nanoparticles decorated by an Aβtargeted antibody on memory recovery in AD-like transgenic animals and supported the great potential of anti-Aβ oriented research. They also represent a promising step towards antibodydecorated nanoparticle-based therapy against AD and all the results that were collected in this study were in good agreement with the "sink effect" hypothesis. Also, not only the conception of such nanocarriers is rather simple, as it only relied on surface functionalization of PEGylated nanoparticles by an anti-Aβ 1-42 monoclonal antibody, but the therapeutic mechanism (peripheral action in the blood following intravenous injection) also avoids the design of complex multifunctional nanoobjects with, for example, specific ligands to cross the BBB.

acknowledged for their help. Carlo Ferrarese (Department of Neurology, San Gerardo Hospital, Monza), Elisa Conti (Laboratory Neurobiology, School of Medicine, University of Milano-Bicocca, Monza, Italy) are acknowledged for providing blood samples from human AD patients and healthy controls. Frederic Taran and Dominique Georgin (CEA, France) are also warmly acknowledged for the synthesis of [START_REF] Maezawa | Amyloid-beta protein oligomer at low nanomolar concentrations activates microglia and induces microglial neurotoxicity[END_REF] C polymers.

Figure 1 .

 1 Figure 1. Formation of anti-Aβ 1-42 monoclonal antibody (mAb)-decorated PEGylated P(HDCA-co-MePEGCA) nanoparticles (anti-Aβ 1-42 -NPs) from biotinylated P(HDCA-co-MePEGCA) copolymer after nanoprecipitation and incubation with streptavidin (SAav)-anti-Aβ 1-42 mAb.

  1-42 mAb. 2.5 µL of iminothiolane hydrochloride (2 mg.mL -1 in borate buffer 0.16 M pH 8 EDTA 0.1 M) were added to 40 µL of streptavidin FITC (2.5 mg.mL -1 in borate buffer 0.16 M pH 8 EDTA 0.1 M) and incubated for 30 min under stirring. In parallel 2.5 µL of N-hydroxysuccimide (1 mg.mL -1 in DMF) were added to 132.5 µL of anti-Aβ mAb (0.7 mg.mL -1 in DPBS) and incubated for 90 min under stirring. The two solutions were purified by Amicon Ultra 0.5 30 KDa filter centrifugation (18,000 g, 5 min, room temperature) and then mixed together. The final volume was adjusted to 100 µL with PBS, gently stirred for 120 min and then purified by column filtration using Supedex Prep Grade 200 gel. Conjugation was assessed by semi-native polyacrylamide gel electrophoresis (PAGE) (Biorad apparatus) (FigureS1) and the conjugation yield was quantified by spectrofluorimetry (Perkin Elmer Luminescence Spectrometer).Anti-Aβ 1-42 -decorated NPs (anti-Aβ 1-42 -NPs). 1 mL of the purified streptavidin FITC-anti-Aβ 1- 42 mAb conjugate was incubated with 1 mL of Bio-NPs (radiolabeled or not) for 30 min. Anti-Aβ 1-42 -NPs were purified by ultracentrifugation (150,000 g, 1 h, 4°C, Beckman Coulter). The supernatant was discarded and the pellet was resuspended in the appropriate volume of nanopure water to obtain a final concentration of 2.5 mg.mL -1 . The amount of anti-Aβ 1-42 mAb on purified nanoparticles was indirectly evaluated by fluorescence quantification of streptavidin FITC since the streptavidin FITC:mAb molar ratio is 1:1. Nanoparticle characterization. The average diameter (D z ) of the nanoparticles was measured by dynamic light scattering (DLS), weighted by intensity, after dilution 1/100 (v/v) in nanopure water with a NanoZS from Malvern (173° scattering angle, Orsay, France) at 25 °C. The nanoparticle surface charge was determined by ζ-potential measurement at 25 °C after dilution with 1 mM NaCl solution applying the Smoluchowski equation and using the same apparatus.

