Misgina Tilahun Tsehaye

Heon Nak

Peter Choi

Jens Fischer

Emilie Tübke

Fannie Planes

Cristina Alloin

Iojoiu

Heon Choi

Peter Fischer

Jens Tübke

Emilie Planes

Fannie Alloin
email: fannie.alloin@lepmi.grenoble-inp.fr

Cristina Iojoiu
email: cristina.iojoiu@lepmi.grenoble-inp.fr

Anion Exchange Membranes Incorporating Multi N -Spirocyclic Quaternary Ammonium Cations via Ultraviolet-Initiated Polymerization for Zinc Slurry-Air Flow Batteries

Keywords: Zinc slurry-air flow battery, Anion exchange membrane, N-spirocyclic quaternary ammonium, Poly(phenylene oxide), UV irradiation Highlights

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Introduction

Driven by the ever-growing greenhouse gas emissions, mainly CO2, research directs to energy productions from 'greener' alternatives, such as wind and solar energy has recently received significant attention in the research community 1 . However, since they generate intermittent electrical power, there is a need for an energy storage device for optimal matching of energy demand and supply [2][3][4][5] . Currently, Li-ion 6 and Pbacid 7,8 batteries are widely used energy storage systems; however, concerns over their safety and limited energy density require alternative energy storage devices [9][10][11] .

Redox-flow batteries (RFBs) are promising alternatives because they decouple energy storage and power output to meet customer demand, have a long lifetime and can operate at ambient temperature and pressure 3 12-15 . Among them is the Zinc (Zn) slurry-air RFB, a type of Zn-air battery which uses a suspension of Zn particles in aq. KOH electrolyte solution as the negative electrode and electrolyte. It is regarded as a potential energy storage technology due to its relatively high specific capacity, eco-friendliness, safety and inexpensive cost (wide abundance of Zn and O2) 16,17 . In addition to the Zn slurry, the battery consists of a membrane and an air electrode.

The membrane in Zn slurry-air flow battery (ZSAFB) is used to conduct charge-carrying ions between the electrodes and avoid short circuit. The membrane should exhibit robust alkaline stability, high ionic (hydroxide, OH -) conductivity and no or low zincate (Zn(OH)4 2-) ion crossover to develop a long-term Zn slurry-air RFB. The utilization of porous membranes in Zn-air batteries 18 has been reported to result in in considerable Zn(OH)4 2-ion crossover [START_REF] Tsehaye | Membranes for Zinc-Air Batteries: Recent Progress, Challenges and Perspectives[END_REF] , resulting in loss of capacity [START_REF] Lee | Electrospun Polyetherimide Nanofiber Mat-Reinforced, Permselective Polyvinyl Alcohol Composite Separator Membranes: A Membrane-Driven Step Closer toward Rechargeable Zinc-Air Batteries[END_REF] and blockage of air electrode by the formed ZnO particles (Zn(OH) 4 2-→ ZnO + 𝐻 2 𝑂 + 2𝑂𝐻 -). To reduce the Zn(OH)4 2-ion crossover, various porous membrane modification strategies, including ion-selective polymer coating [START_REF] Hwang | Selective Ion Transporting Polymerized Ionic Liquid Membrane Separator for Enhancing Cycle Stability and Durability in Secondary Zinc-Air Battery Systems[END_REF][START_REF] Krejčí | Transport of Zn ( OH ) 4 2 -Ions across a Polyolefin Microporous Membrane[END_REF][START_REF] Tsehaye | Pristine and Modified Porous Membranes for Zinc Slurry-Air Flow Battery[END_REF] and nanoparticles filling [START_REF] Kiros | Separation and Permeability of Zincate Ions through Membranes[END_REF] have been attempted. These, on the other hand, have been linked to a decrease in ionic conductivity.

Another possibility is to use anion exchange membranes (AEMs), which are regarded as promising candidates and are frequently proposed for use in Zn-air batteries [START_REF] Fu | Electrically Rechargeable Zinc-Air Batteries: Progress, Challenges, and Perspectives[END_REF][START_REF] Li | Recent Advances in Zinc-Air Batteries[END_REF] . An AEM containing a sulfonium cation was reported to show lower permeation of Zn(OH)4 2-ions than a Celgard ® 5550 [START_REF] Dewi | Cationic Polysulfonium Membrane as Separator in Zinc-Air Cell[END_REF] . The Zn-air cell employing the prepared AEM showed six-fold higher discharge capacity than that of the polypropylenebased cell. Moreover, AEMs based on Poly(2,6-dimethyl-1,4-phenylene oxide) (PPO) and three different amines were fabricated and employed in primary Zn-air batteries [START_REF] Abbasi | Poly(2,6-Dimethyl-1,4-Phenylene Oxide)-Based Hydroxide Exchange Separator Membranes for Zinc-Air Battery[END_REF] . Among them, the PPO-trimethylamine membrane had a Zn(OH)4 2-ion crossover of only 1.9 × 10 -14 m 2 /s, which was considered to be related to the membrane's proper ionic channels. Additionally, commercial AEMs, Tokuyama A201 [START_REF] Fu | A Flexible Solid-State Electrolyte for Wide-Scale Integration of Rechargeable Zinc-Air Batteries[END_REF][START_REF] Zhang | Laminated Cross-Linked Nanocellulose/Graphene Oxide Electrolyte for Flexible Rechargeable Zinc-Air Batteries[END_REF] and Fumatech FAA-3 [START_REF] Kwon | Transparent Bendable Secondary Zinc-Air Batteries by Controlled Void Ionic Separators[END_REF][START_REF] Soni | A 3-D Nanoribbon-like Pt-Free Oxygen Reduction Reaction Electrocatalyst Derived from Waste Leather for Anion Exchange Membrane Fuel Cells and Zinc-Air Batteries[END_REF] have been tested in different Zn-air cells. However, despite the intriguing potential of AEMs for these batteries, it is still a relatively unexplored topic 18,[START_REF] Tsehaye | Membranes for Zinc-Air Batteries: Recent Progress, Challenges and Perspectives[END_REF] . The correlation between AEM characteristics and Zn-air battery performance, in particular, has not been thoroughly investigated.

One of the biggest challenges for using AEMs in electrochemical applications, such as alkaline batteries, is alkaline stability [START_REF] Zhang | Imidazolium Functionalized Polysulfone Anion Exchange Membrane for Fuel Cell Application[END_REF][START_REF] Varcoe | Anion-Exchange Membranes in Electrochemical Energy Systems[END_REF][START_REF] Yang | A Strategy to Construct Alkali-Stable Anion Exchange Membranes Bearing Ammonium Groups via Flexible Spacers[END_REF] . Chemical instability of the fixed functional groups, polymer backbone and/or spacer can all cause chemical degradation of such ionomers. Cationic degradation, in particular, is regarded to be the primary reason why most present-day polymer-based AEMs rapidly degrade in alkaline conditions above 80 °C. Linear polyaromatic AEMs incorporating piperidinium and 6-azonia-spiro [5.5]undecane (ASU), an alkaline stable N-spirocyclic quaternary ammonium (QA) [START_REF] Marino | Alkaline Stability of Quaternary Ammonium Cations for Alkaline Fuel Cell Membranes and Ionic Liquids[END_REF] , cations via superacid-mediated polycondensation have been reported by Jannasch's research group [START_REF] Olsson | Arylene Piperidinium) Hydroxide Ion Exchange Membranes: Synthesis, Alkaline Stability, and Conductivity[END_REF][START_REF] Pham | Arylene Alkylene)s with Pendant N -Spirocyclic Quaternary Ammonium Cations for Anion Exchange Membranes[END_REF] . The poly(terphenyl dimethyl-piperidinium)-based AEMs showed no sign of degradation in 2 M NaOH at 60 °C for two weeks [START_REF] Olsson | Arylene Piperidinium) Hydroxide Ion Exchange Membranes: Synthesis, Alkaline Stability, and Conductivity[END_REF] .

Moreover, Strasser et al. [START_REF] Strasser | Base Stable Poly(Diallylpiperidinium Hydroxide) Multiblock Copolymers for Anion Exchange Membranes[END_REF] prepared a multiblock AEM copolymers containing poly(diallylpiperidinum hydroxide) cation, which lost only 8% of its ionic conductivity after storage under harsh alkaline condition

(1 M KOH, 80 °C) for 120 h. Recently, PPOs with 3,6-diazaspiro [5.5]undecane were prepared following two steps of quaternization method [START_REF] Xue | Phenylene Oxide)s Incorporating N-Spirocyclic Quaternary Ammonium Cation/Cation Strings for Anion Exchange Membranes[END_REF] . However, compared to ASU, the cation was reported to have inferior alkaline stability because of the cation's tertiary N atom which resulted in both Hofmann elimination and SN2 substitution.

Another issue associated with alkaline stability of AEM is that AEMs quaternized in their benzylic positions (close to the polymer backbone) are sensitive towards chemical degradation, thus unstable especially in highly alkaline medium at high temperature [START_REF] Nuñez | Quantitative 1 H NMR Analysis of Chemical Stabilities in Anion-Exchange Membranes[END_REF] . Membranes with tethered benzyltrimethyl ammonium cations were reported to break into small pieces after 3 days, whereas comb-shaped AEMs retained their mechanical strengths in 1 M NaOH at 80 °C for 83 days [START_REF] Li | Highly Stable, Anion Conductive, Comb-Shaped Copolymers for Alkaline Fuel Cells[END_REF] . Moreover, cations placed directly on the benzylic positions limit the ionic functions mobility and prevent phase separation, a prerequisite for highly conductive AEMs [START_REF] Dang | Exploring Different Cationic Alkyl Side Chain Designs for Enhanced Alkaline Stability and Hydroxide Ion Conductivity of Anion-Exchange Membranes[END_REF] .

In this study, methyldiallylammonium groups were attached to a PPO via a 6-C long spacer (PPO-6CH2-Q)

to prevent benzylic position instability and facilitate phase separation. PPO was chosen for its high glass transition temperature, mechanical strength, chemical stability and low cost [START_REF] Xu | Poly(2,6-Dimethyl-1,4-Phenylene Oxide) (PPO)-A Versatile Starting Polymer for Proton Conductive Membranes (PCMs)[END_REF][START_REF] Ran | Anion Exchange Membranes (AEMs) Based on Poly(2,6-Dimethyl-1,4-Phenylene Oxide) (PPO) and Its Derivatives[END_REF] . Furthermore, to overcome the difficulty of grafting N-spirocyclic quaternary ammonium cationic groups onto polymer backbone structures [START_REF] Strasser | Base Stable Poly(Diallylpiperidinium Hydroxide) Multiblock Copolymers for Anion Exchange Membranes[END_REF][START_REF] Pham | Aromatic Polymers Incorporating Bis-N -Spirocyclic Quaternary Ammonium Moieties for Anion-Exchange Membranes[END_REF] , ultraviolet (UV)-curing method is used to graft the N, N-diallylpiperidinium chloride (DAPCl) monomer onto the modified PPO polymer backbone. Multi-cation side chains have been reported

to enhance the chemical stability and OH -conductivity of ion exchange membranes [START_REF] Cheng | Novel Self-Cross-Linked Multi-Imidazolium Cations Long Flexible Side Chains Triblock Copolymer Anion Exchange Membrane Based on ROMP-Type Polybenzonorbornadiene[END_REF] . In order to investigate the impact of ion exchange capacity (IEC) on water state, ionic conductivity, zincate ion crossover and battery performance, the monomer to PPO-6CH2-Q polymer ratio was varied from 0.5 to 3. The prepared AEMs were tested for alkaline and thermal stabilities and characterized using atomic force microscopy and differential scanning calorimetry, which were used to investigate phase separation and water state in the membranes, respectively. Finally, the performance of the AEMs in ZSAFBs was studied by recording and comparing their polarization curves.

Polymer synthesis

PPO-CO-Br: PPO with 6-C pendant chain (PPO-6CO-Br) was synthesized by slightly altering the method discussed elsewhere [START_REF] Hibbs | Alkaline Stability of Poly(Phenylene)-Based Anion Exchange Membranes with Various Cations[END_REF][START_REF] Parrondo | Synthesis and Alkaline Stability of Solubilized Anion Exchange Membrane Binders Based on Poly(Phenylene Oxide) Functionalized with Quaternary Ammonium Groups via a Hexyl Spacer[END_REF] . In a round bottom reaction flask, PPO (5 g) was dissolved in chlorobenzene (200 mL) and flushed with Ar. The reaction flask was kept at 0 °C for 30 mins before adding 6-BHC (9.6 mL) and AlCl3 (2.5 g). Finally, the temperature was increased to room temperature (RT) for 24 h. The alkyl brominated PPO mixture was precipitated by using methanol, filtered and dried at 60 o C to obtain a white solid product (7.1 g). The degree of bromination/functionalization (around 33%) was evaluated using the 1 H NMR spectroscopy.

