Sébastien Roca

Laurent Leclercq
email: laurent.leclercq@umontpellier.fr

Philippe Gonzalez

Laura Dhellemmes

Laurent Boiteau

Gaulthier Rydzek

Hervé Cottet
email: herve.cottet@umontpellier.fr

Modifying last layer in polyelectrolyte multilayer coatings for capillary electrophoresis of proteins

Keywords: protein adsorption, capillary zone electrophoresis, successive multiple ionicpolymer layer, PEGylated polycation, polyelectrolyte multilayers Highlights

Protein adsorption on the inner wall of the fused silica capillary wall is an important concern for capillary electrophoresis (CE) analysis since it is mainly responsible for separation efficiency reduction. Successive Multiple Ionic-polymer Layers (SMIL) are used as capillary coatings to limit protein adsorption, but even low residual adsorption strongly impacts the separation efficiency, especially at high separation voltages. In this work, the influence of the chemical nature and the PEGylation of the polyelectrolyte deposited in the last layer of the SMIL coating was investigated on the separation performances of a mixture of four model intact proteins (myoglobin (Myo), trypsin inhibitor (TI), ribonuclease a (RNAse A) and lysozyme (Lyz)).

Poly(allylamine hydrochloride) (PAH), polyethyleneimine (PEI), ε-poly(L-lysine) (εPLL) and αpoly(L-lysine) (αPLL) were compared before and after chemical modification with polyethyleneglycol (PEG) of different chain lengths. The experimental results obtained by performing electrophoretic separations at different separation voltages allowed determining the residual retention factor of the proteins onto the capillary wall via the determination of the plate height at different solute velocities and demonstrated a strong impact of the polycationic last layer on the electroosmotic mobility, the separation efficiency and the overall resolution. Properties of SMIL coatings were also characterized by quartz microbalance and atomic force microscopy, demonstrating a glassy structure of the films.

Introduction

Theoretical separation efficiency in capillary electrophoresis (CE) is ideally controlled by solute axial diffusion [START_REF] Jorgenson | Zone electrophoresis in open-tubular glass capillaries[END_REF]. For a globular protein of typical size about 1 nm (molecular diffusion coefficient D of about 2 × 10 -10 m 2 s -1 , apparent electrophoretic mobility of 20 TU (Tiselius Units:

1 TU = 10 -9 m 2 V -1 s -1 ) and typical electric field strength of 500 V cm -1 corresponding to "typical" 20 kV per 40 cm total capillary length used in this study), about one million theoretical plates are expected in an ideal CE experiment. In practice, the plate number is generally much lower, with typically about a few tens of thousands plates [START_REF] Pattky | Advantages and limitations of a new cationic coating inducing a slow electroosmotic flow for CE-MS peptide analysis: a comparative study with commercial coatings[END_REF][START_REF] Leclercq | Modulation of the electroosmotic mobility using polyelectrolyte multilayer coatings for protein analysis by capillary electrophoresis[END_REF][START_REF] Roca | Polyelectrolyte multilayers in capillary electrophoresis[END_REF]. Different factors can explain this reduction, among which radial gradient temperature inside the capillary [START_REF] Xuan | Band-broadening in capillary zone electrophoresis with axial temperature gradients[END_REF] due to Joule effect and capillary cooling, electromigration dispersion [START_REF] Beckers | UV detection in capillary zone electrophoresis peaks or dips -that is the question[END_REF][START_REF] Malá | New methodology for capillary electrophoresis with ESI-MS detection: Electrophoretic focusing on inverse electromigration dispersion gradient. High-sensitivity analysis of sulfonamides in waters[END_REF], capillary coiling [START_REF] Kašička | Contribution of capillary coiling to zone dispersion in capillary zone electrophoresis[END_REF], or protein adsorption onto the capillary wall [START_REF] Leclercq | Modulation of the electroosmotic mobility using polyelectrolyte multilayer coatings for protein analysis by capillary electrophoresis[END_REF][START_REF] Khaledi | High-performance capillary electrophoresis theory, techniques, and applications[END_REF][START_REF]Fluid mechanics of electroosmotic flow and its effect on band broadening in capillary electrophoresis[END_REF][START_REF]Electrokinetic flow and dispersion in capillary electrophoresis[END_REF][START_REF] Gaš | Peak broadening in capillary zone electrophoresis[END_REF][START_REF] Schure | Consequences of wall adsorption in capillary electrophoresis: theory and simulation[END_REF]. Monolayer coatings and successive multiple ionic-polymer layer (SMIL) capillary coatings were developed to limit protein adsorption [START_REF] Hajba | Continuous-flow-based microfluidic systems for therapeutic monoclonal antibody production and organ-on-a-chip drug testing[END_REF][START_REF] Dolník | Capillary electrophoresis of proteins 2005-2007[END_REF][START_REF] Righetti | The state of the art of dynamic coatings[END_REF][START_REF] Lucy | Non-covalent capillary coatings for protein separations in capillary electrophoresis[END_REF][START_REF] Mcmillan | Emulsion technologies for multicellular tumour spheroid radiation assays[END_REF][START_REF] Štěpánová | Recent applications of capillary electromigration methods to separation and analysis of proteins[END_REF][START_REF] Štěpánová | Applications of capillary electromigration methods for separation and analysis of proteins (2017-mid 2021) -A review[END_REF][START_REF] Renard | Superhydrophobic capillary coatings: Elaboration, characterization and application to electrophoretic separations[END_REF][START_REF] Bekri | Polyelectrolyte multilayer coatings for the separation of proteins by capillary electrophoresis: Influence of polyelectrolyte nature and multilayer crosslinking[END_REF][START_REF] Pallotta | Analytical strategy for studying the formation and stability of multilayered films containing gold nanoparticles[END_REF][START_REF] Villemet | Protein mapping of peanut extract with capillary electrophoresis[END_REF][START_REF] Horká | Sensitive identification of milk protein allergens using on-line combination of transient isotachophoresis/micellar electrokinetic chromatography and capillary isoelectric focusing in fused silica capillary with roughened part[END_REF][START_REF] Michalikova | Middle-up characterization of the monoclonal antibody infliximab by capillary zone electrophoresis-mass spectrometry[END_REF][START_REF] Šolínová | Covalent anionic copolymer coatings with tunable electroosmotic flow for optimization of capillary electrophoretic separations[END_REF][START_REF] Konášová | Covalent cationic copolymer coatings allowing tunable electroosmotic flow for optimization of capillary electrophoretic separations[END_REF]. Pioneered in the late 1960 [START_REF] Iler | Multilayers of colloidal particles[END_REF], multilayered polyelectrolyte coatings were rediscovered at the beginning of the 90s by Decher et al. [START_REF] Decher | Buildup of ultrathin multilayer films by a self-assembly process: II. Consecutive adsorption of anionic and cationic bipolar amphiphiles and polyelectrolytes on charged surfaces[END_REF][START_REF] Decher | Fuzzy Nanoassemblies: Toward layered polymeric multicomposites[END_REF] and firstly applied to CE in 1998 [START_REF] Katayama | Stable cationic capillary coating with successive multiple ionic polymer layers for capillary electrophoresis[END_REF]. SMIL coatings are built inside the capillary by successive adsorption of oppositely charged polyelectrolytes (PE) onto the capillary wall. SMIL coatings greatly improve the separation repeatability / reproducibility compared to bare fused silica capillary [START_REF] Roca | Polyelectrolyte multilayers in capillary electrophoresis[END_REF][START_REF] Bekri | Polyelectrolyte multilayer coatings for the separation of proteins by capillary electrophoresis: Influence of polyelectrolyte nature and multilayer crosslinking[END_REF][START_REF] Nehmé | Use of coated capillaries for the electrophoretic separation of stereoisomers of a growth hormone secretagogue[END_REF][START_REF] Nehmé | Stability of capillaries coated with highly charged polyelectrolyte monolayers and multilayers under various analytical conditions-Application to protein analysis[END_REF][START_REF] Graul | Capillaries modified by polyelectrolyte multilayers for electrophoretic separations[END_REF], but even very small residual protein adsorption can still reduce the separation efficiency [START_REF] Leclercq | Quantification of adsorption and optimization of separation of proteins in capillary electrophoresis[END_REF], with a plate height H increasing with the power 2 of the retention factor k. For a fair comparison of the coating performances between different capillary coatings, and for a valuable quantitative estimation of their performances, it was thus crucial to quantify residual protein adsorption by the determination of the protein retention factor k. We recently demonstrated that this determination can be performed by quantifying protein separation efficiency at different separation voltages. Experimentally, the plate height H obtained for each protein at different separation voltages can be plotted as a function of the protein linear velocity u according to the following law [START_REF] Schure | Consequences of wall adsorption in capillary electrophoresis: theory and simulation[END_REF][START_REF] Golay | Theory of chromatography in open and coated tubular columns with round and rectangular cross-sections, Gas Chromatography[END_REF]:

𝐻 = 2𝐷 𝑢(1 + 𝑘) + 𝑑 ! " 𝑢 𝐷 𝑘 " 16(1 + 𝑘) + 2 3 𝑑 𝑓 2 𝑢 𝐷 𝑠 𝑘 1 + 𝑘 + 𝐴 ( 1
)
where dc is the internal diameter of the capillary, u is the linear velocity of the analyte, df is the film thickness, Ds is the diffusion coefficient in the stationary phase and A is a constant related to extra column contributions. Knowing that 𝑑 ! " ≫ 𝑑 # " , the third term related to the resistance in mass transfer in stationary phase can be neglected in equation [START_REF] Jorgenson | Zone electrophoresis in open-tubular glass capillaries[END_REF] as a first approximation. Curve fitting of the plate height experimental data is an excellent way to determine k value for each protein on a given capillary coating. In equation [START_REF] Jorgenson | Zone electrophoresis in open-tubular glass capillaries[END_REF], D was obtained experimentally by Taylor dispersion analysis. The determination of k values allows a ranking of the capillary coating performances, whatever the electroosmotic mobility values, and thus, whatever the range of protein migration times which can greatly vary with the nature/surface charge of the coating.