  Tg2576 14 month-old AD-like transgenic male mice (B6;SJL-Tg(APPSWE)2576Kha, Taconic Europe) and 9 wild type (WT) age-matched littermates were used for anti-A 1-42 -NPs evaluation. Tg2576 mice carry a transgene coding for the 695-amino acid isoform of human Alzheimer Aβ precursor protein carrying the Swedish mutation. All animals were single housed in a specific pathogen free facility in standard mouse cages containing sawdust with food (2018S Harlan diet) and water ad libitum, under conventional laboratory conditions (room temperature: 20 ± 2 °C; humidity: 60%) and a 12/12 h light/dark cycle (7:00 am -7:00 pm). Drug treatment and behavioral tests were all conducted during the light cycle. No environmental enrichment was used, since it is well known to improve AD mouse pathology.[START_REF] Lazarov | Environmental Enrichment Reduces Aβ Levels and Amyloid Deposition in Transgenic Mice[END_REF][START_REF] Valero | Short-term Environmental Enrichment Rescues Adult Neurogenesis and Memory Deficits in APP Sw, Ind Transgenic Mice[END_REF] All procedures involving animals and their care were conducted according to European Union (EEC Council Directive 86/609, OJ L 358,1; 12 December 1987) and Italian (D.L. n.116, G.U. suppl. 40, 18 February 1992) laws and policies and in accordance with the United States Department of Agriculture Animal Welfare Act and the National Institutes of Health (Bethesda, MA, USA) policy on Human Care and Use of Laboratory Animals. They were reviewed and approved by the Mario Negri Institute Animal Care and Use Committee that includes ad hoc members for ethical issues. Animal treatment. For novel object recognition tests, all animals (Tg or WT) were IV injected with 100 L (40 mg/kg of polymer) of fluorescently-labelled anti-A 1-42 -NPs (0.8 mg/kg of antibody) or PBS as vehicle once every other day for three weeks. To prevent the effect of subjective bias, animals were allocated to treatment by an operator not involved in the study and animal groups were named with numbers. Mice were treated always at the same time of the day (9-11 am), in a specific room inside the animal facility, following a randomized order.Novel Object Recognition (NOR) test. NOR is a memory test which relies on spontaneous animal behavior without the need of stressful elements such as food or water deprivation or electric foot shock.47 In the NOR mice are introduced into an arena containing two identical objects that they can explore freely. Twenty-four hours later mice are reintroduced into the arena, containing two different objects one of which previously presented (familiar) and a new completely different one (novel). At the end of treatment mice were tested in an opensquare grey arena (40 x 40 cm), 30 cm high, with the floor divided into 25 squares by black lines, placed in a specific room dedicated to behavioral analysis and separated from the operator's room. The following objects were used: a black plastic cylinder (4 x 5 cm), a glass vial with a white cup (3 x 6 cm) and a metal cube (3 x 5 cm). The task started with a habituation trial during which the animals were placed in the empty arena for 5 minutes and their movements recorded as the number of line-crossings, which provide an indication of both WT and Tg mice motor activity. The next day, mice were again placed in the same arena containing two identical objects (familiarization phase). Exploration was recorded in a 10minute trial by an investigator blinded to the genotype and treatment. Sniffing, touching and

( 1 :

 1 2 w/v) in ice-cold extraction buffer containing 50 mM TRIS-HCl (pH 7.4), 150 mM NaCl, 50 mM EDTA, 1% Triton X-100 (TX) and 2% protease inhibitor cocktail (Roche) with a manual tissue homogenizer (Eppendorf, Italy). Brain homogenates were centrifuged at 21 000 g, 4 °C for 30 min and the supernatants, referred to as the Triton-soluble fractions (soluble A), were removed with care taken not to disturb the pellet and stored at -80 °C until A dosage. The pellets remaining after the TX extract were re-suspended in 70% formic acid (FA) (1:10 w/v), homogenized and centrifuged at 100.000 g, 4 °C, 1 h in Beckman TL-100 ultracentrifuge. The resulting FA-extracted supernatants, referred to as FA-soluble fraction (insoluble A) were neutralized with 1 M TRIS buffer (pH 11) and stored at -80°C until A dosage. Levels of Aβ 1-40 and Aβ 1-42 in each fraction were quantified by sandwich enzymelinked immunosorbent assay (ELISA kit, IBL, Germany) following the manufacturer's instruction. The total protein content in each fraction was quantified by bicinchoninic acid assay (Sigma-Aldrich, Milano, Italy). All samples were analyzed at least in triplicate.

fractions containing 50

 50 µg of total proteins (quantified by bicinchoninic acid assay, Sigma-Aldrich, Milano, Italy) were combined with NuPAGE® Sample Buffer, heated at 70 °C for 10 min before electrophoresing on NuPAGE Novex 4-12% Bis-tris gels at 200 V for 35 min (Invitrogen corporation, Milano, Italy) and electro-transferred onto nitrocellulose membrane (GE Healthcare). The membrane was blocked in 5% skim milk in TRIS buffered salinetween-20 (TBS-T) (10 mM Tris-HCl, 150 mM NaCl, 0.1% Tween-20, pH 8.0) for 30 min at 37 °C, incubated with the 6E10 anti-Aβ antibody (1:1000 dilution) in blocking buffer overnight at 4°C. Blot was rinsed in TBS-T and then incubated with secondary antibody (1.5 h, RT, 1:20 000 dilution), followed by further rinsing. Bands were visualized with enhanced chemiluminescence (ECL) Reagent (Amersham, USA) and then imaged using ImageQuant LAS4000 (GE Healthcare). Molecular weight of A bands was determined by comparison to the SeeBlue®Plus2 Pre-stained Standard (Invitrogen corporation, Milano, Italy). The content of soluble A assemblies was quantified by measuring the intensity of the chemiluminescent bands using ImageJ Software.Quantification of Aβ in plasma. After sacrifice, mice blood was collected from the hearth and processed for plasma separation. Levels of Aβ 1-40 and Aβ 1-42 in plasma were quantified by sandwich enzyme-linked immunosorbent assay (Wako Chemicals GmbH, Germany)

Figure 2 .