PPO-6CH2-Br:

The ketone groups in the PPO-6CO-Br polymer were reduced as follows. In a 350 mL heavy wall pressure vessel (pressure reactor purchased from chemglass), PPO-6CO-Br (1 g, 6.85 mmol ketone groups) was dissolved in 1,2-dichloroethane (40 mL), and then trifluoroacetic acid (40 mL, 685 mmol) and triethylsilane (4 mL, 68.5 mmol) were added. The reaction was performed, at 100 o C for 24 h, under pressure to prevent the solvent from evaporation and to raise the reaction temperature beyond the solvent-reactive mixture boiling point. The reaction mixture was neutralized with aq. KOH solution (30 wt.

%, 100 mL) until the pH becomes neutral. The organic phase was then separated and precipitated using methanol. Subsequent dissolving in chloroform and re-precipitating in methanol were performed to remove the remaining triethylsilane and purify the obtained product. The collected polymer product (PPO-6CH2-Br) was dried at 60 o C and its degree of ketone reduction was calculated using the 1 H NMR peak areas.

PPO-6CH2-Q:

The PPO-6CH2-Br polymer was first dissolved in 5 wt. % NMP at 40 o C. To completely replace the bromine groups, methydiallylamine was added in excess (2.5 molar) and the reaction was allowed to run for 72 h. The quaternized polymer (PPO-6CH2-Q) was collected by precipitation with diethyl ether and dried under vacuum (35 o C). The complete replacement of bromine atoms with the amine was confirmed using 1 H NMR. The protocol followed to prepare PPO-6CO-Br, PPO-6CH2-Br and PPO-6CH2-Q is shown in scheme I.

Scheme I: Schematic representation of PPO-6CO-Br, PPO-6CH2-Br and PPO-6CH2-Q polymers preparation.

N, N-Diallylpiperidinum chloride

DAPCl was prepared following a procedure previously published with slight modifications (Scheme II) [START_REF] Strasser | Base Stable Poly(Diallylpiperidinium Hydroxide) Multiblock Copolymers for Anion Exchange Membranes[END_REF][START_REF] Olsson | Poly( N , N -Diallylazacycloalkane)s for Anion-Exchange Membranes Functionalized with N -Spirocyclic Quaternary Ammonium Cations[END_REF] .

First, 20 mL deionized (DI) water was added to a 250 mL reaction flask, followed by NaOH (12 g) at 0 °C.

Once the mixture was completely dissolved, 25.54 g piperidine was added. Next, 26.3 mL allyl bromide was added dropwise, after which it was reacted at RT for 6 h. Separatory funnel was used to collect the oil layer and chloroform was added to the aqueous layer remaining for one-time extraction. The chloroform was then removed by rotary vacuum evaporator. The product was neutralized by washing it with DI water, then drying it over Na2SO4 and filtering it with filter paper, with the remaining liquid product (Nallylpiperidine, 32.7 g) was collected for the following step. The yield was calculated to be 86%.

In the second step, the prepared N-allylpiperidine, allyl chloride (25.1 g) and DI water (60 mL) were added into a reaction flask. The reaction, protected with copper filament, was heated at 65 °C for two days. The oil part was then separated and poured into excess acetone, which was thoroughly stirred. Following a 48 h of drying at 40 °C, the solid DAPCl (32.8 g, 62% yield) was obtained.

Scheme II. Schematic representation of DAPCl synthesis.

Membrane preparation

Four AEMs with different IECs were prepared using UV-initiated polymerization in a similar way to our previous study [START_REF] Tsehaye | Study of Anion Exchange Membrane Properties Incorporating N-Spirocyclic Quaternary Ammonium Cations and Aqueous Organic Redox Flow Battery Performance[END_REF] . The membrane preparation was performed as follows. The appropriate amounts of each species (polymer, DAPCl monomer and initiator) were solubilized in a mixture of 1, 2-dichloroethane and NMP and the solutions are cast on a petri dish. The mixture solution was covered with aluminum foil with small holes to keep it out of the light and permit the 1,2-dichloroethane evaporation at RT. The filmcontaining petri dish was degassed to remove O2. The viscous homogenous solution was cross-linked via UV-irradiation for 3 min under Ar. The membrane-containing petri dish was put in a 60 °C oven overnight.

Following NMP evaporation, the AEM was collected and immersed in DI water to wash out any remaining solvents, monomers and non-grafted Poly(DAPCl). As shown in the Figure 1, the AEMs consists of pyrrolidinium and poly(5-azoniaspiro[4.5]decane) cations.

UV-induced radical polymerization

Atomic force microscopy (AFM)

The surface morphology of the AEMs (OH -form) was studied by tapping mode AFM analysis using Si probes (with a spring constant 5 N/m) and resonance frequency 60-80 kHz. The surface of the membranes was imaged immediately after adding a drop of DI water at RT and 40-60% relative humidity using Nanowizard (JPK) of CIME Nanotech platform (Grenoble, France). The height and phase images were taken at the same time in a hard tapping conditions, with a 0.7-0.8 ratio of Asp to A0 controlling the tip's interaction with the specimen. In these operating settings, phase images are recognized to contribute the most to morphology in these types of samples. The image processing was done with "Gwyddion" and the image analysis was done with "ImageJ" open-source software. Previously reported procedure [START_REF] James | Interpretation of Contrast in Tapping Mode AFM and Shear Force Microscopy. A Study of Nafion[END_REF] was used to identify the ionic and hydrophobic domains. Five images of PPO-6CH2-3x membrane were evaluated.

Ion exchange capacity (IEC) and water uptake (WU)

The IECs of the AEMs were determined through Mohr titration method. The prepared AEMs were first converted to Br -form by immersing in 1 M NaBr at 50 o C for two days. The membrane samples were then rinsed several times with DI water and dried at 50 °C overnight. The AEMs were weighed and transferred to NaNO3 solution (0.2 M, 25 mL) for two days at RT. Three samples of the resultant solution were titrated with 0.01 M aq. AgNO3, with K2CrO4 indicator. Finally, the IECs were calculated using equations (1 &2):

𝐼𝐸𝐶 𝐵𝑟-= 𝑉𝑜𝑙𝑢𝑚𝑒 × 𝐶𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝐴𝑔𝑁𝑂 3 𝑔 𝑑𝑟𝑦 𝑚𝑒𝑚𝑏𝑟𝑎𝑛𝑒 = 0.01 × 𝑉𝑜𝑙𝑢𝑚𝑒 𝐴𝑔𝑁𝑂 3 𝑔 𝑑𝑟𝑦 𝑚𝑒𝑚𝑏𝑟𝑎𝑛𝑒 (1)
𝐼𝐸𝐶 𝑂𝐻-= 𝐼𝐸𝐶 𝐵𝑟- 1- 𝐼𝐸𝐶 𝐵𝑟-(𝑀 𝐵𝑟--𝑀 𝑂𝐻-) 1000 = 𝐼𝐸𝐶 𝐵𝑟- 1-0.0629 × 𝐼𝐸𝐶 𝐵𝑟- (2)
The water uptake (WU) capacity of the membranes was determined gravimetrically from their wet and dry weights, as shown in equation (3):

𝑊𝑈(%) = (𝑊𝑒𝑡 𝑚𝑒𝑚𝑏𝑟𝑎𝑛𝑒 𝑤𝑒𝑖𝑔ℎ𝑡-𝑑𝑟𝑦 𝑚𝑒𝑚𝑏𝑟𝑎𝑛𝑒 𝑤𝑒𝑖𝑔ℎ𝑡) 𝑑𝑟𝑦 𝑚𝑒𝑚𝑏𝑟𝑎𝑛𝑒 𝑤𝑒𝑖𝑔ℎ𝑡 × 100 (3)
The reported IEC titration and WU of the AEMs are an average of at least three measurements.

The hydration number (λ) was estimated as follows equation ( 4 (5)

Differential scanning calorimetry (DSC)

The number (N) of bound or non-freezable water and bulky or freezable water in the AEMs were determined by DSC (Mettler-Toledo DSC1 instrument). AEM samples were immersed in water for a week. Wet membrane samples (approximately 10 mg), weighed after mopping their surface with tissue paper, were sealed in Al sample container. The thermal cycle was programmed to lower from RT to -50 °C. It was followed by heating up to 25 o C at 5 K/min scan rate under N2 (50 mL/min). The Nnon-freezable and Nfreezable were determined using equations 6 and 7:

𝑁 𝑓𝑟𝑒𝑒𝑧𝑎𝑏𝑙𝑒 = 𝑀 𝑓𝑟𝑒𝑒 𝑀 𝑡𝑜𝑡 × 𝜆 = Δ𝐻 𝑓𝑟𝑒𝑒 /Δ𝐻 𝑖𝑐𝑒 𝑀 𝑡𝑜𝑡 × λ (6)
Where Mtot and Mfree are the masses of the total and freezable water molecules, respectively. ΔHfree enthalpy is determined by integrating the area under the DSC peak. ΔHice is taken to be 334 J/g.

𝑁 𝑛𝑜𝑛-𝑓𝑟𝑒𝑒𝑧𝑎𝑏𝑙𝑒 = 𝜆 -𝑁 𝑓𝑟𝑒𝑒𝑧𝑎𝑏𝑙𝑒 (7)

Alkaline stability

The chemical degradation of the samples was determined by monitoring the changes in Cl -ion conductivity after immersion in different alkaline solutions (6 M KOH at RT and 1 M KOH at 60 °C) during two weeks.

Moreover, to simulate the practical ZSAFB application, the stability of the membranes in alkaline Zn slurry electrolyte at RT was studied.

Thermogravimetric analysis

Thermal properties of the membranes were determined by thermogravimetric analysis under N2. Dry membrane samples were heated from RT to 800 °C with 10 °C/min heating rate.

Zincate ions crossover

A home-made diffusion cell was employed to quantitatively determine the zincate ion crossing through the prepared AEMs [START_REF] Lee | Electrospun Polyetherimide Nanofiber Mat-Reinforced, Permselective Polyvinyl Alcohol Composite Separator Membranes: A Membrane-Driven Step Closer toward Rechargeable Zinc-Air Batteries[END_REF][START_REF] Hwang | Selective Ion Transporting Polymerized Ionic Liquid Membrane Separator for Enhancing Cycle Stability and Durability in Secondary Zinc-Air Battery Systems[END_REF][START_REF] Abbasi | Poly(2,6-Dimethyl-1,4-Phenylene Oxide)-Based Hydroxide Exchange Separator Membranes for Zinc-Air Battery[END_REF][START_REF] Kim | Artificially Engineered, Bicontinuous Anion-Conducting/-Repelling Polymeric Phases as a Selective Ion Transport Channel for Rechargeable Zinc-Air Battery Separator Membranes[END_REF] . In this diffusion cell design, the enriched chamber contained 6 M KOH and Zn(OH)2 (0.3 M, solubility limit at RT) and the depleted chamber contained only 6 M aq. KOH. The 0.5 cm 2 membrane was put between the chambers and atomic absorption spectroscopy (AAS) was used to quantify the amount of zincate ion on the depleted side.

2.5.9 Single Zn slurry-air cell assembly and electrochemical performance A single Zn slurry-air cell containing air cathode, membrane and Zn slurry with a CuNi plate was used. The catalyst ink consists of Pt/C catalyst, FAA-3 ionomer (23 wt. %. of the catalyst layer, Fumatech), DI water and isopropanol. The Pt loading was 1 mg/cm 2 . After 15 min of sonication in an ultrasonic water bath, the prepared suspension was sprayed over a gas diffusion layer (5 cm 2 ) (SGL Carbon, 29BC, Germany). The Zn slurry was prepared in the same manner as described in our previous study [START_REF] Choi | Development of Flow Fields for Zinc Slurry Air Flow Batteries[END_REF] , and the battery electrochemical performance was tested as stated in that study.

The ZSAFB (Figure 2), employing the four prepared AEMs, performance was studied by current-voltage characteristic curves. EIS measurements were performed to determine the ohmic resistance of the battery with the various AEMs. The flow rates of the Zn slurry in the negative electrode and synthetic air in the air cathode were 160 and 100 mL/min, respectively.