Reducing protein adsorption, and thus the retention factor, is probably one of the most critical parameter to improve capillary coatings in CE enabling decreasing the dependency of the peak broadening with increasing separation voltage. The aim of this work was to study the impact of were systematically determined to compare all the tested capillaries.

Experimental section

Chemicals and Materials

Reagents.

Copper sulfate (99.99% purity), sodium ascorbate (98% purity), tris(benzyltriazolylmethyl) amine (THPTA, 95% purity), sodium nitrate (99.0% purity), 4-(2hydroxyethyl)-1-piperazineethane sulfonic acid (HEPES, 99.5% purity) and sodium acetate anhydrous (99% purity) were purchased from Sigma-Aldrich (Saint-Quentin-Fallavier, France).

Sodium hydroxide (98 % purity) and sodium chloride (99.5% purity) were purchased from Fluka (Saint-Quentin-Fallavier, France). Triethylamine (99% purity) was obtained from Acros Organics (Geel, Belgium). Acetic acid (99.99% purity) was supplied by VWR Chemicals (Rosny-sous-Bois, France). Sodium azide (99.5% purity) was purchased from Merck (Darmstadt, Germany).

Dimethylformamide (99.9% purity) was purchased from Carlo Erba Reagents (Val-de-Rueil, (PSS, Mw = 70 kDa) was purchased from Acros Organics (Geel, Belgium). Except for PAH and PEI, all reagents and polymers were used as received without any further purification. All aqueous solutions were prepared using deionized water (18 MΩcm -1 ) delivered by a Synergy UV water purification system (Millipore, Fontenay-sous-Bois, France).

0.1 μm cut-off Durapore membrane filters were purchased from Millipore (Molsheim, France).

Dialysis tubings (cut-off: 1, 3.5, 8 and 12-14 kDa) were purchased from Spectrum Labs (San Francisco, CA, USA). SI1) according to the calculation method described in Figure SI 8.

Proton NMR

Preparative Size Exclusion Chromatography (SEC)

Automated preparative SEC experiments were performed on a General Electric Healthcare Äkta Purifier 100 (Chicago, IL, USA) including a P-900 pump unit and a UV-900 multi wavelength UV-visible detector unit. Separations were carried out on an OHpak SBG Shodex column guard (50 × 6 mm) followed by a 300 × 8 mm SB-805M-HQ Shodex column. The eluent was an aqueous solution of acetic acid (50 mM), sodium acetate (50 mM), sodium chloride (500 mM) and sodium azide (300 mgL -1 ), pH 4.8. Prior to use, the eluent was filtered through a 0.1 μm cut-off Durapore membrane. Polymers samples dissolved at 4 gL -1 in the eluent were injected by 500-µL successive batches and eluted at 1.0 mLmin -1 flow rate. The collected polymer fractions were dialyzed against deionized water, concentrated on a rotatory evaporator, and finally freeze-dried. The resulting powder was stored under inert atmosphere at -20°C.

Prior to use for PEG grafting, commercial PAH and PEI (of polydispersity indexes PDI = 3.5 and 6.0, respectively) were fractionated by preparative SEC, affording samples with final PDI = 1.5 and 1.9, respectively (see Table SI SI1.

Capillary coating procedure

Bare fused silica capillary (purchased from Polymicro Technologies (Phoenix, AZ, USA)) was first activated according to the following procedure: successive flushes (at 96.5 kPa (14.5 psi)) of

NaOH (1 M, 10 min, 1 mL in a 1.5 mL glass vial), pure water (5 min, 1 mL in a 1. and ±20 kV, with a polarity depending on the capillary surface charge (as indicated on the Figure caption). DMF (0.001% m/v in the BGE) was hydrodynamically injected as EOF marker at 2.7 kPa (0.4 psi) for 2 s, followed by a protein mixture (0.25 gL -1 each in the BGE) at 2.7 kPa for 4 s.

RNAse A, Lyz, Myo and TI were kept separately in pure water (at 2 gL -1 ) in the freezer before dilution in the BGE for the preparation of the sample mixture. The temperature of the cartridge was set at 25°C. Detection wavelength was 214 nm.