 2 Figure 2. Biodistribution of radio-labelled anti-Aβ 1-42 -NPs in wild type mice. Quantification of 14 C anti-Aβ 1-42 -NPs in the main organs at 5 min, 1 h, 4 h and 8 h after intravenous injection of 40 mg/kg in wild type mice (n = 4).

Figure 3 .

 3 Figure 3. Impact of nanoparticle-based treatments on Tg2576 mouse memory. Memory evaluation after PBS or anti-Aβ 1-42 -NPs (40 mg/kg) administration 3 injections/week for 3 weeks: (A) time % investigation on the novel vs. the familiar object of WT mice (n = 9) and Tg2576 mice (n = 9) at the end of treatment; (B) quantification of discrimination index. Histograms are mean ± SEM. One-way ANOVA found a significant effect of treatment: F 2,23 = 8.0, p = 0.002. **p < 0.01 by Tukey's post-hoc test.

Figure 4 .

 4 Figure 4. Soluble and insoluble Aβ levels in the brain of Tg2576 mice after treatment with anti-Aβ 1-42 NPs. (A) Insoluble Aβ 1-40 and Aβ 1-42 levels in the brain of Tg2576 mice after 3 weeks' treatment with anti-Aβ  -NPs or PBS. Data (mean ± SD) are expressed as pmol of Aβ/mg of total proteins. (B) Triton-soluble Aβ (i.e., oligomers) levels in the brain of Tg2576 mice are reduced after treatment with anti-Aβ 1-42 -NPs. Triton-soluble Aβ levels in the brain of Tg2576 mice after 3 weeks' treatment with anti-Aβ 1-42 -NPs or PBS. Data (mean ± SD) are expressed as a % of change of Aβ levels. p values were calculated by Student's t test.

Figure 5 .

 5 Figure 5. Amount of Aβ oligomers of different MW in the brain of Tg2576 mice after treatment with anti-Aβ  -NPs. (A) Representative WB of Aβ oligomers probed with 6E10 anti-Aβ 1-42 mAb and visualized by ECL. (B) Estimation of Aβ assemblies sorted by MW present in the brain of Tg2576 treated with PBS taken as reference or with anti-Aβ 1-42 -NPs obtained by quantifying the intensity of the chemiluminescent bands using the ImageJ Software. Data (mean ± SD of triplicate experiments) are expressed as % of change of Aβ levels. p values were calculated by Student's t test.

Figure 6 .

 6 Figure 6. Aβ levels in plasma of Tg2576 mice after treatment with anti-Aβ 1-42 -NPs. Tg2576 mice treated with PBS or with anti-Aβ 1-42 -NPs. Data (mean ± SD) are expressed as a % of change of Aβ levels. p values were calculated by Student's t test.

  NIHS adult male mice between 17 and 27 g (Harlan Laboratories) were used for

	this purpose according to European Union (EEC Council Directive 86/609, OJ L 358,1; 12
	December 1987) laws and policies and in accordance with the Principles of Laboratory
	Animal Care and legislation in force in France (Decree No. 2013-118 of February 1, 2013).

  1-40 and Aβ 1-42 levels (15.40  1.72 and 98.59  9.13 pmols of A/mg total proteins, respectively) was detected by comparison with the soluble Aβ 1-40 and Aβ 1-42 levels in mice treated with PBS (21.73  3.38 and 114.07  10.11 pmols of Aβ/mg of total proteins, respectively) (Figure

Acknowledgements

The research leading to these results has received funding from the European Community's Seventh Framework Program (FP7/2007-2013) under agreement n°212043. The CNRS and the French Ministry of Research are also warmly acknowledged for financial support. Hélène Chacun, Juliette Vergnaud, Gillian Barratt and Céline Boursier (IFR141-IPSIT) are

Supporting Information

Supplementary Figure S1.


K Lippera

O Dauchot

S Michelin

M Benzaquen
email: michael.benzaquen@polytechnique.edu

No net motion for oscillating near-spheres at low Reynolds numbers

We investigate the flow around an oscillating nearly-spherical particle at low, yet nonvanishing, Reynolds numbers, and the potential resulting locomotion. We analytically demonstrate that no net motion can arise up to order one in Re and order one in the asphericity parameter, regardless of the particle's shape. Therefore, geometry-induced acoustic streaming propulsion, if any, must arise at higher order.