Results and discussion

Polymers and monomer preparation

The reaction protocol followed to prepare the polymer is summarized in scheme I. The bromination, ketone reduction and quaternization reaction products involved during the preparation of PPO-6CH2-Q were confirmed using 1 H NMR. The bromination of PPO polymer with a 6-C spacer containing a ketone group in the benzylic position using 6-bromohexanoyl chloride in a Friedel-Crafts acylation was performed based on slight modification of the method discussed in the literature [START_REF] Hibbs | Alkaline Stability of Poly(Phenylene)-Based Anion Exchange Membranes with Various Cations[END_REF][START_REF] Parrondo | Synthesis and Alkaline Stability of Solubilized Anion Exchange Membrane Binders Based on Poly(Phenylene Oxide) Functionalized with Quaternary Ammonium Groups via a Hexyl Spacer[END_REF] . The peak at around 3.0 ppm (denoted as peak i) represents to the protons bonded to the carbon next to the ketone group. The bromination degree was determined from the proportion of the 1 H NMR integrals of protons in -CH2Br and the CH3 groups from the PPO (at 2.1 ppm) (Figure 3a). The functionalization ratio was fixed at 33% bromo-alkylation of PPO structural unit. The bromination was followed by ketone reduction i.e., peak i was shifted from 3 ppm to 1.7 ppm and a new peak appeared (denoted k, 2.8 ppm). The reduction of ketone groups is important because the proton in the α-position of the ketone can form enolate anions, which are strong nucleophiles and can degrade the functional groups of the membrane [START_REF] Parrondo | Synthesis and Alkaline Stability of Solubilized Anion Exchange Membrane Binders Based on Poly(Phenylene Oxide) Functionalized with Quaternary Ammonium Groups via a Hexyl Spacer[END_REF] . The ketone reduction degree (96-98%) was calculated from 1 H NMR spectra (Figure 3b). In the next step, the bromine alkyl side chain was reacted with diallylmethylamine and the quaternized PPO with methyldiallylammonium was obtained (PPO-6CH2-Q).

As shown in the NMR of the PPO-6CH2-Q (Figure 3c), where new peaks matching with the protons of diallylmethylammonium appeared. The QA cationic monomer based on piperidine containing allyl groups, i.e., DAPCl, was synthesized in two steps following a procedure reported in the literature [START_REF] Strasser | Base Stable Poly(Diallylpiperidinium Hydroxide) Multiblock Copolymers for Anion Exchange Membranes[END_REF][START_REF] Olsson | Poly( N , N -Diallylazacycloalkane)s for Anion-Exchange Membranes Functionalized with N -Spirocyclic Quaternary Ammonium Cations[END_REF] . The 1 H NMR spectra of DAPCl is shown in Figure S1.

Ion exchange capacity

The crosslinked AEMs with thickness between 45 and 65 µm were obtained by copolymerizing PPO-6CH2-Q with DAPCl. The ratio between PPO quaternized ammonium side chain and DAPCl was varied between 0.5 to 3 in order to prepare membranes with poly-ionic side chains, with IEC varying between 2.3 and 3.6 mmol OH -/g. The prepared AEMs titrated IEC (Table 1) was found to vary between 2 and 3.4 mmol OH - /g, slightly lower than the theoretical ones due to (i) incomplete polymerizations of DAPCl and (ii) the formation of poly(DAPCl), not-bonded to PPO-6CH2-Q, which were removed from the membrane during the water washing step. Based on the titrated IEC values, the per cent (%) of DAPCl incorporated into the PPO-6CH2-Q was calculated (DAPCl yield, Table 1). The DAPCl yield (%) increased with increasing of DAPCl to polymer ratio due to, probably, the higher number of double bonds and/or low viscosity of the solution. It can be noted that the conversion for this class of copolymers is higher than for our previous results in which the CH2Br was directly bonded to the PPO backbone [START_REF] Tsehaye | Study of Anion Exchange Membrane Properties Incorporating N-Spirocyclic Quaternary Ammonium Cations and Aqueous Organic Redox Flow Battery Performance[END_REF] )feed × 100.

Water uptake, ionic conductivity and surface morphology

An optimized membrane water uptake content is required for developing hydrated ionic domains as well as maintenance of membrane mechanical strength and selectivity. The water uptakes of the PPO-6CH2-0.5x and PPO-6CH2-3x membranes in OH -at RT were 32 wt.% and 105 wt.%, respectively (Table 2). The increase of IEC of the membranes induces a large water-uptake because of the lengthening of the hydrophilic chains and decreases of the crosslinking density and stiffness. The water uptake of the prepared membranes is lower when compared to membranes based on PPO containing poly(DAPCl) with similar IEC, in which the allyl function was directly bonded to PPO backbone [START_REF] Tsehaye | Study of Anion Exchange Membrane Properties Incorporating N-Spirocyclic Quaternary Ammonium Cations and Aqueous Organic Redox Flow Battery Performance[END_REF] .

AEM with a high OH -ion conductivity is essential for the development of high power density batteries. As the Zn slurry-air flow battery operates at RT, an AEM with high OH -ion conductivity at RT is needed.

Table 2 shows the OH -and Cl -(measured for comparison purpose) ions conductivities of the crosslinked AEMs. The OH -ions conductivity of the AEMs seems to be about 2.7-4 times higher than that of their respective Cl -ions conductivity mainly due to the difference in molar conductivities of the ions (Cl -= 7.631 and OH -= 19.86 mS m 2 /mol), in agreement with findings reported in the literature [START_REF] Hou | Recyclable Cross-Linked Anion Exchange Membrane for Alkaline Fuel Cell Application[END_REF] . As shown in Table 2, the AEMs ionic conductivity and water uptake varied significantly depending on their IEC. Generally speaking, ionic conductivity is dependent on the charge carrier concentration and their mobility. The increase in the IEC causes an increase in the amount of the charge carrier as well as their mobility. Indeed, the mobility of the charge carrier increases with an increase in water uptake. For instance, a fully hydrated PPO-6CH2-3x membrane has a OH -conductivity of 19 mS/cm at RT. This high value proves an efficient transportation of OH -ion and well-connected ion channels [START_REF] Xue | Phenylene Oxide)s Incorporating N-Spirocyclic Quaternary Ammonium Cation/Cation Strings for Anion Exchange Membranes[END_REF] , as shown in the AFM images of PPO-6CH2-3x membrane (Figure 4).

Hydration number and water states in the membrane

The λ of the membranes was found to increase with the IEC. As shown in Table 3, PPO-6CH2-1x membranes have approximately three more water molecules per functional group than PPO-6CH2-0.5x membranes, with averages of 7.8 and 11.6, respectively. PPO-6CH2-2x and PPO-6CH2-3x, on the other hand, had higher hydration numbers, around 14 and 16.4, respectively. Similar values have been reported for other AEMs in the literature [START_REF] Xue | Phenylene Oxide)s Incorporating N-Spirocyclic Quaternary Ammonium Cation/Cation Strings for Anion Exchange Membranes[END_REF][START_REF] Weiber | Ion Distribution in Quaternary-Ammonium-Functionalized Aromatic Polymers: Effects on the Ionic Clustering and Conductivity of Anion-Exchange Membranes[END_REF] .

To determine whether the water molecules are in interaction with the ionic species or in water clusters, DSC measurements were performed. The DSC curves are given in Figure S2. There is no freezable water detected in the PPO-6CH2-0.5x membrane, as shown in Table 3. Similarly, on the PPO-6CH2-1x membrane, a very small amount of freezable water was detected. It can be concluded that nearly all of the water absorbed in these two membranes are in strong interaction with the ionic function [START_REF] Liu | Anion Conducting Multiblock Copolymers with Multiple Head-Groups[END_REF] . Contrarily, the other two membranes, PPO-6CH2-2x and PPO-6CH2-3x, showed a freezable water molecule of about 2 and 6 water molecules, respectively, suggesting the presence of bulk water which are not associated with the ionic functions [START_REF] Lue | Water States in Perfluorosulfonic Acid Membranes Using Differential Scanning Calorimetry[END_REF] . Overall with the exception of PPO-6CH2-0.5x membrane, regardless of the membranes IEC and water uptake, it appears that about 11 water molecules per ionic function are in strong interaction with the polymer. Excess water molecules, on the other hand, are freely available in the membrane.

Similarly, we recently reported the amounts of freezable and non-freezable water molecules in benzylic substituted PPO-based AEMs containing N-spirocyclic QA cations [START_REF] Tsehaye | Study of Anion Exchange Membrane Properties Incorporating N-Spirocyclic Quaternary Ammonium Cations and Aqueous Organic Redox Flow Battery Performance[END_REF] . For example, a membrane with an IEC of 2.84 mmol Cl -/g was found to have 7 ± 0.8 freezable water molecules per ionic function.

S20

Zincate ions crossover

Minimizing zincate ion crossover is critical in rechargeable ZSAFB. To determine the best compromise between OH -ion conductivity and ion selectivity, the zincate ion crossover through these AEMs was measured and compared. As shown in Figure 5, the Zn ions concentration in the depleted side increased over time for all the membranes. For e.g., for PPO-6CH2-3x membrane, less than 5% of the Zn species crossed to the second compartment after one week. The low Ds of zincate ion of the prepared AEMs (Table 4) could be explained by the selectivity of the ionic channels resulting from the nanophase separated morphology [START_REF] Abbasi | Poly(2,6-Dimethyl-1,4-Phenylene Oxide)-Based Hydroxide Exchange Separator Membranes for Zinc-Air Battery[END_REF] . The membranes with higher amount of freezable or bulky water exhibited a higher Zn(OH)4 2-ions crossover. A larger amount of water in the ionic domains is directly related to larger ionic channel sizes and presence of free water, resulting in lower ionic species selectivity. As a result, the membranes permselectivity decreases as the IEC increases. The diffusion coefficients of zincate ions through various types of membranes, including AEMs, have been reported in the literature [START_REF] Tsehaye | Membranes for Zinc-Air Batteries: Recent Progress, Challenges and Perspectives[END_REF] .

Membrane stability

Long-term AEM stability in a highly alkaline solution is required for long-term use in rechargeable

ZSAFBs. The PPO-6CH2-3x (membrane with the highest IEC) was chosen as a representative sample since it is easier to correctly measure changes in ionic conductivity (and/or IEC) when membranes with high IEC are used [START_REF] Parrondo | Synthesis and Alkaline Stability of Solubilized Anion Exchange Membrane Binders Based on Poly(Phenylene Oxide) Functionalized with Quaternary Ammonium Groups via a Hexyl Spacer[END_REF] .

As shown in Table 2, a fully hydrated pristine PPO-6CH2-3x membrane has a Cl -ion conductivity of 6.6 mS/cm at RT. The Cl -ion conductivity of the PPO-6CH2-3x remains constant after 14 days of immersion in both 6 M KOH (aq.) solution and Zn-slurry at RT. Moreover, the membrane maintained above 92% of its Cl -ion conductivity after two weeks under an accelerated degradation test (1 M KOH, 60 °C). Loss in ionic conductivity of AEMs in alkaline media is mainly caused by chemical degradation of the cation and/or polymer backbone used [START_REF] Wang | Alkali-Stable Partially Fluorinated Poly(Arylene Ether) Anion Exchange Membranes with a Claw-Type Head for Fuel Cells[END_REF] . However, it must be also noted that a minor decline in conductivity can also be caused by AEM morphology change instead of chemical degradation [START_REF] Dang | Alkali-Stable and Highly Anion Conducting Poly(Phenylene Oxide)s Carrying Quaternary Piperidinium Cations[END_REF] . The mechanical flexibility and integrity of the investigated membrane seem not to be affected after the alkaline stability tests. Chemically degraded membranes are often reported to become brittle and cracked into pieces [START_REF] You | Highly Conductive and Chemically Stable Alkaline Anion Exchange Membranes via ROMP of Trans -Cyclooctene Derivatives[END_REF] . When compared to the base-stable N-spirocyclic-based AEMs reported in the literature, the present AEMs demonstrated comparable [START_REF] Strasser | Base Stable Poly(Diallylpiperidinium Hydroxide) Multiblock Copolymers for Anion Exchange Membranes[END_REF][START_REF] Xue | Phenylene Oxide)s Incorporating N-Spirocyclic Quaternary Ammonium Cation/Cation Strings for Anion Exchange Membranes[END_REF] or superior stability [START_REF] Shen | Multi-Block Poly(Ether Sulfone) for Anion Exchange Membranes with Long Side Chains Densely Terminated by Piperidinium[END_REF][START_REF] Shang | Anion Exchange Membranes Based on Poly (Ether Ether Ketone) Containing N-Spirocyclic Quaternary Ammonium Cations in Phenyl Side Chain[END_REF] under similar testing conditions. A comparison of N-spirocyclic QA-based AEMs alkaline stability is summarized in Table S1.