Single channel Quartz Crystal Microbalance with Dissipation (QCM-D)

QCM-D experiments were performed on a X1 Single Channel device (from AWSensors, Valencia, Spain), using an "in-flow" QCM cell at 23°C, equipped with a 14 mm diameter quartz sensor coated by SiO2 and with a fundamental resonance frequency of 5 MHz. The coating procedure was performed on bare sensors first activated by O2 plasma (5 min, P(O2) = 0.5 mbar, 80W). Normalized frequency shifts (Δfn/n in Hz) and corresponding normalized dissipation shifts (ΔDn/n) for the 3 rd , 5 th , 7 th and 9 th resonance harmonics (n) were first measured in air, then monitored throughout the experiment. Before starting the SMIL buildup, pure water was flushed at 0.05 mLmin ) [START_REF] Reviakine | Hearing what you cannot see and visualizing what you hear: interpreting quartz crystal microbalance data from solvated interfaces[END_REF] are typical of rigid glassy film structures. In the case of thin rigid films, the mass deposited on the crystal can be calculated from the Sauerbrey equation [START_REF] Bruckenstein | Experimental aspects of use of the quartz crystal microbalance in solution[END_REF]:

∆𝑚 = -𝐶 × ∆𝑓 $ /𝑛 (2)
where Δm (in ngcm -2 ) corresponds to the variation of areal mass, C (in ngcm -2 Hz -1 ) is the masssensitivity constant that depends on crystal property, n is the overtone number and Δf is the frequency shift (in Hz). A typical mass sensitivity constant C = -17.7 ngcm -2 Hz -1 value can be used for a 5 MHz quartz crystal resonator at room temperature.

The film buildup kinetics was evaluated by plotting the values of Δf3/3 and ΔD3/3 for each layer after HEPES rinsing, as well as the corresponding calculated areal masses deposited on the crystal according to Saurebrey's model. Determination of the film thickness was performed using the QTM software, taking account of all frequency overtones and approximating the SMIL mass density as ρ = 1 gcm -3 [START_REF] El Haitami | Effect of the supporting electrolyte anion on the thickness of PSS/PAH multilayer films and on their permeability to an electroactive probe[END_REF].

Atomic Force Microscopy (AFM)

Activation and coating of QCM-D sensor were realized as described in Section 2.9. QCM-D sensors were prepared overnight before AFM analysis and kept in water to avoid SMIL retraction.

Atomic force microscope measurements were performed using a Multimode 8 Instrument controlled by Nanoscope 5 electronic software. Peak Force Tapping mode cantilevers (Nanosensors) with scanasyst fluid spring (constant near 0.7 N/m) were used for the imaging at approximately 60 kHz in air and water. Images were analyzed using Gwydion software.

Results and discussion

In this work, a series of cationic polyelectrolytes, namely PAH, PEI, εPLL or αPLL, were grafted with PEG (1kDa or 5 kDa) at different molar proportions. These polycations were selected because they contain primary amine groups that can be derivatized with lateral PEG chains. PAH, PEI, εPLL were derivatized using the NHS / primary amine reactivity, while αPLL were derivatized using a click-chemistry approach. The PEG grafted polycations are named "Polycation-g-(mPEGy)x" where y corresponds to the mPEG length (1kDa or 5 kDa) and x is the calculated PEG/polycation molar ratio. Various PEGylation molar ratios were achieved up to 20%. All details about the synthesis, chemical structure and characterization, are given in the experimental sections 2.2 to 2.6 and in the SI (see Table SI 1 and subsequent sections in Supporting Information).

Non-PEGylated and PEGylated PAH, PEI, εPLL or αPLL were used as a last layer deposited on a 4-layer PDADMAC/PSS SMIL sub-structure for capillary coatings, as depicted in Figure 1. The PDADMAC / PSS system was retained as a suitable sub-structure since it was described as a prime SMIL coating for CE applications [START_REF] Graul | Capillaries modified by polyelectrolyte multilayers for electrophoretic separations[END_REF]. To study the impact of the last polycationic layer on separation performances in CE, a mixture of 4 model intact proteins (Myo, RNAse A, Lyz, TI at 0.25 gL -1 each in 2 M acetic acid, pH 2.2) was selected.

Influence of the polycation nature on the last layer on the H=f(u) curve and resolution

Before than 0.6%) on migration times at -10 kV and -20 kV (see Table 1 for numerical data). The repeatability followed this ranking: PEI > PDADMAC ~ PAH > αPLL ~ εPLL. Lower stability was observed for PLL, and especially for εPLL at -10 kV. To get more information about the performances of the capillary coatings, the H=f(u) plots were systematically investigated by measuring the separation efficiency (plate height) at different separations voltages. Figure 2B shows typical H=f(u) plots in the case of Myo for all the polycations tested as a last layer (see 1): PDADMAC (6.5 × 10 -2 ) ~ PAH (6.9 × 10 -2 ) < PEI (7.3× 10 -2 ) < αPLL (7.5 × 10 -2 ). Regarding εPLL, poor curve fitting was obtained together with much lower repeatabilities on separation efficiency (see larger error bars in Figure 2B) which can be considered as a signature of a lower stability of the coating. This finding can be explained by the low polymer molar mass (ca. 4000 g mol -1 ) of εPLL compared to that of the other polymers (10 to 100 times longer chains) that generated less robust SMIL coatings. The low k value obtained for εPLL (3.8 × 10 -2 ) should be balanced by the much higher A value (6.9 µm). The A value is supposed to incorporate all constant extra-column effects but the contribution to the A value in regard to the coating properties remains unclear.