Introduction

Solid bodies forced to oscillate in a fluid may, as a result, undergo a net motion, provided their shape breaks an appropriate symmetry. Yet, in the absence of inertia (i.e. when the Reynolds number Re is strictly zero), no net motion can arise from timereciprocal actuation due to the linearity of Stokes' equations [START_REF] Purcell | Life at low-Reynolds number[END_REF]. Above a critical Re c = O(1), a symmetric rigid body can achieve unidirectional locomotion as a result of symmetry-breaking instability resulting from the nonlinear inertial contribution to the Navier-Stokes equations [START_REF] Alben | Coherent locomotion as an attracting state for a free flapping body[END_REF]. The purpose of the present work is to analyze the emergence of self-propulsion at small but finite Re (i.e. the effect of inertia is weak but non-negligible) for oscillating asymmetric particles. Indeed one could expect that asymmetric flows, resulting from asymmetric boundary conditions, shall push the particle, thereby inducing nonzero average motion [START_REF] Nadal | Asymmetric steady streaming as a mechanism for acoustic propulsion of rigid bodies[END_REF].

Artificial microswimmers have received much recent attention, thanks to their potential application to drug delivery or water treatment [START_REF] Sundararajan | Catalytic motors for transport of colloidal cargo[END_REF][START_REF] Martinez-Pedrero | Magnetic propulsion of self-assembled colloidal carpets: efficient cargo transport via a conveyor-belt effect[END_REF][START_REF] Tiwari | Application of nanoparticles in waste water treatment[END_REF], or their fundamental interest to study active matter (see e.g. [START_REF] Palacci | Living Crystals of Light-Activated Colloidal Surfers[END_REF][START_REF] Buttinoni | Dynamical Clustering and Phase Separation in Suspensions of Self-Propelled Colloidal Particles[END_REF][START_REF] Bechinger | Active Particles in Complex and Crowded Environments[END_REF]. Among the many possible routes to self-propulsion, swimming in self-generated physico-chemical gradients i.e. autophoresis [START_REF] Moran | Phoretic self-propulsion[END_REF], as well as bubblegenerating [START_REF] Wang | Selecting the swimming mechanisms of colloidal particles: bubble propulsion versus self-diffusiophoresis[END_REF][START_REF] Li | Rocket science at the nanoscale[END_REF] or magnetically-actuated microswimmers [START_REF] Dreyfus | Microscopic artificial swimmers[END_REF]) have received a particular attention. In these examples, a front-back asymmetry in the design of the system is necessary. Yet, symmetry-breaking and self-propulsion can also be achieved by exploiting an instability [START_REF] Bricard | Emergence of macroscopic directed motion in populations of motile colloids[END_REF][START_REF] Michelin | Spontaneous autophoretic motion of isotropic particles[END_REF][START_REF] Izri | Self-propulsion of pure water droplets by spontaneous marangoni-stress-driven motion[END_REF] or flexibility [START_REF] Wiggins | Flexive and propulsive dynamics of elastica at low reynolds number[END_REF].

Recently, passive rigid particles levitating in the nodal planes of an acoustic stationary wave have been observed to self-propel in a plane orthogonal to their direction of excitation [START_REF] Wang | Autonomous motion of metallic microrods propelled by ultrasound[END_REF]. To explain such findings, [START_REF] Nadal | Asymmetric steady streaming as a mechanism for acoustic propulsion of rigid bodies[END_REF] proposed an acoustic streaming mechanism, suggesting that near-spherical particles with asphericity parameter ǫ can achieve a net O(ǫRe) propulsion, in the low frequency limit. Several studies have since stood upon the results of Nadal & Lauga to account for their observations (see e.g. [START_REF] Sabrina | Shape-directed microspinners powered by ultrasound[END_REF][START_REF] Ahmed | Density and shape effects in the acoustic propulsion of bimetallic nanorod motors[END_REF][START_REF] Soto | Acoustically propelled nanoshells[END_REF].

In contrast, we here demonstrate analytically that no net motion can arise at O(ǫRe) from a time-reciprocal oscillation and that self-propulsion, if any, must arise at higher order. In Section 2, the governing equations for an oscillating particle are presented. In Section 3, we introduce the particle geometry and the Taylor expansions of the velocity fields in Re and ǫ. In Sections 4 and 5, we compute the net motion of the particle at the first two orders in Re. In Section 6, we discuss our results and conclude.