The high base-stability of the prepared AEMs could be due to: (i) the constrained ring conformation of the cation used inducing high transition energies for Hofmann elimination and substitution degradations [START_REF] Marino | Alkaline Stability of Quaternary Ammonium Cations for Alkaline Fuel Cell Membranes and Ionic Liquids[END_REF] , (ii) the grafting of the ionic side chain through a hydrophobic spacer to the polymer backbone which simultaneously removes the poor stability of PPO benzylic function [START_REF] Xue | Phenylene Oxide)s Incorporating N-Spirocyclic Quaternary Ammonium Cation/Cation Strings for Anion Exchange Membranes[END_REF][START_REF] Dang | A Comparative Study of Anion-Exchange Membranes Tethered with Different Hetero-Cycloaliphatic Quaternary Ammonium Hydroxides[END_REF] and improves the microphase separation, making the PPO chain less susceptible to chemical attack 67 and (iii) the crosslinking, which improves the mechanical properties and limits the mobility of OH -into the hydrophilic domain [START_REF] Zhang | Ether Spaced N-Spirocyclic Quaternary Ammonium Functionalized Crosslinked Polysulfone for High Alkaline Stable Anion Exchange Membranes[END_REF] .

Jannasch et al., [START_REF] Olsson | Poly( N , N -Diallylazacycloalkane)s for Anion-Exchange Membranes Functionalized with N -Spirocyclic Quaternary Ammonium Cations[END_REF] reported the excellent alkaline stability of poly(DAPCl) at 120 °C for 14 days.

The thermal stability of PPO-6CH2-3x membrane was also investigated with TGA under N2 atmosphere (Figure S3). The membrane showed two stages degradation behaviour. The first weight loss, which begins at around 300 °C is due to the heterocyclic QA groups [START_REF] Olsson | Poly( N , N -Diallylazacycloalkane)s for Anion-Exchange Membranes Functionalized with N -Spirocyclic Quaternary Ammonium Cations[END_REF] , which occurs at higher temperature than that of common QA groups [START_REF] Xue | Phenylene Oxide)s Incorporating N-Spirocyclic Quaternary Ammonium Cation/Cation Strings for Anion Exchange Membranes[END_REF] . The last degradation which occurred at above 400 °C is ascribed to the PPO polymer backbone, and remaining Poly(DAPCl). Two stages thermal degradation of poly(DAPCl) has been reported in the literature [START_REF] Olsson | Poly( N , N -Diallylazacycloalkane)s for Anion-Exchange Membranes Functionalized with N -Spirocyclic Quaternary Ammonium Cations[END_REF] .

In summary, the prepared AEMs are thermally and chemically stable enough to be employed in alkaline ZSAFB, which typically use 6 M (aq.) KOH and operate at RT.

Single ZSAFB performance

The prepared AEMs are suitable candidates for ZSAFB applications due to their high OH -ion conductivity, alkaline stability and low zincate ion crossover. To investigate the impact of IEC of AEMs on power capability, ZSAFB single cells were built and run with the four prepared AEMs.

The cell resistances incorporating the different membranes were measured and are summarized in Figure 6(a). As expected, the IEC of the membranes, had a major impact on the cell resistance. For e.g., PPO-6CH2-0.5x membrane-based cell exhibited the highest cell resistance (4.5 Ω cm 2 ) among the tested systems. Whereas, the battery using PPO-6CH2-3x membrane displayed the lowest cell resistance, only 0.8 Ω.cm 2 , despite the fact that this membrane is the thickest. Similar IEC dependency and pattern was established in the OH -ion conductivity of the membranes. The cell resistance determined from the current density-voltage curve (Figure 6b) was about 2-3 times higher than that of cell resistances obtained through EIS measurement. This indicates the presence of additional resistance under current which may be due to the ion transport limitations. that the AEMs are strong enough to separate the Zn-slurry electrolyte from the air electrode [START_REF] Zhang | Ether Spaced N-Spirocyclic Quaternary Ammonium Functionalized Crosslinked Polysulfone for High Alkaline Stable Anion Exchange Membranes[END_REF] . As predicted, both the cell voltage and the power density were found to depend on the discharge current.

PPO-6CH2-0.5x and 1x membranes delivered maximum power densities of 35 mW/cm 2 and 39 mW/cm 2 , respectively. The high ohmic resistance of the membranes is detrimental for the cell, as very high current density is used. The cell assembled with PPO-6CH2-3x membrane exhibited an excellent maximum power density (153 mW/cm 2 ), which is higher than that of cell assembled with PPO-6CH2-2x (137 mW/cm 2 ) membrane due its higher OH -ion conductivity. The peak power density obtained is much higher than that of the cells employing commercial porous membranes [START_REF] Tsehaye | Pristine and Modified Porous Membranes for Zinc Slurry-Air Flow Battery[END_REF] . However, the cell was further optimized here, with the addition of an ionomer inside the electrode, which improves the electrochemical reaction kinetics. Furthermore, the power density attained is one of the highest values reported for Zn-air batteries employing AEMs. As summarized elsewhere [START_REF] Tsehaye | Membranes for Zinc-Air Batteries: Recent Progress, Challenges and Perspectives[END_REF] It's also worth noting that the maximum power density is not the only factor to consider when selecting a membrane. Other parameters, such as the degree of zincate species crossover (i.e., closely related to capacity fade), should be evaluated as well. Overall, because the peak power densities of the single cells employing PPO-6CH2-2x and PPO-6CH2-3x membranes are similar, and the PPO-6CH2-2x membrane has a much lower Zn(OH)4 2-ion crossover, it can be concluded that the PPO-6CH2-2x is more promising for rechargeable ZSAFB because it is expected to have much lower capacity fade during cyclability testing. The peak power density and Zn(OH)4 2-ion crossover tradeoff are shown in Figure 7. The cyclability of the developed system will be investigated in future works. with PPO-6CH2-3x delivered a maximum power density of 153 mW/cm 2 . On the other hand, given its low zincate ion crossover and high power density (137 mW/cm 2 ), the PPO-6CH2-2x is a very promising membrane for alkaline batteries, including rechargeable ZSAFB. In future research, we believe it is vital to look into the long-term durability of the membranes in a ZSAFB and their mechanical strengths.

Supporting information

Figure 1 :

 1 Figure 1: Schematic diagram of the AEM fabrication protocol.

4

 4 Hydroxide and chloride ions conductivitiesThe Cl -and OH -ions conductivities (σ) of the membranes were determined by exchanging the resulting membranes in a saturated NaCl (1 M) and KOH (1 M), respectively, at RT for 24 h. The membranes were then washed with DI water. The ionic conductivities were determined (at RT) via a through-plane electrochemical impedance spectroscopy (EIS) in the frequency range 13 MHz -5 Hz (equation 5). To avoid carbonation process, the AEM (OH -form) resistance measurement was performed in a home-made glove-box under continuous Ar flow. membrane thickness (cm) Membrane active area (𝑐𝑚 2 )× 𝑀𝑒𝑚𝑏𝑟𝑎𝑛𝑒 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 (Ω)

Figure 2 :

 2 Figure 2: Image of the ZSAFB.

Figure 3 : 1 H

 31 Figure 3: 1 H NMR spectrum of PPO-6CO-Br (a), PPO-6CH2-Br (b) and PPO-6CH2-Q (c).

Figure 4 :

 4 Figure 4: AFM phase image of PPO-6CH2-3x membrane (scan size 1 µm x 0.5µm) (left) and post-treatment of AFM phase image (right); darker and lighter regions represent cationic and non-ionic groups, respectively 56 .

Figure 5 :

 5 Figure 5: Plots of zincate ion crossover as a function of time/thickness. The figure lines are drawn to guide the eye.

Figure 6 :

 6 Figure 6 : Zn slurry-air flow battery (using the prepared AEMs) resistance (a) and power density (b) measured at RT, with flow rates of the Zn slurry and synthetic air of 160 and 100 mL/min, respectively.

Figure 7 :

 7 Figure 7: The impact of membrane IEC on peak power density and Zn(OH)4 2-ion crossover.

  have successfully prepared four AEMs, in which the multi N-spirocyclic QAs were grafted to the PPO polymer backbone with 6-C spacer through UV-irradiation. The IEC of each AEM was PPO-6CH2-0.5x PPO-6CH2-1x PPO-

1 H

 1 Figure S1: 1 H NMR spectrum of (i) N-allylpiperidine in CDCl3 and of (ii) N,N-diallylpiperidinum chloride (DAPCl) in H2O.

Figure S2 :

 S2 Figure S2: DSC curves of (a) PPO-6CH2-0.5x, (b) PPO-6CH2-1x, (c) PPO-6CH2-2x and (d) PPO-6CH2-3x membranes.

Figure S3 :

 S3 Figure S3: TGA curves of PPO-6CH2-3x membrane.

  The diffusion coefficient (D) of zincate ion of each membrane was determined from the slope of time vs 𝑙𝑛(

		𝐶 𝐴 𝐶 𝐴 -𝐶 𝐵	) plot. The zincate ion concentrations in the
	depleted and enriched sides are CB (mol/L) and CA (mol/L), respectively. Additionally, the ratio of ionic
	conductivity to zincate ion diffusion coefficient is used to estimate the membranes' permselectivity as
	follows (equation 8):
	Permselectivity =	𝜎 OH-D Zn(OH) 4 2-

Table 1 .

 1 . Thickness, IEC and DAPCl yield of the prepared membranes

	AEM	Thickness (µm)	IEC (mmol Br -/g) Theoretical Titration	IEC (mmol OH -/g)*	𝐷𝐴𝑃𝐶𝑙 𝑃𝑃𝑂-6𝐶𝐻 2 -Q ** Feed Titration	DAPCl yield (%)***
	PPO-6CH2-0.5x	45 + 3			2.00	1.80 ± 0.04 2.03 ± 0.04	0.5	0.27	53
	PPO-6CH2-1x	49 + 3			2.26	2.03 ± 0.05 2.33 ± 0.05	1	0.59	59
	PPO-6CH2-2x	55 + 3			2.66	2.42 ± 0.03 2.85 ± 0.03	2	1.34	67
	PPO-6CH2-3x	60 + 3			2.91	2.80 ± 0.02 3.4 ± 0.02	3	2.52	84
	*Calculated from titration IECBr-using equation (2).
			(	𝐷𝐴𝑃𝐶𝑙 𝑃𝑃𝑂-6𝐶𝐻 2 -Q	)𝑡itration
				(	𝐷𝐴𝑃𝐶𝑙 𝑃𝑃𝑂-6𝐶𝐻 2 -Q

** Molar ratio of DAPCl to PPO-6CH2-Q. *** DAPCl yield (%) =

Table 2 :

 2 WU and ionic conductivity of the prepared membranes at 20 °C

	Membranes	WU (OH -form) (%)	Ionic conductivity (mS/cm)
			Cl -	OH -
	PPO-6CH2-0.5x	32 (± 0.7)	0.94 (±0.15)	3.3 (± 0.2)
	PPO-6CH2-1x	55 (± 1.5)	1.9 (± 0.10)	5.2 (± 0.25)
	PPO-6CH2-2x	80 (± 3)	3.1 (± 0.2)	13 (± 0.32)
	PPO-6CH2-3x	105 (± 4)	6.6 (± 0.2)	19 (± 0.4)

Table 3 :

 3 Water state in the AEMs: Bulky and bound water molecules

	AEM	λOH-		ΔHf (J/g wet	ΔHf	Bulky water (wt.%) c	λ in the membrane
				membrane) a
							(J/g water) b	Bulky	Non-freezable
	PPO-6CH2-0.5x	7.8 ± 0.3	0			0	0	0	7.8 ± 0.3
	PPO-6CH2-1x	11.6 ± 0.2	4.1 ± 0.7		11.6 ± 2	1.9 ± 0.33	0.22 ± 0.04	11.4 ± 0.1
	PPO-6CH2-2x	13.9 ± 0.5	31 ± 2.3			69.2 ± 5.1	16.6 ± 1.3	2.3 ± 0.16	11.6 ± 0.3
	PPO-6CH2-3x	16.4 ± 0.4	62 ± 3			121± 5.8	38.1 ± 1.8	6.23 ± 0.3	10.2 ± 0.4
	a. Obtained from the DSC by integrating the DSC freezing curve area.
	b. ΔH 𝑓 ( 𝑔 𝑤𝑎𝑡𝑒𝑟 ) = ΔH 𝑓 ( 𝐽 𝑔 𝑤𝑒𝑡 𝑚𝑒𝑚𝑏𝑟𝑎𝑛𝑒 ) × 𝐽	𝑚𝑒𝑚𝑏𝑟𝑎𝑛𝑒 𝑤𝑒𝑖𝑔ℎ𝑡 (𝑔) 𝑤𝑎𝑡𝑒𝑟 𝑤𝑒𝑖𝑔ℎ𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑚𝑒𝑚𝑏𝑟𝑎𝑛𝑒 (𝑔)
	c. Freezable water (wt. %) =	ΔH 𝑓 ( ΔH 𝑖𝑐𝑒 , 𝑝𝑢𝑟𝑒 𝑤𝑎𝑡𝑒𝑟 𝐽 𝑔 𝑤𝑎𝑡𝑒𝑟 )	× 𝑊𝑎𝑡𝑒𝑟 𝑢𝑝𝑡𝑎𝑘𝑒 (𝑤𝑡. %).