SMIL coatings induced differences in the electroosmotic mobility and thus migration time.

Therefore, the impact of the last polycationic layer was investigated on peak resolution (Rs) based on the Myo / RNAse A pair (see Figure 2C). Rs depends both on the migration times and the separation efficiency, as shown in equation 3:

𝑅 % = 1 4 × ∆µ &' < µ (') > × √𝑁 ( 3
)
where <µapp> is the average apparent electrophoretic mobility of the two analytes (in TU), Dµep is the difference in effective electrophoretic mobility between the two analytes and N is the average plate number. The highest Rs values were obtained with the PAH polycation followed by PDADMAC or εPLL (see Figure 2C). Despite the lowest separation efficiencies obtained with εPLL, the resolution loss due to higher peak broadening was balanced by the decrease of the electroosmotic flow (µeo(εPLL) = -40.7 TU vs µeo(PDADMAC) = -45.5 TU) which increased the apparent selectivity and migration times. A diminution of the resolution was observed for all the coatings when the applied voltage increased (increasing u) since the experimental data are located in the ascending part of the H=f(u) curve. As a conclusion of this first part, it seems that PAH is the best polycation for the last layer, and therefore the best candidate for subsequent PEGylation.

Investigating SMIL growth by QCM-D

The SMIL film growth was investigated by QCM-D. ). As the Sauerbrey equation is applicable for thin rigid films (see Section 2.9, eq. 2), an approximated deposited areal mass was calculated for the (PDADMAC/PSS)2 SMIL coating: a frequency shift of ≈ -26.98 Hz was found, leading to 478 ngcm -2 deposited areal mass (corresponding to a film thickness of 4.5 nm), as presented in Table 2.

Next, the influence of the last deposited cationic layer on the film growth was investigated. εPLL deposition showed the smallest final frequency shift and final dissipation, corresponding to 98 ngcm -2 deposited areal mass. In the case of PDADMAC, the deposited areal mass was higher (121 ngcm -2 ). In contrast, the deposition of PAH as a last layer led to harmonic differentiation and a concomitant increase in dissipation values (Figure SI 33), indicating a change in the viscoelastic properties of the film, attributed to a more hydrated structure. Yet, the ratio of final Δf3/3 divided by final ΔD3/3 was around 1.5 × 10 -7 , indicating that the Sauerbrey equation was still valid for this film, with a calculated deposited areal mass of 301 ngcm -2 . For all systems, and with the film volumic mass (ρ) approximated to 1 gcm -3 , the thickness was determined using QTM software as about 7 nm (see exact values in Table 2).

AFM analyses were performed on bare and (PDADMAC/PSS)2.5 SMIL coated QCM-D sensor in air and water medium to characterize SMIL covering. As shown in Figure SI 37, no significant impact was observed in the case of bare sensor when analysis was performed either in air (roughness of 0.9 nm) or in water (roughness of 1.1 nm). Comparing with the SMIL coated sensor in water (Figure SI 38), an increase of 2.3 nm in rugosity was observed, related to the presence of the SMIL at the surface. Finally, AFM experiment was performed on the same QCM-D sensor, after complete evaporation of water (Figure SI 39). Bare silica (dark area) was observed on the left of the picture, while the dried SMIL coating remained on the right. The in-between light zone corresponds to SMIL retractation. To measure SMIL height, topographic measurement was performed on the bare silica zone and on the SMIL coating zone, leading to average 8.7 ± 3.8 nm and 7.2 ± 5.0 nm differences (see Table SI2), in agreement with the value obtained by QCM-D experiment (6.8 nm).

To conclude, with average film thickness of around 7 nm and final dissipation below 7 × 10 -6 , all films showed rigid properties. PDADMAC and ePLL yielded thinner film coating, with lower deposited areal mass and dissipation shift than PAH. The glassy nature of these films contributes to refraining any diffusion of proteins within the multilayer, while the SMIL's polycationic last layer grants antifouling properties, as experimented in CE (Figure 2). However, the low molar mass of εPLL led to less robust films and thus to lower repeatability in CE measurements [START_REF] Leclercq | Quantification of adsorption and optimization of separation of proteins in capillary electrophoresis[END_REF][START_REF] Pei | Insight into the stability of poly(diallydimethylammoniumchloride) and polybrene poly cationic coatings in capillary electrophoresis[END_REF]. When PAH was used as the last layer, higher dissipation shift and higher deposited masses were observed, but similar or slightly higher CE performances were obtained compared to PDADMAC.