Governing equations

We consider here a rigid and homogeneous particle of typical size R oscillating with frequency ω and amplitude a in an incompressible and Newtonian fluid of kinematic viscosity ν. Using R, aω and 1/ω respectively as reference length, velocity and time scales, the dimensionless Navier-Stokes and continuity equations read [START_REF] Zhang | Oscillatory motions of circular disks and nearly spherical particles in viscous flows[END_REF]:

λ 2 ∂ t u + Re∇u • u = ∇ • σ , ∇ • u = 0 , (2.1)
with σ = -pI + (∇u + ∇ ⊤ u), the dimensionless stress tensor. The Reynolds number and reduced frequency are respectively defined as Re = aωR/ν and λ 2 = (R/δ) 2 with δ = ν/ω the viscous penetration depth. More precisely, a translational oscillation is imposed to the particle along the e x direction, Ũ = e it e x , and the particle is free to move along the other directions, and is thus force-free along the yz plane and torque-free about any axis. The longitudinal and angular velocities of the particle resulting from its imposed oscillation are U = U y e y + U z e z and Ω = Ω x e x + Ω y e y + Ω z e z . In the frame of reference of the laboratory, the boundary conditions read:

u| S = Ũ + U + Ω × r , u| r→∞ = 0 . (2.2)
In order to determine U and Ω following an approach analogous to that of Lorentz' reciprocal theorem [START_REF] Happel | Low Reynolds number hydrodynamics: with special applications to particulate media[END_REF], the auxiliary problem of a particle of the same instantaneous geometry in a steady Stokes flow is considered:

∇ • σ = 0 , ∇ • u = 0 , (2.3)
with boundary conditions,

u| S = U + Ω × r , u| r→∞ = 0 . (2.4)
Let us stress that the particle is rigid so that by instantaneous geometry one should understand that the surface boundary of the auxiliary problem matches that of its real counterpart at each time. Using (2.1) and ( 2.3), one obtains:

V u • (∇ • σ) -u • (∇ • σ) dV = Re V u • ∇u • u dV + λ 2 V u • ∂ t u dV . (2.5)
Using the divergence theorem together with the continuity equations, (2.5) reduces to:

S∞-S ( u • σ -u • σ) • n dS = Re V u • ∇u • u dV + λ 2 V u • ∂ t u dV . (2.6)
Because u, u ∼ 1/r and σ, σ ∼ 1/r 2 when r → ∞ (see e.g. [START_REF] Happel | Low Reynolds number hydrodynamics: with special applications to particulate media[END_REF], the surface integral at infinity in (2.6) vanishes. The boundary conditions (2.2) and (2.4) then yield:

( Ũ + U ) • F + Ω • L -U • F -Ω • L = Re V u • ∇u • u dV + λ 2 V u • ∂ t u dV , (2.7)
with F = S σ • ndS and L = S (r × σ) • ndS (resp. F and L), the force and torque in the real (resp. auxiliary) problem. For the real problem, F and L derive from Newton's laws:

F = ρ∂ t U , L = ∂ t (J • Ω) , (2.8)
with ρ the particle-to-fluid density ratio and J the particle's inertia tensor. For the auxiliary problem, F and L are linearly related to Û and Ω through the possibly nondiagonal resistance matrix [START_REF] Kim | Microhydrodynamics: principles and selected applications[END_REF]. In order to compute the particle motion (U , Ω), we shall consider in (2.7) either (i) an auxiliary steady propulsion ( Û , 0) with Û U to determine U , or (ii) an auxiliary steady rotation (0, Ω) with Ω Ω to determine Ω. Note that finding the contribution at O(Re n ) of the first term on the righthand side of (2.7) relies on the knowledge of the velocity field u at O(Re n-1 ) only, hence the possibility of a recursive calculation order by order in Re. Conversely, computing the second term shall rely on peculiar symmetry and time-average considerations to be made explicit below. Note that for a homogeneous particle, the above formulation also applies to the motion of a particle exposed to a uniform oscillating flow -Ũ , once inertial forces are accounted for as a modified pressure.

Nearly spherical particles in low Reynolds flows

We consider a nearly-spherical particle of volume V and center-of-mass O. By choosing R = (3V /4π) 1/3 and taking O as the origin of the system of axes, one can define the particle's geometry through r = 1 + ǫf (n) with ǫ ≪ 1. By construction f satisfies:

S f (n)dS = 0 , S f (n)ndS = 0 . (3.1)
The governing equations are first linearised with respect to Re ≪ 1, e.g. defining u = u 0 +Reu 1 +O(Re 2 ), and each order is further expanded as a regular perturbation problem in ǫ ≪ 1, e.g.

u k = u 0 k + ǫu ǫ k + O(ǫ 2
) with k = 0, 1 †. In the following sections, we shall consider the problems at O(Re k ǫ ℓ ), and successively look into the two leading orders in Re.