Table 4 :

 4 Diffusion coefficient (D) of zincate ion

	AEM	D (m 2 /s)	Permselectivity (S.s/m 3 )	Ref.
	PPO-6CH2-0.5x	8.9 × 10 -15	3.7 x 10 13
	PPO-6CH2-1x	2.8 × 10 -13	1.9 x 10 12	This study
	PPO-6CH2-2x	9.7 × 10 -13	1.3 x 10 12
	PPO-6CH2-3x	2.3 × 10 -12	8.2 x 10 11
	PPO-TMA	1.9 × 10 -14	9.2 × 10 13	28
	AEM-Ia3‾d	1.6 × 10 -13	7.9 × 10 12	60
	PPO-3.45 + Celgard ® 3501	5.2 × 10 -13	-	23

  controlled by changing the ratio between DAPCl and PPO-6CH2-Q. AFM analysis clearly showed welldefined phase separation in PPO-6CH2-3x membrane, which results in excellent OH -ion conductivity at room temperature. The membranes displayed an excellent alkaline stability in 1 M KOH at 60 °C, with less than 8% drop in ionic conductivity after 360 h of the stability test. Furthermore, the membranes showed no signs of chemical degradation in a typical electrolyte used in ZSAFBs. The OH - ion conductivity, amount of bulky or freezable water, permselectivity and battery performances of the AEMs were mostly influenced by the IEC of the AEMs. AEMs with low IECs, and thus little or no freezable water molecules exhibited negligible zincate ion crossover, whereas AEMs with high IECs (freezable water molecules) had much higher zincate ion crossover, implying that bulky water may play a role in increasing unwanted zincate ion crossover. Even though the IEC sacrifices the zincate ion selectivity of the AEMs, however, it merits the peak power densities of the ZSAFBs. A single ZSAFB

		IEC (mmol OH -/g)
	2.03	2.33	2.85	3.4	2,5×10 -12
					1,5×10 -12 2,0×10 -12	diffusion coef. (m 2 /s)
						2-
						Zn(OH) 4

Table S2 .

 S2 Retention in σ and/or IEC are the remaining proportions of membrane σ and/or IEC after the alkaline test at the specified conditions. Anion exchange membranes used in Zn-air batteries. Reproduced from ref13 .

	AEM		Battery type				Electrode		Electrolyte	Peak power	Ref
					Zn electrode	Air electrode		density (mW/cm 2 )
	FAA-3-based AEM		Rechargeable Zn-air	Stainless-steel mesh coat	Cobalt oxide based commercial	6 mol KOH	9.8	14
			battery		with Zn species				PAA gel
	A201-based		Solid	flexible	Zn film and poly	Co3O4 nano-particles (<50 nm	33	15
			rechargeable	Zn-air	(vinylidene	fluoride-	particle size) Gas diffusion layer:
	Laminated cross-linked	battery		cohexafluoropropene)	(catalyst loading of 1.0 mg/cm 2 )	44
	nanocellulose/GO				polymer binder
	(QAFCGO)
	PPO-ASU-40 PPO-MPY		ASU Primary Zn-air battery Pure Zn plate 10 M NaOH at 80 °C, 250 h diffusion layer. Mixture of MnO2 and 53.8% (σ) Ni-foam current collector and gas	7 M KOH	70	16
	ASU-PPO		ASU		1 M NaOH(aq) at 80 °C, 720 h VXC72 as catalyst.	96.7% (σ)
	[PAPi][OH] AEM composed	of	ASU All-solid-state Zn-air	1 M NaOH 80 °C, 168 h Polished Zn foil IrO2 and 40 wt.% Pt/C, catalyst ink, 5 98.4% (σ)	-	50	8	17
	PES-NS-10% hydroxypropyltrimonium PVA/guar	Dense battery		2 M NaOH at 80 °C, 864 h wt % Nafion and ethanol 84.3% (σ) and 86.2% ( IEC)	9
	chloride (PGG-GP)		N-spirocyclic
	CP1 PPO-3.45 + 3501		Dimethyl pyrrolinidium Zn slurry-air flow	1 M KOH at 80 °C, 720 h Zn slurry containing Zn, Catalyst ink (Pt/C catalyst with Pt 60% (σ) and 55% (IEC) Zn slurry	66	10	18
	PPO-7QPi-1.7		Piperidinium battery		1 M, NaOH at 90 C, 192 h ZnO, Carbopol and loading at 1 mg/cm 2 , isopropanol and 94% (σ) aqueous KOH. deionized water)	11
	QPEEK-spiro-pyr PPO-6CH2-3x		Pyrrolinidium Zn slurry-air flow battery	1 M NaOH at 60 °C, 480 h Zn slurry containing Zn, Catalyst ink (Pt/C catalyst mixed 81.0% (σ) ZnO, Carbopol and with Fumion FAA-3, deionized water	Zn slurry	153	This work
	QPEEK-spiro-pip		Piperidinium		aqueous KOH.		and isopropanol)	74.7% (σ)
	PPO-6CH2-3x		Poly(DAPCl)		1 M KOH at 60 °C, 336 h	92.4% (σ)	This study
	*S37

Electronic Supplementary Information (ESI)

Anion exchange membranes incorporating multi

Corresponding authors*: Fannie.Alloin@lepmi.grenoble-inp.fr Cristina.iojoiu@lepmi.grenoble-inp.fr

S36


Nathalie Forgeard

Marine Baron

Jonathan Caron

Clémentine Boccon-Gibod

Nayara Guedes

Véronique Morel

Nathalie Jacque

Maya Ouzegdouh

Sylvain Choquet

Clotilde Bravetti

Daphné Krzisch

Florence Nguyen-Khac

Elise Chapiro

Véronique Leblond

MD Damien Roos-Weil
email: damien.roosweil@aphp.fr.

Inflammation in Waldenström macroglobulinemia is associated with 6q deletion and need for treatment initiation

Keywords: Waldenström's macroglobuilnemia, prognostic factors, C-Reactive Protein, inflammation, cytogenetics

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Waldenström's macroglobulinemia (WM) is a rare mature B-cell lymphoproliferative disorder, characterized by bone marrow (BM) infiltration of lymphoplasmacytic cells and the presence of a monoclonal IgM. MYD88 L265P mutation has been identified as the main driver event, with more than 90% of patients harbouring this mutation [START_REF] Braggio | Molecular pathogenesis of Waldenström's macroglobulinemia[END_REF]. MYD88 is an adaptor protein of the Toll-like receptors (TLR) and IL-1 receptors signalling cascades, and is consequently key in the activation of the NF-kB and JAK/STAT pathways [START_REF] Braggio | Molecular pathogenesis of Waldenström's macroglobulinemia[END_REF]. The second most commonly mutated gene is CXCR4 (30-40%) [START_REF] Hunter | The genomic landscape of Waldenström macroglobulinemia is characterized by highly recurring MYD88 and WHIMlike CXCR4 mutations, and small somatic deletions associated with B-cell lymphomagenesis[END_REF], including around 50% of mutations located in the S338 codon (referred to as CXCR4 WHIM ). The most frequent cytogenetic abnormality is 6q deletion (del6q), found in 30 to 55% of cases [START_REF] Nguyen-Khac | Chromosomal aberrations and their prognostic value in a series of 174 untreated patients with Waldenström's macroglobulinemia[END_REF]. Other chromosomal abnormalities are, in decreasing order of frequency : trisomy 18 (tri18), del13q, del17p, tri4 and del11q [START_REF] Krzisch | Cytogenetic and molecular abnormalities in Waldenström's macroglobulinemia patients: Correlations and prognostic impact[END_REF]. Thirty percent of patients are asymptomatic at diagnosis. Of those, 68%

will require treatment at 10 years [START_REF] Pophali | Prevalence and survival of smouldering Waldenström macroglobulinaemia in the United States[END_REF]. Main reasons for therapy initiation are anaemia (67%), increasing IgM levels, peripheral neuropathy (20%), hyperviscosity (15%), and organomegaly (10%) [START_REF] Bustoros | Progression Risk Stratification of Asymptomatic Waldenström Macroglobulinemia[END_REF].

A significant association between WM and dysimmune conditions has been reported [START_REF] Kristinsson | Immune-related and inflammatory conditions and risk of lymphoplasmacytic lymphoma or Waldenstrom macroglobulinemia[END_REF], supporting the hypothesis that chronic immune stimulation could contribute to lymphomagenesis. Clinical practitioners caring for WM patients also notice that a fraction presents with chronic inflammatory syndrome, for which there is no obvious cause besides WM itself. Chronic inflammation may be the cause of and/or contribute to the development of anaemia, the leading cause of WM treatment initiation. However, very few data exist on the association between WM and inflammatory syndrome. One prospective clinical trial published in 2001 including 231 WM patients showed that 51% of the cohort had a CRP value of at least 1 mg/L at enrolment, which was associated with a higher risk of treatment initiation and lower progression-free (PFS) and overall survival (OS) rates [START_REF] Dhodapkar | Prognostic factors and response to fludarabine therapy in patients with Waldenström macroglobulinemia: results of United States intergroup trial (Southwest Oncology Group S9003)[END_REF].

Authors have postulated that the association between WM and inflammation could be mediated by interleukin 6 (IL-6), an inductor of CRP synthesis (9), with higher levels in WM patients compared to healthy controls, and significant reduction after treatment [START_REF] Hatzimichael | Serum levels of IL-6 and its soluble receptor (sIL-6R) in Waldenström's macroglobulinemia[END_REF]. IL-6 production is mediated through NF-kB activation, itself the consequence of MYD88 L265P mutation. However, not all patients present with inflammation, suggesting that other mechanisms are at work. Main objectives of our study were to estimate the prevalence of inflammatory WM and to compare clinical and biological characteristics between inflammatory and non-inflammatory patients.

We conducted a retrospective analysis including all WM patients who consulted in the Clinical Haematology Unit of Pitié-Salpêtrière hospital (Paris, France) between 1988 and 2020 (4) with available information regarding CRP, at diagnosis and/or before treatment initiation. We defined "inflammatory" status as a CRP value equal or superior to 5 mg/L (according to local normal values) and confirmed once at a minimum interval of one month, with no obvious cause besides WM itself.

Written consent for clinical, biological and bone marrow (BM) analyses were obtained in accordance with the Helsinki declaration and with ethical approval from national (CNIL 2212382) Those therapies consisted of chemoimmunotherapy, chemotherapy, anti-CD20 monoclonal antibody alone or in combination with proteasome inhibitor in 117 (70%), 33 (20%), 13 (8%) and four (2%) of cases, respectively. Median follow-up for the whole WM cohort was 7.2 years. Median treatmentfree survival (TFS) was 3.5 years. Five-year PFS for patients receiving 1L therapy was 57%, and 5-year OS for the whole cohort was 92% (data not shown). Median CRP level was 16.7 mg/L (range [0-263]), including 103 (46%) patients with CRP values strictly below 5 mg/L, and 119 (54%) patients with CRP values ≥ 5 mg/L (thereafter referred to as "inflammatory"). Among the 119 patients with inflammatory WM, patients harbouring CRP levels between 5-19, 20-49 and ≥ 50 mg/L were 66 (55%), 31 (26%) and 22 (18%) respectively (Figure 1A). Mean albumin value was significantly lower in the inflammatory group, as expected. Inflammatory WM patients more frequently presented with lymphadenopathies and anaemia (Table 1 and Supplementary Table S2). All other routine clinical and biological parameters were similar between inflammatory and non-inflammatory WM groups (Table 1 and Supplementary Table S2, whether CRP was studied as a negative/positive value or a continuous value respectively). The frequencies of main cytogenetic abnormalities and gene alterations in the whole cohort are represented in Supplementary Table S1. The most frequent mutations were identified in MYD88 (90%), CXCR4 (24%), MLL2/KMT2D (11%), ARID1A (10%), TP53 (7%), and CD79A/B (7%). Main cytogenetic abnormalities identified by karyotype and/or FISH were, in decreasing order of frequency, del6q (28%), tri4 (11%), tri12 (7%) and del17p (7%). Complex karyotype (CK) was observed in 17% of cases. TP53 abnormalities (either del17p and/or TP53 mutation) were present in 11% of 190 evaluable patients. In univariate analysis, del6q was the only cytogenetic or molecular abnormalities to be significantly associated with inflammatory status (Table 1), with del6q occurring in 39% of inflammatory versus 11% of non-inflammatory WM (p<10 -3 , Table 1). The proportion of del6q patients increased along with CRP values (Figure 1B). The level of CRP as a continuous variable was significantly associated with the presence of del6q (HR 1.014 [1.004-1.027], p=0.015) meaning the probability of del6q increased by 14% for a 1-point increase in CRP (IC95% [0.4%-2.7%], Supplementary Table S2). Eighty-three percent of del6q patients had CRP values ≥ 5 mg/L compared to 49% non-del6q WM patients. In multivariate analysis, only del6q (p=0.02) and albumin (p<10 -2 ) were found to be significantly associated with inflammatory status, whereas lymphadenopathies and anaemia were not (Table 1).