Due to thin film growth, rigid behavior, excellent analytical properties and ease of functionalization, PAH was selected as the capping layer reference. In the following, PEGylation of PAH, which is known to improve antifouling properties [START_REF] Hedayati | Protein adsorption measurements on low fouling and ultralow fouling surfaces: A critical comparison of surface characterization techniques[END_REF][START_REF] Chen | 50th Anniversary Perspective : Polymer brushes: novel surfaces for future materials[END_REF][START_REF] Xing | Quantitative fabrication, performance optimization and comparison of PEG and zwitterionic polymer antifouling coatings[END_REF][START_REF] Zhang | From neutral to zwitterionic poly(α-amino acid) nonfouling surfaces: Effects of helical conformation and anchoring orientation[END_REF][START_REF] Leigh | Antifouling photograftable zwitterionic coatings on PDMS substrates[END_REF][START_REF] Lin | Ultralow fouling and functionalizable surface chemistry based on zwitterionic carboxybetaine random copolymers[END_REF][START_REF] Smith | Polyelectrolyte multilayer nanocoating dramatically reduces bacterial adhesion to polyester fabric[END_REF][START_REF] Boulmedais | Polyelectrolyte multilayer films with pegylated polypeptides as a new type of anti-microbial protection for biomaterials[END_REF][START_REF] Zhang | Effect of salt on phosphorylcholine-based zwitterionic polymer brushes[END_REF][START_REF] Wang | Click poly(ethylene glycol) multilayers on RO membranes: Fouling reduction and membrane characterization[END_REF], was investigated, and the impact of PEGylation on electroosmotic mobility and CE separations was discussed.

Influence of the PEGylation density and length on electroosmotic flow mobility (µeo)

PAH was grafted with PEG chains of two different lengths (1 kDa and 5 kDa) at various ratios (see Section 2.6 for polymer synthesis). PEGylated PAH were prepared from 1 kDa PEG chains (targeting 10% and 20% grafting ratios) and from 5 kDa PEG chains (targeting 1%, 2%, 3.5%, 5%, 7.5%, 10%, 20% and 30% grafting ratios). Effective % of PEGylation was measured by 1 H NMR or SEC/MALLS, as described in Sections 2.2 and 2.4, and results were gathered in Table SI 1.

As shown in Figure 4, no influence of the PEGylation of the PAH last layer on the EOF was observed for 1 kDa PEG lateral chains: µeo was found stable at ≈ -47 TU, even at 23.5% grafting proportion. In contrast, in the case of PAH grafted with 5 kDa mPEG, PEGylation strongly impacted the EOF, even at the lowest PEG ratio. This is in good agreement with the hydrodynamic diameter of a 5kDa PEG (≈ 6 nm [START_REF] Liu | Nanoscale lubricating film formation by linear polymer in aqueous solution[END_REF]) which is higher than the Debye length (4 nm at I = 6.4 mM ionic strength, 2 M acetic acid BGE), while the typical size of 1kDa PEG chain (around 1 nm [START_REF] Liu | Nanoscale lubricating film formation by linear polymer in aqueous solution[END_REF]) is significantly smaller than the Debye length. The impact of PEGylation on the EOF is based on the hydrodynamic screening of the charges contained in the double layer by the PEG chains. This hydrodynamic screening is only effective when the PEG dimension is larger than the Debye length in the experimental conditions.

Since the electrophoretic mobility (µep) of the four proteins tested were 23.7 TU for TI, 27.2 TU for Myo, 28.2 TU for RNAse and 31.6 TU for Lyz, the electrical polarity must be reversed from 5kDa mPEG PAH with grafting ratio equal or higher than 5%. This leads to a complete inversion of the protein migration order. A plateau was reached when PEGylation density was higher than 11%, where essentially a neutral coating behavior was observed (µeo ≈ -2 TU). PEGylation was also performed on PEI and similar impact on EOF was observed (see Figure 4), but with a faster EOF decrease with the grafting density of the 5 kDa mPEG.

Impact of the PEGylation of PAH on k and Rs

Figure 5A shows the impact of PEGylation for the PAH grafted with 1kDa PEG chains in the last layer. A slight reduction of µeo was observed (≈ -41.5 TU). However, intra-capillary repeatability was lower (see RSDtm values in Table 3). In contrast, little impurity peaks around the Lyz peak were detected in the case of PAH-g-(mPEG1kDa)0.235 in the last layer unlike PAH, due to better selectivity due to the slight EOF reduction.

Figure 5B shows the impact of PEGylation for the PAH grafted with 5kDa PEG chains in the last layer. Interestingly, SMIL with PAH-g-(mPEG5kDa)0.021 gave similar µeo than PAH-g-(mPEG1kDa)0.138 and 0.235, but with better repeatability. SMIL with PAH-g-(mPEG5kDa)0.11 showed a quasi-neutral behavior coated capillary with µeo at ≈ -5.5 TU. The order of detection of the proteins was thus reversed using positive polarity, and the analysis time was drastically reduced (from 12 min for the non-PEGylated PAH to only 7 min for the PAH-g-(mPEG5kDa)0.11) in the last layer.