Zeroth-order in Re

At leading order O(Re 0 ), (2.1) and (2.2) become:

λ 2 ∂ t u 0 = -∇p 0 + ∇ 2 u 0 , ∇ • u 0 = 0 , (4.1) u 0 | S = Ũ + U 0 + Ω 0 × r , u 0 | r→∞ = 0 , (4.2)
and (2.7) reduces to:

( Ũ + U 0 ) • F + Ω 0 • L -U • F 0 -Ω • L 0 = λ 2 V u • ∂ t u 0 dV , (4.3)
and this result is expanded as a linear perturbation in ǫ below.

4.1. Perfect sphere -O(Re 0 ǫ 0 ) While it is quite clear that no net motion can arise at O(ǫ 0 Re 0 ) (i.e. unsteady Stokes flow around a spherical particle), we briefly rederive this result to provide the reader with the general methodology. At leading order O(ǫ 0 ), (4.3) becomes:

( Ũ + U 0 0 ) • F 0 + Ω 0 0 • L 0 -U • F 0 0 -Ω • L 0 0 = λ 2 V0 u 0 • ∂ t u 0 0 dV , (4.4)
where V 0 denotes the volume of fluid outside the reference unit sphere. First, recalling U U 0 0 provides U • Ũ = 0. Second, the velocity field u 0 (resp. u 0 0 ) is linear with respect to U (resp. Ũ ), and axisymmetric about the axis holding the vector U (resp. Ũ ) and passing through the centre of mass of the particle. As a result, using the expression of u 0 and u 0 0 (Appendix A) shows that the RHS of (4.4) does not include any contribution from the forcing Ũ . There is therefore no net motion at this order, i.e. U 0 0 = 0. A similar reasoning shows that Ω 0 0 = 0 as well. This last result imposes the rotation velocity of the particle to be at least first order (either in ǫ or Re). The forcing and induced rotation act therefore on two separate time scales. As a consequence, at leading order, the geometry of the particle, f , can be considered constant over the O(1) period of the forcing (fast time scale). 4.2. Near-sphere correction -O(Re 0 ǫ 1 ) At O(ǫ 1 ) (4.3) becomes:

U ǫ 0 • F 0 + Ω ǫ 0 • L 0 + Ũ • F ǫ -U • F ǫ 0 -Ω • L ǫ 0 = λ 2 V0 u ǫ • ∂ t u 0 0 + u 0 • ∂ t u ǫ 0 dV -λ 2 S0 f u 0 • ∂ t u 0 0 dS , (4.5)
where the surface integral is the O(ǫ) contribution from the difference of the volume integrals on V and V 0 (e.g [START_REF] Zhang | Oscillatory motions of circular disks and nearly spherical particles in viscous flows[END_REF]. The analysis of [START_REF] Zhang | Oscillatory motions of circular disks and nearly spherical particles in viscous flows[END_REF] shows that the rotation of a torque-free homogeneous near-sphere resulting from an O(1) imposed translation is O(ǫ 2 ) and thus Ω ǫ 0 = 0. Consequently the torque L ǫ 0 linked to Ω ǫ 0 through Newton's law (2.8) vanishes as well. Using (2.4) and (3.1), the last term in (4.5) vanishes exactly:

S0 f u 0 • ∂ t u 0 0 dS = ( U × Ω) • S0 f n dS = 0 . (4.6)
Since we are interested in the net motion of the particle, we take the time-average over the fast time scale (forcing period) of (4.5). The RHS t can be shown to vanish because u ǫ 0 and u 0 0 are periodic in time, and the integration domains are time-independent. Therefore:

U ǫ 0 t • F 0 -U • F ǫ 0 t = 0 . (4.7)
Equation (4.7) is linear with no net contribution of the forcing Ũ : no net motion can occur at O(Re 0 ǫ 1 ), U ǫ 0 t = 0.

First-order in Re

At O(Re 1 ), (2.1) and (2.2) become:

λ 2 ∂ t u 1 + ∇u 0 • u 0 = -∇p 1 + ∇ 2 u 1 , ∇ • u 1 = 0 , (5.1)
u 1 | S = U 1 + Ω 1 × r , u 1 | r→∞ = 0 , (5.2)
and (2.7) reduces to:

U 1 • F + Ω 1 • L -U • F 1 -Ω • L 1 = V u • ∇u 0 • u 0 dV + λ 2 V u • ∂ t u 1 dV . (5.3)
does not contribute to particle motion, as demonstrated in Section 5.1. As a consequence, only the first contribution to u ǫ 0 | r=1 in (5.10a) provides a net contribution to v ǫ 1 . For clarity, we now distinguish the cases of pure translation and pure rotation.