Inflammatory WM patients more often required treatment (85% vs. 64%, p<10 -3 ) (Table 1) but this did not translate in TFS difference (Figure 2A). Types of therapies were comparable between the two subgroups except for a slightly higher proportion of WM patients receiving rituximab monotherapy in the non-inflammatory subgroup (12% vs. 5%). Mean CRP values were significantly lower after 1L treatment in inflammatory WM (Table 1) and correlated with IgM response (Figure 1C). We did not observe any difference in terms of response to 1L therapy nor PFS (Table 1 andFigure 2B; 5-year PFS of 52 and 65% for inflammatory and non-inflammatory group respectively). Median OS was 10 years in the inflammatory group versus not reached in the non-inflammatory group, however it did not reach statistical significance (p=0.06, Figure 2C). Survival analyses according to CRP groups (5-19, 20-49 and ≥ 50 mg/L) or as a continuous variable did not yield significant differences (data not shown and Supplementary Table S3). Univariate and multivariate analyses of clinical and biological variables associated with TFS, PFS and OS in the entire cohort are summarized in Supplementary Table S3.

Prognostic factors associated with shorter OS included ISSWM (p<10 -2 ), anaemia (p=0.02), tri4 (p=0.03), del6q (p<10 -2 ) and TP53abn (p<10 -2 ), while only TP53abn (p=0.04) retained significant pejorative impact in multivariate analysis.

In this large retrospective WM cohort, considering a positivity threshold of 5 mg/L, we observed WM-associated inflammation in more than half of cases. Inflammatory status was significantly associated with anaemia and consequently more frequent need for treatment initiation.

Inflammatory status was not associated with higher IgM spike or BM infiltration, which is consistent with the hypothesis of cytokine-mediated anaemia in this context. One of the main findings of our work was the strong association between inflammation and del6q, corroborating two previous retrospective studies focusing on del6q status [START_REF] Ocio | 6q deletion in Waldenström macroglobulinemia is associated with features of adverse prognosis[END_REF] [START_REF] García-Sanz | 6q deletion in Waldenström macroglobulinaemia negatively affects time to transformation and survival[END_REF]. It is of particular interest given that the minimally deleted region of 6q chromosome in WM contains IBTK, HIVEP2 and TNFAIP3 genes, negative regulators of NF-kB (13) which mediates IL-6 production. Furthermore, it has been shown that TNFAIP3 specifically exerts an inhibitory effect on the L265P mutated MYD88 receptor [START_REF] Wenzl | Loss of TNFAIP3 enhances MYD88L265P-driven signaling in non-Hodgkin lymphoma[END_REF], allowing us to postulate that one of the mechanisms through which del6q is associated with inflammation is the loss of TNFAIP3. However, more than half of the inflammatory patients from our cohort did not harbour del6q, indicating that other mechanisms are at play. While inflammatory WM patients more frequently required therapy, we did not observe significant association with TFS, response to therapy nor PFS. A trend for poorer OS was observed although not significant (p=0.06). Inflammation has many deleterious effects, such as fostering malnutrition, cardiovascular disease and cancer. Rare cases of WM associated with AA amyloidosis have been described [START_REF] Terrier | The clinical spectrum of IgM-related amyloidosis: a French nationwide retrospective study of 72 patients[END_REF]. Sharp decrease in CRP values at the end of 1L treatment in our cohort shows that haematological and inflammatory responses were well correlated. Although our study did not explore this hypothesis, it could be speculated that persistence of elevated CRP after 1L therapy could contribute to morbi-mortality in these patients. This raises the question of early therapy initiation and/or complementary therapies directed against either the WM clone or IL-6 in order to prevent long-term consequences of chronic inflammation. Indeed, IL-6 plays an autocrine role in lymphoplasmacytic differentiation ( 9) and an anti-tumoral effect of anti-IL6 therapy has been demonstrated in a WM murine model in which anti-IL6 receptor antibodies were associated with a decrease in tumoral syndrome and M spike [START_REF] Han | Targeting IL-6 receptor reduces IgM levels and tumor growth in Waldenström macroglobulinemia[END_REF].

In conclusion, our work highlights clinical and biological specificities of WM patients with chronic inflammation, notably a higher prevalence of del6q, more frequent need for therapy initiation and a trend for poorer OS, which will have to be confirmed by further studies. These findings may have implications for the understanding of inflammation in WM as well as for further therapeutic developments.

Figure 1 .Figure 2 . 3 (

 123 Figure 1. Description of inflammatory patients. CRP distribution (density plot) (A), proportion of del6q patients according to CRP levels (B), correlation between CRP and haematological response (C)

  and local (CPP Ile-De-France 05/21/2014) ethics committees. Main characteristics of the whole WM cohort are detailed in Supplementary Table S1. No significant difference was observed between WM patients with available CRP or not. The study population comprised 222 patients, of which 66% were male. Median age at WM diagnosis was 64.5 years old (range [28.4-88.2]). During follow-up, 167/222 (75%) WM patients required first-line (1L) therapy.

Table 1 . Patients characteristics associated with inflammatory status

 1 Diffuse Large B Cell Lymphoma ; IPSSWM, International Prognostic Scoring System of WM. Subgroup comparisons were performed using Chi-square of Fisher's exact test for categorical variables, and Student's t test for continuous variables. Multiple hypothesis correction was performed with Holm's method for univariate analysis. A multivariate logistic regression was performed for multivariate analysis.

	9

Acknowledgements: We thank Dr Nicolas Gilliers for his assistance with the statistical analysis, and Luce Smagghe for his technical help (financially supported by FILO). This study was supported by grants from INCA-DGOS-Inserm_12560 (SiRIC CURAMUS is financially supported by the French National Cancer Institute, the French Ministry of Solidarity and Health and Inserm with financial support from ITMO Cancer AVIESAN), by prix Force Hémato 2020 and by the Fondation ARC pour la recherche sur le cancer (to DRW). Luce Smagghe and Jonathan Caron were financially supported by SiRIC CURAMUS.

Due to the nature of this research, participants of this study did not agree for their data to be shared publicly, so supporting data is not available.


Yoann Athiel

Justine Nasone

Lousineh Arakelian

Lionel Faivre

Anaïs Dugas

Jean-Marie Jouannic

Jérôme Larghero

Lucie Guilbaud

Biodistribution of allogenic umbilical cord-derived mesenchymal stromal cells after fetal repair of myelomeningocele in an ovine model

Keywords: Mesenchymal stromal cells, Myelomeningocele, Fetal surgery, Biodistribution, Tracking, Ovine model

Background: Myelomeningocele (MMC) is a spinal cord congenital defect that leads to paraplegia, sphincter disorders and potential neurocognitive disabilities. Prenatal surgery of MMC provides a significant benefit compared to surgery at birth. Mesenchymal stromal cell (MSC) therapy as an adjuvant treatment for prenatal surgery showed promising results in animal experiments which could be considered for clinical use in human fetuses. Despite numerous reassuring studies on the safety of MSCs administration in humans, no study focused on MSCs biodistribution after a local MSCs graft on the fetal spinal cord.

Aim:

The purpose of our study was to assess the biodistribution of umbilical cord-derived mesenchymal stromal cells (UC-MSCs) at birth in lambs who had a prenatal myelomeningocele repair using a fibrin patch seeded with allogenic UC-MSCs.

Methods:

After isolation, UC-MSCs were tagged using a green fluorescent protein (GFP)-containing lentiviral vector. MMC defects were surgically created at 75 days of gestation and repaired 15 days later using UC-MSCs patch. Lambs were delivered at 142 days and sacrificed. DNA extraction was performed among biopsies of the different organs and q-PCR analysis was used to detect the expression of GFP (GFP DNA coding sequence).

Results: In our 6 surviving lambs grafted with UC-MSCs, GFP lentivirus genomic DNA was not detected in the organs.

Conclusion:

These reassuring data will support translational application in humans, especially since the first human clinical trial using mesenchymal stromal cells for in-utero treatment of MMC started recently in U.S.A.

Introduction

Myelomeningocele (MMC) is a spinal cord congenital defect which leads to paraplegia, sphincter disorders and cognitive disabilities. Prenatal repair surgery of MMC improves motor function and neurological outcomes compared to postnatal repair [START_REF] Adzick | A randomized trial of prenatal versus postnatal repair of myelomeningocele[END_REF][START_REF] Guilbaud | Open fetal surgery for myelomeningocele repair in France[END_REF]. However, this benefit remains limited since 71% of children are not able to walk independently at an average age of 7.8 years [START_REF] Houtrow | Prenatal repair of myelomeningocele and school-age functional outcomes[END_REF][START_REF] Farmer | The management of myelomeningocele study: full cohort 30-month pediatric outcomes[END_REF]. Several international studies investigated the use of stem cells as an adjuvant therapy of MMC prenatal surgery [START_REF] Dugas | Cell therapy for prenatal repair of myelomeningocele: a systematic review[END_REF]. Thus, very promising results using human placental-derived mesenchymal stromal cells grafted on the fetal spinal cord during surgery in the MMC ovine model

Open Access

*Correspondence: lucie.guilbaud@aphp.fr have been reported [START_REF] Kabagambe | Placental mesenchymal stromal cells seeded on clinical grade extracellular matrix improve ambulation in ovine myelomeningocele[END_REF][START_REF] Wang | Placental mesenchymal stromal cells rescue ambulation in ovine myelomeningocele[END_REF][START_REF] Galganski | In utero treatment of myelomeningocele with placental mesenchymal stromal cells-selection of an optimal cell line in preparation for clinical trials[END_REF][START_REF] Vanover | High density placental mesenchymal stromal cells provide neuronal preservation and improve motor function following in utero treatment of ovine myelomeningocele[END_REF]. Our group experimented the use of allogenic umbilical cord-derived mesenchymal stromal cells (UC-MSCs) in the same MMC ovine model and showed similar findings [START_REF] Guilbaud | In utero treatment of myelomeningocele with allogeneic umbilical cord mesenchymal stromal cells in an ovine model[END_REF]. However, demonstrating the safety of MSCs graft in fetuses is essential before considering clinical application in humans. For this reason, the European Medicines Agency recommends the study of the cells kinetics, migration and persistence [START_REF]Guideline on human cell-based medicinal products[END_REF]. Despite numerous reassuring studies on the safety of MSCs administration in humans [START_REF] Xiao | Significant Improvement of acute complete spinal cord injury patients diagnosed by a combined criteria implanted with neuroregen scaffolds and mesenchymal stem cells[END_REF][START_REF] Deng | Collagen scaffold combined with human umbilical cord-mesenchymal stem cells transplantation for acute complete spinal cord injury[END_REF][START_REF] Lalu | Safety of cell therapy with mesenchymal stromal cells (SafeCell): a systematic review and meta-analysis of clinical trials[END_REF], the study of their kinetics and migration after a fetal spinal cord grafting has never been reported so far.

The objective of our study was to study the biodistribution of UC-MSCs at birth after fetal MMC surgical repair using an allogenic UC-MSCs patch in an ovine model.

Materials and methods

Ethics statement

This study protocol was approved by the French national committee on animal research (APAFIS#2845-2015100520053611v10) and all animals received care in strict compliance with institutional guidelines, and guidelines for the provision of standard care to laboratory animals. The study was carried out in compliance with the ARRIVE guidelines.

Ovine MSCs production

UC-MSCs were collected from lambs delivered by cesarean section at 139 days of gestation and isolated using the explant method as previously described [START_REF] Guilbaud | In utero treatment of myelomeningocele with allogeneic umbilical cord mesenchymal stromal cells in an ovine model[END_REF].