Regarding the impact of PEGylation on H, 1kDa PEGylated PAH led to lower average k values but slightly higher A values compared to the unmodified PAH, as shown for instance in Figure 6A for Myo, Figure 6B for Lyz and Table 3. Average k value (tested on the four proteins) was 7 × 10 -2 , 4.4 × 10 -2 and 6.05 × 10 -2 for PAH, (PAH-g-(mPEG1kDa)0.138)1 and (PAH-g-(mPEG1kDa)0.235)1, respectively. Average A values changed from 1.9 µm for PAH to 4.4 µm for 13.8% mPEG grafted PAH and 3.3 µm for 23.5% mPEG grafted PAH. On the whole, the separation efficiencies were very similar to those obtained with unmodified PAH, but the resolution was significantly higher with 1kDa PEGylation (see Figure 6C), especially at high voltage due to flatter H=f(u) curve and slightly lower EOF.

Increasing mPEG length from 1kDa to 5kDa was first detrimental on the average k value at intermediate grafting ratio (k = 9.4 × 10 -2 for 2.1% PEG) but reached similar value (k = 4.8 × 10 -

2 ) for higher ratios (11% PEG). Higher average A values were obtained, i.e. 4.65 µm at 2.1% and 5.3 µm at 11%. This non-linear effect of PEGylation on both the slope (k) and the constant value A of the H=f(u) curve may be not only related to residual protein adsorption. The coating homogeneity may also affect the characteristics of H=f(u) curve.

On the whole, the PEGylation of the PAH last layer was found to lead to much flatter H=f(u)

curve, except at intermediate 5 kDa grafting ratio (2.1%). Therefore, the resolution was much less affected by increasing voltages on PEGylated SMIL compared to the non-PEGylated PAH last layer (see the variation of the resolution between Myo and RNAse A with the voltage on Figure 6 C). indicate that thicker films are promoted by PAH bearing longer pendant chains with a moderate grafting degree while thinner films are formed with shorter pendant chains and higher grafting degree. Since the final dissipation and frequency shifts obtained for all films remained in the rigid domain, the Sauerbrey equation was applied to estimate deposited areal masses and the corresponding thicknesses were calculated (Table 4). While films capped with PAH and 1 kDa PEGylated PAH chains (23% grafting rate) reached similar thicknesses (≈ 7 nm), demonstrating that less chain adsorption had occurred with the latter, the SMIL capped with 5 kDa PEGylated PAH (11% grafting rate) exhibited a deposited areal mass 50 % higher, leading to a ≈ 11 nm thick film. In that case, the thickness of the last deposited layer accounts thus for around 7 nm, a result that confirms that the mPEG impact on EOF was observed for PEG chains longer than the Debye length (Figure 4). Such a significant increase in the film thickness when 5 kDa PEGylated PAH (11% grafting rate) was deposited, suggests that mPEG chains adopt a non-collapsed conformation on top of the coating. Combination of the rigid film structure of (PDADMAC/PSS)2, known for its high stability, robustness and analytical separation efficiency [START_REF] Leclercq | Modulation of the electroosmotic mobility using polyelectrolyte multilayer coatings for protein analysis by capillary electrophoresis[END_REF][START_REF] Roca | Polyelectrolyte multilayers in capillary electrophoresis[END_REF][START_REF] Leclercq | Quantification of adsorption and optimization of separation of proteins in capillary electrophoresis[END_REF], with a slightly hydrated polycation in the last layer where the water thin layer reduced adhesion or adsorption phenomena [START_REF] Peng | Protein fouling resistant membrane prepared by amphiphilic pegylated polyethersulfone[END_REF][START_REF] Tang | Simultaneously covalent and ionic bridging towards antifouling of GO-imbedded nanocomposite hollow fiber membranes[END_REF][START_REF] Shih | Tunable blood compatibility of polysulfobetaine from controllable molecularweight dependence of zwitterionic nonfouling nature in aqueous solution[END_REF], is thus key to coatings with antifouling properties.

Conclusion

The first step of this work was to investigate the impact of the last polycationic layer on separation performances in CE using SMIL coated capillaries. For that purpose, five different polyelectrolytes were tested on a (PDADMAC/PSS)2 SMIL coating, namely PDADMAC, PAH, PEI, εPLL and αPLL. With similar properties as PDADMAC, PAH was found the best candidate to investigate the impact of PEGylation in the last layer. 5 kDa PEGylation generated reduced EOF, even at low PEG ratios (typically between 1 and 5 % grafting), while the impact of 1 kDa PEGylation on EOF was much lower even at high grating rates (i.e. 10-25%), with a decrease of the electroosmotic mobility by only about 5%. This finding can be explained by the hydrodynamic screening of the PEG chains in the electrical double layer by sufficiently long PEG chains. An electroosmotic mobility plateau was reached for 5 kDa PEG chains, when the PEGylation rate was above 11%. The SMIL coating then adopts a neutral behavior with very low electroosmotic mobility (~ 2 TU) and a reverse migration order of detection of the proteins.