Translation

Setting Ω = 0, (5.9) simplifies after some algebraic calculations using the definitions of G 1 , G 2 , u ε and u ε 0 (Appendices A and B):

v ǫ 1 = K(λ) [f nnn] n . . . e x e x U , (5.11)
where [•] n denotes the average over the unit sphere: [•] n = S0 •(n)dS, and . . . denotes the three-fold tensorial contraction. Quite remarkably, (5.11) provides the expression of the net translational velocity as a product of a function of λ and a functional of f . The tensorial contraction, together with the angular symmetry properties of the inertial forcing, ensure that only a limited set of the spherical harmonic components of the shape function f can contribute to a net motion. Further algebraic calculations show that K(λ) conveniently reduces to

K(λ) = ∞ r=1 dJ λ (r)
dr dr with:

J λ (r) = - 1 4 ℜ 27(1 -r 2 ) 16λ 4 0 r 8 -3|Λ 0 | 2 + 2Λ 0 3 + 3λ 0 r -λ 2 0 r 2 e λ0(1-r) -3 + 3λ 0 r + (λ 0 r) 2 1 + λ 0 r -λ 2 0 r 2 e 2ℜ[λ0](1-r) , (5.12)
where an overbar denotes the complex conjugate, ℜ[z] is the real part operator of z and Λ 0 = 1 + λ 0 + λ 2 0 /3 with λ 0 = λe -iπ/4 . Therefore, using J λ (∞) = J λ (1) = 0, one finds the central result of the present communication:

U ǫ 1 t = 0 .
(5.13)

No translational net motion can arise at first order (both in Re and non-sphericity ǫ) from geometric asymmetry. This result stems from the fact that the near-field (r = O(1)) and far-field (r ≫ 1) contributions to the inertial forcing compensate exactly.

Rotation

Considering now U = 0, the same method provides:

v ǫ 1 = L(λ) f nn n : e x (e x × Ω) , (5.14)
with: (5.15) with F (z) = [Chi(z) -Shi(z)]e z where Chi/Shi are the hyperbolic cosine/sine integral functions respectively [START_REF] Abramowitz | Handbook of mathematical functions: with formulas, graphs, and mathematical tables[END_REF]. We note from (5.8) and (5.14) that (i) no rotation is obtained along the direction of oscillation (i.e. Ω ǫ 1 t • Ũ = 0) and that (ii) the particle dynamics is an overdamped rotation toward an equilibrium position. The oscillation direction Ũ is aligned with a principal direction of the symmetric and trace-less second-order tensor f nn n with positive or negative eigenvalue depending on the sign of L. Further, the function L t changes sign for λ c ≈ 3.6, resulting in a shift in the equilibrium orientation between λ < λ c and λ > λ c . This transition confirms fundamental differences in the streaming flow and associated forcing between small and large frequencies, as already observed by [START_REF] Collis | Autonomous propulsion of nanorods trapped in an acoustic field[END_REF] when studying numerically the propulsion of an oscillating asymmetric dumbbell.

L(λ) = 1 256 ℑ 1 Λ 0 -48(λ 0 (λ 0 (λ 0 (λ 0 + 6) + 18) + 30) + 24)|λ 0 | 2 F (2ℜ(λ 0 )) +3i(λ 0 (λ 0 (λ 0 (λ 0 + 9) + 27) + 42) + 30)λ 2 0 Λ0 F (λ 0 ) +3i(λ 0 (λ 0 (λ 0 (λ 0 + 3) + 33) + 78) + 66)λ 2 0 Λ 0 F (iλ 0 ) +(1 -i)λ 7 0 + (3 -7i)λ 6 0 -(5 + 35i)λ 5 0 -(6 + 108i)λ 4 0 -(60 + 210i)λ 3 0 -(264 + 306i)λ 2 0 -(348 + 360i)λ 0 -132i ,

Conclusion

In this work, we analysed the translation and rotation resulting from the oscillation of a homogeneous near-sphere up to O(ǫRe), showing analytically that no net translation occurs regardless of the oscillation frequency and despite the geometric asymmetry of the particle. This result, which contradicts the conclusions of [START_REF] Nadal | Asymmetric steady streaming as a mechanism for acoustic propulsion of rigid bodies[END_REF], stems from the exact cancellation of the streaming flow forcing in the immediate vicinity of the particle and far away from it, making it difficult to capture numerically as any discretisation introduces necessarily a truncation error. We also show that a transient rotation can stir back the particle towards one of its equilibrium positions.

Notwithstanding, our results do not contradict the numerical observations of [START_REF] Collis | Autonomous propulsion of nanorods trapped in an acoustic field[END_REF] for which a weak front-back asymmetry of a dumbbell was sufficient to produce a net motion at that order: in that case, the elongated shape of the particle combined with the small asymmetry of the two spheres leads to an O(1) periodic rotation of the system, which is at the heart of the self-propulsion, when coupled to the oscillating translation -in contrast, such rotation is absent at O(ǫRe) in the case of a near-sphere. All together, developing net motion around an asymmetric particle appears to require an O(ǫ) rotation/translation coupling, as obtained for instance using density inhomogeneities.