GFP transduction

UC-MSCs were transduced using a green fluorescent protein (GFP)-containing lentiviral vector (Dharmac-onTM GIPZTM Lentiviral shRNA, Horizon Discovery, Lafayette, USA) as previously described [START_REF] Guilbaud | In utero treatment of myelomeningocele with allogeneic umbilical cord mesenchymal stromal cells in an ovine model[END_REF]. Three days after transduction, we confirmed the presence of at least 90% GFP-tagged cells by flow cytometry analysis. Cell selection was performed using puromycin for about 10 days. GFP-tagged UC-MSCs were then cultured to obtain the required number of cells in complete medium Minimum Essential Medium alpha (MEM α) Glu-taMAX ™ supplement, no nucleosides (Gibco, Grand Island, USA), supplemented with 10% of Fetal Bovine Serum (FBS) (Cytiva, South Logan, USA) and 1% penicillin/streptomycin (Dutscher, Bernolsheim, France).

Characterization of ovine UC-MSC

Characterization of UC-MSCs was previously described [START_REF] Guilbaud | In utero treatment of myelomeningocele with allogeneic umbilical cord mesenchymal stromal cells in an ovine model[END_REF]. Briefly, it included (1) a growth analysis until passage 5 to assess the doubling time for each passage, (2) flow cytometry assay to identify the typical MSCs membrane antigen expression (CD29, CD73, CD90, CD105, CD45) and antigen expression of ovine MSCs isolated from bone marrow (CD31, CD44, CD166), and (3) analysis of multipotency to evaluate the adipogenic and osteogenic differentiations.

Preparation of the UC-MSCs fibrin patch

Sixteen million GFP-tagged UC-MSCs were seeded into a fibrin patch containing fibrinogen (20 mg/ml) and thrombin (4 NIH units/ml) from the EVICEL ® kit (Ethicon, NJ, USA) as previously described [START_REF] Guilbaud | In utero treatment of myelomeningocele with allogeneic umbilical cord mesenchymal stromal cells in an ovine model[END_REF].

MMC defect creation and repair in the ovine model

The MMC creation and repair were performed under general anesthesia as previously described [START_REF] Guilbaud | In utero treatment of myelomeningocele with allogeneic umbilical cord mesenchymal stromal cells in an ovine model[END_REF]. Briefly, the MMC defect creation was performed at 75 days of gestation after laparotomy and hysterotomy. A laminectomy from L1 to L5 was performed before a removal of the dura-mater at the same level. The MMC defect was repaired at 90 days of gestation. The UC-MSCs patch was placed on the spinal cord and the skin was closed over the patch, using a Vicryl 2-0 running suture. Lambs were delivered by cesarean section at 142 days of gestational age, that is 52 days after UC-MSCs graft. Lambs' clinical evaluation was performed at 2 h of life and the animals were sacrificed for histopathological and immunohistochemical analysis. Macroscopic examinations of brain, lungs, live, spleen and intestines were performed by cross sections of the formaldehyde fixed organ.

Lambs tissue samples

A 1 cm by 1 cm biopsy was taken from the following organs: heart, liver, kidney, intestines, spleen, lungs, bone marrow, umbilical cord and placental cotyledons. All samples were immediately stored at -80 °C.

DNA extraction

Tissue samples were prepared by cryogenic grinding method using liquid nitrogen, then stored at -20 °C. Genomic DNA was extracted using PureLink ® Genomic DNA Kit (Invitrogen) according to manufacturer's instructions with an elution volume of 100 μL. DNA was stored at -20 °C before PCR analysis. DNA isolated from GFP-tagged UC-MSCs and non-tagged UC-MSCs served as positive and negative controls, respectively.

Quality and quantity of extracted DNA were estimated by spectrophotometry at 260 and 280 nm, respectively (A 260 /A 280 ) using Nanodrop Lite ® Spectrophotometer (Thermo Fischer Scientific).

Real-time quantitative PCR analysis

RT-qPCR was performed using QuantStudio 7 Flex Real-Time PCR System (Thermo Fischer Scientific). Presence of GFP in extracted DNA was evaluated using the GFP Applied Biosystems ™ TaqMan ™ Gene Expression Assay (Thermo Fisher Scientific) (Assay ID: Mr03989638_mr). To ensure the accuracy of the qPCR detection, presence of an ovine reference gene, ras homolog family member B (RHOB) (Assay ID: Oa04654852_s1) was also checked. RT-qPCR was performed in triplicate in 25-µl reactions containing 2.5 μL genomic DNA (corresponding to 10 ng of genomic DNA), 12.5 μL PCR Master Mix, 8.75 μL RNase-Free Distilled Water and 1.25 μL of the Applied Biosystems ™ TaqMan ™ Gene Expression Assay (Thermo Fisher Scientific). DNA amplification was performed according to the manufacturer's instructions: an initial activation and denaturation step of 20 s at 95 °C followed by 45 cycles consisting of 3 s at 95 °C and 30 s at 60 °C.

Dilution analysis

Lower limit of quantitation was determined using a dilution method. DNA coming from GFP-tagged UC-MSCs (DNA GFP+) was diluted in DNA coming from nontagged UC-MSCs (DNA GFP-) to reproduce in vivo dilution of UC-MSCs in the different organs. We defined the threshold for detection at the dilution for which the GFP + DNA was not detected.

DNA extraction from cellular patch

To ensure that UC-MSCs seeded in the fibrin patch kept their DNA GFP+, we performed QT-PCR analysis among DNA extracted directly from the cellular patch which contained sixteen million of GFP-tagged UC-MSCs.

Results

Characterization of ovine UC-MSC

Flow cytometry analysis was performed and confirmed typical MSCs antigen expression according to the International Society for Cellular Therapy definition (CD29, CD73, CD90, CD105, CD45) and antigens expression usually analyzed in the study of ovine MSCs (CD31, CD44, CD166) [START_REF] Adamzyk | Different culture media affect proliferation, surface epitope expression, and differentiation of ovine MSC[END_REF]. Analysis of MSCs multipotency was performed by confirmation of adipogenic and osteogenic differentiations of isolated cells, as previously described [START_REF] Guilbaud | In utero treatment of myelomeningocele with allogeneic umbilical cord mesenchymal stromal cells in an ovine model[END_REF].

Efficacy

Results of previous experimentations, demonstrating the benefit of using MSCs as adjunctive treatment for MMC fetal surgery, were already reported in the original publication [START_REF] Guilbaud | In utero treatment of myelomeningocele with allogeneic umbilical cord mesenchymal stromal cells in an ovine model[END_REF]. The fetal loss rate was 40%, consistent with the completion of two in utero surgeries during gestation. Fetal demises usually occurred after the second fetal surgery and the condition of the aborted fetuses was not suitable for valuable tissue preservation.

Immunohistochemical analysis

We performed an immunohistochemistry analysis showing the presence of few GFP-tagged cells located in the dermis of lambs at location of the patch (Fig. 1) [START_REF] Guilbaud | In utero treatment of myelomeningocele with allogeneic umbilical cord mesenchymal stromal cells in an ovine model[END_REF].

Macroscopic and microscopic examinations

No tumors were observed on macroscopic and microscopic examinations of the brain, lungs, liver, spleen, and intestines of any of the six lambs [START_REF] Guilbaud | In utero treatment of myelomeningocele with allogeneic umbilical cord mesenchymal stromal cells in an ovine model[END_REF].

DNA extraction

DNA was purified from 2.10 6 GFP-tagged UC-MSCs. We obtained 24,600 ng of DNA in 100 µL elution volume (246 ng/ µL). DNA was obtained from 44 samples from 6 lambs. The amount of total DNA isolated vary from 1,500 to 34,230 ng according to the sample. The DNA purityrepresented by the 260/280 ratio-varies from 1.62 to 2.0.

Validation of primers and probes

Among DNA extracted from GFP-tagged UC-MSCs, the qPCR detection was positive for GFP and RHOB probes. Among the DNA extracted from non-tagged UC-MSCs, the qPCR detection was positive for the RHOB probe but not for the GFP probe.

Dilution analysis

Mean CT is reported in Table 1. We confirmed that the mean CT of RHOB expression was stable as the dilution proceeds but detection of GFP expression decreases gradually and finally was not detected for the dilution 1/1.10 6 . Lower limit of quantitation is 1/10 5 .

DNA extraction from cellular patch

Among DNA extracted directly from the cellular patch containing GFP-tagged UC-MSCs, detection of GFP was positive.

Q-PCR analyses

Genomic DNA GFP + was not detected in any of the 44 samples but genomic RHOB DNA was detected in all the cases. Mean CT of QPCR analysis is reported in Table 2. All the mean CT of RHOB were in the threshold of sensibility.

Discussion

In the nine organs screened in each of the six lambs grafted with allogenic UC-MSCs, no GFP lentivirus genomic DNA was detected at 52 days of the graft. These results support the safety of UC-MSCs use as an adjuvant therapy in MMC fetal surgery.

Risk of MSCs

UC-MSCs were already used as therapeutic option in various diseases in humans [START_REF] Mebarki | Human umbilical cord-derived mesenchymal stem/stromal cells: a promising candidate for the development of advanced therapy medicinal products[END_REF]. Historically, stem cell therapy has been associated with an increased risk of genetic instabilities and transformation process after long-term culture [START_REF] Barkholt | Risk of tumorigenicity in mesenchymal stromal cell-based therapies-bridging scientific observations and regulatory viewpoints[END_REF]. However, no tumor formation has been reported in our previous experimentations and in clinical studies that focused on in vivo tumorigenesis after UC-MSCs transplantation. Recently, a large metaanalysis of 55 randomized studies with 2696 patients confirmed the safety of administrated MSCs. Especially, on longer term, there was no significant increase risk of malignancy for the MSC as compared to control groups [START_REF] Thompson | Cell therapy with intravascular administration of mesenchymal stromal cells continues to appear safe: an updated systematic review and meta-analysis[END_REF]. Focusing on our way of administration, two studies reported administration of UC-MSCs directly into the spinal cord in humans after traumatic spinal cord injury. Data concerning the safety did not show any adverse event after transplantation [START_REF] Xiao | Significant Improvement of acute complete spinal cord injury patients diagnosed by a combined criteria implanted with neuroregen scaffolds and mesenchymal stem cells[END_REF][START_REF] Deng | Collagen scaffold combined with human umbilical cord-mesenchymal stem cells transplantation for acute complete spinal cord injury[END_REF]. Although UC-MSCs are not very immunogenic and the fetus has an immature immunity, the allogeneic nature of the cells may explain why they do not persist in the host (immune rejection).

MSCs in prenatal myelomeningocele therapy

Different ways of MSCs administration have been studied in prenatal myelomeningocele therapy [START_REF] Dugas | Cell therapy for prenatal repair of myelomeningocele: a systematic review[END_REF]. Fauza's group experimented intra-amniotic injections of MSCs from amniotic fluid in a retinoic acid murine model of MMC [START_REF] Shieh | Transamniotic stem cell therapy (TRASCET) in a rabbit model of spina bifida[END_REF]. MSCs, previously labelled with a luciferase gene, were found after birth in umbilical cord, placenta, spleen and brain by luminometric analysis [START_REF] Shieh | Donor mesenchymal stem cell linetics after transamniotic stem cell therapy (TRASCET) for experimental spina bifida[END_REF]. Although this technique is less invasive, these results could preclude application in humans.

Farmer's team experimented human placental-derived MSCs, seeded in an extracellular scaffold, in the same surgical ovine model of MMC presented here. Through several published studies, their results suggested the benefit of MSCs in the motor function improvement [START_REF] Kabagambe | Placental mesenchymal stromal cells seeded on clinical grade extracellular matrix improve ambulation in ovine myelomeningocele[END_REF][START_REF] Wang | Placental mesenchymal stromal cells rescue ambulation in ovine myelomeningocele[END_REF][START_REF] Galganski | In utero treatment of myelomeningocele with placental mesenchymal stromal cells-selection of an optimal cell line in preparation for clinical trials[END_REF][START_REF] Vanover | High density placental mesenchymal stromal cells provide neuronal preservation and improve motor function following in utero treatment of ovine myelomeningocele[END_REF]. Despite their use of MSCs transduced with GFP, no cell tracking was reported in their studies. Recently, they conducted safety evaluation of their human placentalderived MSCs in a murine model. They implanted the heterologous cells seeded on extracellular matrix into subcutaneous murine pocket. No tumor was found and MSCs did not seem to persist at the implantation site or at distance at 4 weeks and 6 months after grafting [START_REF] Jackson | Placental mesenchymal stromal cells: preclinical safety evaluation for fetal myelomeningocele repair[END_REF]. This slightly differs from our findings as we observed a survival of the MSCs 52 days after grafting [START_REF] Guilbaud | In utero treatment of myelomeningocele with allogeneic umbilical cord mesenchymal stromal cells in an ovine model[END_REF].