† Note that ǫ must remain small compared to all other dimensionless length scales, i.e. ǫ ≪ 1 (particle radius) and ǫ ≪ 1/λ (viscous boundary layer thickness).

Acknowledgements

We are grateful to F. Nadal and E. Lauga for insightful discussions on this problem. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under Grant Agreement 714027 (SM).

Note that here, in addition to the unsteady forcing, the non-linear convective term acts as a source term in (5.3). Because it is quadratic in velocity, one might expect that its average in time is nonzero, which could in turn yield net particle motion. 5.1. Perfect sphere -O(Re 1 ǫ 0 ) At leading order O(ǫ 0 ) (5.3) becomes:

(5.4)

The symmetry properties of u 0 and u 0 0 (Appendix A) impose that the first term on the RHS of (5.4) vanishes. The second term on the RHS vanishes as well because it is the integral of the scalar product between two axisymmetric fields about orthogonal principal directions. Therefore (5.4) becomes:

implying that, very much like for O(Re 0 ǫ 0 ), U 0 1 = 0 and Ω 0 1 = 0.

5.2.

Near-sphere correction -O(Re 1 ǫ 1 ) At O(ǫ 1 ), and using (2.8) together with (5.5), (5.3) becomes:

Taking the average in time of (5.6) over the forcing period, and using that u ǫ 1 and u 0 1 are periodic and that F ǫ 1 and L ǫ 1 are temporal derivatives of periodic functions (2.8), one finally obtains:

where we have used F 0 = -6π U and L 0 = -8π Ω. Integrating by parts, and using the expressions of u 0 and u 0 0 (Appendix A), one obtains:

with the vector fields G 1 and G 2 defined as

0 , whose expressions are provided in Appendix A. Using domain perturbation, the velocity field u ǫ 0 (resp. u ǫ ) is solution of (4.1) (resp. (2.3)) with the following boundary conditions on the unit sphere (see Appendix B):

(5.10b)

A first simplification comes from recalling that Ω ǫ 0 = 0. A second one arises from the fact that the Stokes problem with the uniform boundary condition U ǫ 0 on the unit sphere Appendix A. Unsteady Stokes flow past a spherical particle

A.1. Oscillating flow

The complex velocity field around a sphere oscillating at velocity Ũ reads [START_REF] Kim | Microhydrodynamics: principles and selected applications[END_REF]

and where λ 0 = λe -iπ/4 and Λ 0 = 1 + λ 0 + λ 2 0 /3. Recalling that:

. Steady Translation

The particular case of a steady translating sphere (λ = 0) at velocity U is given by u 0 = A U + B( U • n)n where:

. Steady Rotation

The velocity field around a steady rotating sphere reads u 0 = Ω × n/r 2 . Computing:

one obtains the expression of G 2 = ∇ u 0 0 • u 0 0 + u 0 0 • ∇ u 0 0 as:

Appendix B. Unsteady Stokes flow past a nearly-spherical particle

Here we compute the velocity field solution of the unsteady Stokes problem around a nearly-spherical particle:

In Fourier space the boundary condition on the surface of the particle (B 2) takes the form [START_REF] Zhang | Oscillatory motions of circular disks and nearly spherical particles in viscous flows[END_REF]:

Following [START_REF] Sani | Convective instability[END_REF], we perform a reconstruction of the velocity field from its radial component and associated vorticity:

where ∇ s = ∇ -n∂ r and where u r,n denotes the n th mode in the spherical harmonics basis of the radial component of u ǫ 0 . The latter satisfies in time-Fourier space the equation ∇ 2 (iλ 2 + ∇ 2 )(ru ǫ 0,r ) = 0. The function χ r,n is the n th mode of the radial component of ∇ × u ǫ 0 , satisfying (∇ 2 + iλ 2 )(rχ ǫ r ) = 0. Defining p and q through:

one finally obtains:

) where p m n and q m n denote respectively the modes of p and q in the spherical harmonics basis (Y m n ), and U n , V n and X n follow:

with h n the spherical Hankel function of the first kind and order n [START_REF] Abramowitz | Handbook of mathematical functions: with formulas, graphs, and mathematical tables[END_REF]. In (B 5), the functions p and q defined on the surface of the unit sphere are directly related to the shape function f through (B 2). Using (3.1), they further satisfy : Note that these results can be transposed to obtain u ǫ taking λ = 0 for the translation problem. And a similar approach can be used in the rotating case.