Farmer's promising results suggested the benefit of MSCs on the motor function improvement and they announced the first human clinical trial using mesenchymal stromal cells for in-utero treatment of MMC.

Strengths and limitations

To our knowledge, this is the first biodistribution study of MSCs after local administration to the fetal spinal cord in ovine model. UC-MSCs patches were directly applied into the spinal cord in an experimental ovine model of myelomeningocele.

Several techniques exist to track UC-MSCs in transplantation experiments: luminometry, immunohistochemistry, imaging or nucleic acid amplification testing (qPCR) [START_REF] Shieh | Donor mesenchymal stem cell linetics after transamniotic stem cell therapy (TRASCET) for experimental spina bifida[END_REF][START_REF] Chalphin | Transamniotic stem cell therapy for experimental congenital diaphragmatic hernia: structural, transcriptional, and cell kinetics analyses in the nitrofen model[END_REF][START_REF] Han | Define mesenchymal stem cell from its fate: Biodisposition of human mesenchymal stem cells in normal and Con-A induced liver injury mice[END_REF]. PCR analysis which is based on the amplification of the DNA is a method of choice due to its high sensitivity [START_REF]Guideline on the quality, non-clinical and clinical aspects of gene therapy medicinal products[END_REF]. Our experiment of dilution shows that GFP would be detected at very low concentrations (lower limit of quantitation = 1/10 5 ), below which the presence of migrating cells would probably not have negative consequences.

We recognize some limitations to our study. Lambs were sacrificed shortly after birth (2 h) which did not allow for long-term studies. Furthermore, only one biopsy of the different organ was collected so they are not analyzed in their entirety. Finally, in previous experimentations, we performed an immunohistochemistry analysis showing the presence of few GFP-tagged cells located in the dermis of lambs at location of the patch. Unfortunately, this sample was not available to perform DNA extraction and GFP screening as a positive control of the RT-qPCR analysis. This should prompt systematic dermal biopsies in further studies.

Conclusion

This biodistribution study of grafted UC-MSCs was essential before considering a clinical application in humans, especially in the context of a fetal administration. Within the limits of our experimentations, we have shown that UC-MSCs, administered in a fibrin patch applied to the MMC defect, do not appear to disseminate to distant organs. This study is consistent with the possibility of use in humans.

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Fig. 1

 1 Fig. 1 Immunohistochemical analysis of the spinal cord of a lamb who received UC-MSC patch. Few GFP-tagged were revealed in red by a primary anti-Turbo GRP antibody

Table 1 Q

 1 -PCR experiment among DNA from GFP-tagged UC-MSCs diluted in DNA from non GFP-tagged UC-MSCs CT, cycle threshold; SD, standard deviation; ND, not detected; GFP, green fluorescent protein, RHOB, Ras homolog gene family

	Dilution		Concentration of DNA GFP + in DNA GFP-(ng/µL)	GFP applied biosystems ™ TaqMan ™ gene expression assay	RHOB applied biosystems ™ TaqMan ™ gene expression
						assay
				Mean CT ± SD	CV (%)	Mean CT ± SD	CV (%)
	1 2 3 4 5 6 7 8 9	1 1/2 1/5 1/10 1/100 1/1000 1/10000 1/100000 1/1000000	4 2 0.8 0.4 0.04 0.004 0.0004 0.00004 0.000004	26.5 ± 0.3 27.9 ± 0.1 30.8 ± 0.3 31.3 ± 1.4 33.9 ± 0.6 35.9 ± 0.8 36.2 ± 0.5 36.5 ± 0.9 ND	1.3 0.4 1.0 4.4 1.7 2.2 1.4 2.5 ND	27.6 ± 0.07 27.5 ± 0.06 27.4 ± 0.08 27.2 ± 0.05 27.1 ± 0.09 27.2 ± 0.06 27.1 ± 0.05 26.9 ± 0.01 27.0 + /0.04	0.2 0.2 0.3 0.2 0.3 0.2 0.2 0.03 0.1

Table 2

 2 QPCR analysis NC, not collected; ND, not detected

	Organ	Lamb 1		Lamb 2		Lamb 3		Lamb 4		Lamb 5		Lamb 6
		GFP	RHOB	GFP	RHOB	GFP	RHOB	GFP	RHOB	GFP	RHOB	GFP	RHOB
	Heart	ND	28.6	ND	29.4	ND	24.0	ND	28.2	ND	27.6	NC	NC
	Lung	ND	30.0	ND	28.7	ND	27.1	ND	30.2	ND	28.9	ND	28.2
	Liver	ND	27.9	ND	28.1	ND	27.8	ND	28.4	ND	29.1	ND	29.2
	Spleen	ND	27.8	ND	26.7	ND	31.6	ND	28.9	ND	27.5	ND	27.5
	Kidney	NC	NC	ND	26.1	ND	24.3	ND	26.8	ND	31.2	ND	25.4
	Bowel	ND	31.1	ND	26.9	ND	29.4	ND	26.2	ND	29.9	NC	NC
	Bone marrow	NC	NC	NC	NC	NC	NC	NC	NC	ND	27.4	ND	27.0
	Umbilical cord	ND	28.3	ND	27.5	NC	NC	ND	31.3	ND	31.1	ND	33.8
	Placenta	ND	26.1	ND	28.2	NC	NC	ND	28.3	ND	29.7	NC	NC

Department of Fetal Medicine, APHP, Trousseau Hospital, DMU ORIGYNE, Sorbonne University, Paris, France Full list of author information is available at the end of the article

Acknowledgements

We thank the "Fer à Moulin" School of Surgery of Paris.

Funding

This work was supported by a funding from the Fondation de l' Avenir pour la Recherche Médicale (Grant #AP-RMA-18-028).

Availability of data and materials

All data generated or analyzed during this study are included in this published article.

Abbreviations

Declarations Ethics approval and consent to participate

This study protocol was approved by the French national committee on animal research (APAFIS#2845-2015100520053611v10).

Consent for publication

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Motivation of the work

     Ṡ = -βSI İ = βSI -γI Ṙ = γI      Ṡ = -β(1 -u)SI İ = β(1 -u)SI -γI Ṙ = γI with ∞ 0 u(τ ) dτ ≤ Q How to reduce at best max i I(t) ? Optimal control with L ∞ criterion ẋ = f (x , u) → inf u(•) max t∈[0,T ] h(x (t))
is not in the classical form of a Mayer or Bolza problem... Characterization of the value function (Barron, Ishii, Jensen, Liu, Gonzalez, Aragone...) in terms of a variational inequality :

min ∂ t V + inf u ∂ x V .f (x , u) , V -h(x ) = 0 . Approximation of the L ∞ norm by the L p norm. (D) : ẋ = f (x , y , u) ẏ = g(x , y , u) with x ∈ R n y ∈ R u ∈ U ⊂ R p
Assumptions. U is compact ; f and g are C 1 with linear growth.

P : inf u(•) max t∈[0,T ] y (t)
Question. Is it possible to write an equivalent problem in Mayer or Bolza form (and use numerical tools such as Bocop) ?

Remark : ẋ = f (x , u) y = h(x ) ⇒ ẋ = f (x , u) ẏ = g(x , u) := ∇h(x ) T • f (x , u)
A first formulation with state constraint

     ẋ = f (x , y , u) ẏ = g(x , y , u) ż = 0
The problem

P 0 : inf u(•) z(T )
under the state constraint

z(t) ≥ y (t), t ∈ [0, T ]
and the two boundaries condition

z(T ) = z(0)
is equivalent to P.

Remark. Here z(0) is free.

Another formulation as a Cauchy problem

(D a ) :      ẋ = f (x , y , u) ẏ = g(x , y , u) ż = max(g(x , y , u), 0)(1 -v ) v ∈ [0, 1]
Proposition. With z 0 = y 0 , the problem

P 1 : inf u(•),v (•) z(T )
under the state constraint

C : z(t) ≥ y (t), t ∈ [0, T ]
admits an optimal solution, and is equivalent to problem P.

I := t ∈ (0, T ); y (t ) > y (t) for some t < t .
Sun Rising Lemma :

I ∅ ⇒ int I = n (a n , b n ) with i) y (a n ) = y (b n ) if b n T , ii) if b n = T , then y (a n ) ≥ y (b n ). Then, the control v (t) = 1, t ∈ int I 0, t int I is optimal
A formulation with mixed constraint

(D a ) :      ẋ = f (x , y , u) ẏ = g(x , y , u) ż = max(g(x , y , u), 0)(1 -v ) v ∈ [0, 1]
Proposition. For z 0 = y 0 , the problem

P 2 : inf u(•),v (•) z(T )
under the mixed constraint

C m : max(y (t) -z(t), 0)(1 -v (t)) + z(t) -y (t) ≥ 0, t ∈ [0, T ]
is equivalent to P.

Formulation without constraint

Consider Z = (x , y , z) , Ż ∈ F (Z ) := (u,v )∈U×[0,1]    f (x , y , u) g(x , y , u) h(x , y , z, u, v )    with h(x , y , u, v ) = max(g(x , y , u), 0)(1 -v 1 R + (z -y )) Let S := {Z (•) a.c. ; Ż (t) ∈ F (Z (t)) a.e. and Z (0) = (x 0 , y 0 , y 0 )} Remark. F is only upper semi-continuous. Proposition. Assume G(x , y ) := u∈U f (x , y , u) g(x , y , u)
is convex. Then

P 3 : inf Z (•)∈S z(T )
admits a solution Z (•) and (x (•), y (•)) is optimal for problem P.

1. Consider the augmented dynamics :

Ż ∈ F † (Z ) := (u,v ,α)∈U×[0,1] 2    f (x , y , u) g(x , y , u) h † (x , y , z, u, v , α)    h † (x , y , z, u, v , α) = (1 -α)h(x , y , z, u, v ) + α max w ∈U h(x , y , z, w , 0) ⇒ S † compact ⇒ ∃(x (•), x (•), z (•)) optimal. 2. Any Z (•) sol. of D a with C m belongs to S ⊂ S † ⇒ z (T ) ≤ inf{z(T ); (x (•), y (•), z(•)) sol. of D a with C m }. Sketch of proof 3. If E := {t ∈ (0, T ); z(t) < y (t)} ∅, then one has ż(t) -ẏ (t) ≥ 0 for a.e. t ∈ E ⇒ z(0) -y (0) < 0 : contradiction. ⇒ z(T ) ≥ max t∈[0,T ] y (t) 4. Filipov's Lemma applied to G : ⇒ z (T ) ≥ max t∈[0,T ] y (t) ≥ inf u∈U max t∈[0,T ] y (t); (x (•), y (•)) sol. of D where (x (•), y (•)) is solution of D for a certain u (•).

Approximation from below

(D θ a ) :      ẋ = f (x , y , u) ẏ = g(x , y , u) ż = h θ (x , y , z, u, v ) with h θ (x , y , z, u, v ) = max(g(x , y , u), 0) 1 -v e -θ max(y -z,0)
Proposition. For any increasing sequence θ n → +∞, the problem

P θn 3 : inf Z (•)∈S θn z(T ) admits an optimal solution (x n (•), y n (•), z n )(•)). (x n (•), y n (•)) converges (
λ 2 = log ε log(2) log(1-) -1 ε 2 θ = - log(1 -ε) log (

Conclusions

We

I

  up to a sub-sequence) uniformly to an optimal solution (x (•), y (•)) of problem P, and ( ẋ (•), ẏ (•)) converges weakly to ( ẋ (•), ẏ (•)) in L 2 .Moreover, z n (T ) is an increasing sequence that converges to max t∈[0,T ] y (t). β = 0.21, γ = 0.07, Q = 28, I(0) = 10 6 , 0) log (exp(λ 1 ẏ ) + 1) /λ 1 exp(-θ max(yz, 0)) exp (-θ log(exp(λ 2 (yz)) + 1)/λ 2 )

  have proposed equivalent formulations that allow to use existing numerical methods, but...

with state constraint : not qualified... without state constraint : discontinuous differential inclusion... one cannot use (available) Maximum Principles... Open problem : How to derive necessary optimality conditions ?

is optimal for problem P.

An example

Optimal solution for T = 5 : The SIR problem

The SIR problem

Posit R 0 = β/γ and S h = R -1 0 . Proposition. 1 The "0-singular-0" feedback control