Regarding the analytical behavior of PEGylated last layered SMIL, flatter H=f(u) curves were generally obtained leading to lower k values in agreement with antifouling properties. However, a slight increase of the A values was observed, for unknown reason. On the whole, a lower impact of increasing applied separation voltage (electric field strength) on the separation performances could be obtained using the PEGylated SMIL. The use of (PDADMAC/PSS)2-(PAH-g-(mPEG5kDa)0.11)1 SMIL coating is a very convenient way to build a performant and almost neutral coating, which is not possible to obtain with classical SMIL coatings terminating with positive or negative surface charges. In conclusion, SMILs with PEGylated polycations as the last layer can be used to improve the CE resolution via EOF modulation without compromising the separation efficiency; or as an alternative to neutral coatings for 5kDa PEGylated last layer. As a perspective, it would be interesting to study and to better understand the contribution coming from capillary coatings in the constant term of the H=f(u) curve. Table 1. Impact of the chemical nature of the last polycationic layer on protein retention factor (k) and on RSD on migration times at -10kV and -20kV. Experimental conditions: see Figure 2.

Captions to Figures and Tables:

Table 2. Impact of last polycationic layer on SMIL construction and characteristics. Experimental conditions: see Figure 3.

Figures and Tables

  the chemical nature of the last polycationic layer deposited on a 4-layers poly(diallyldimethylammonium chloride) (PDADMAC) / poly(sodium styrene sulfonate) (PSS) sub-structure, on the CE separation performances of a mixture of 4 model intact proteins (myoglobin (Myo), Trypsin inhibitor (TI), ribonuclease A (RNAse A) and lysozyme (Lyz)). PDADMAC, polyallylamine (PAH), polyethyleneimine (PEI), ε-poly(L-lysine) (εPLL) and αpoly(L-lysine) (αPLL) were compared as a last layer, either used as it or after chemical grafting modification with PEG chains of two different lengths (1 kDa and 5 kDa) at various molar proportions. The different CE characteristic parameters and performances (electroosmotic mobility, protein retention factor k, constant A, separation efficiency, RSD of migration times)

France) .

 . Trypsin Inhibitor (TI, from soybean, isoelectric point (pI): 4.5, purity not indicated by the supplier), Myoglobin (Myo, from equine skeletal muscle, pI: 7.3, 95% purity), Ribonuclease A (RNAse A from bovine pancreas, pI: 9.3, 60% purity) and Lysozyme (Lyz, from chicken egg white, pI: 11.1, 90% purity) were obtained from Sigma-Aldrich (Saint-Quentin-Fallavier, France). Polymers. Methoxy-(polyethylene glycol)-succinimidyl carboxyl methyl ester (mPEG-NHS-SCM) of 1 kDa (ref.: PLS-215) and 5 kDa (PLS-213) average molar masses; and methoxy-(polyethylene glycol)-alkyne (mPEG-Alkyne) of 1 kDa (PLS-2036) and 5 kDa (PLS-2034) molar masses, were purchased from Creative PEGWorks (Durham, NC, USA). e-poly(L-Lysine hydrochloride) (εPLL, Mw = 3.5-4.5 kDa) was purchased from Biosynth Carbosynth (Staad, Switzerland). Poly(allylamine hydrochloride) (PAH, Mw = 120-200 kDa) was supplied by Alfa Aesar (Kandel, Germany). Branched poly(ethyleneimine) (PEI, Mw = 750 kDa) and poly(diallyldimethylammonium chloride) (PDADMAC, Mw = 450 kDa) were purchased from Sigma Aldrich (Saint-Quentin-Fallavier, France). a-poly(L-Lysine hydrobromide)-graft-(azidoalkylamide) 10% (αPLL-10%, Mw = 57 kDa) and 20% (αPLL-20%, Mw = 62 kDa) were purchased from Alamanda Polymers (Huntsville, AL, USA). Poly(sodium styrenesulfonate)

  with 5 mL of buffer to transfer the leftover reagent. The flask was closed, and the solution was stirred overnight. The reaction medium was then dialyzed over 48 h against water (1 kDa cutoff tubing) to remove ungrafted mPEG, meanwhile the outer water was renewed 7 times. The solution was then concentrated on a rotatory evaporator and finally freeze-dried overnight. The resulting solid was then characterized by 400 MHz 1 H NMR (see Figure SI 8 to SI 12) and by SEC-MALLS (see Figure SI 13 to SI 20). Quantitative data for all examples are gathered in Table

  5 mL glass vial) and construction buffer (HEPES 20 mM, NaOH 10 mM, pH 7.4, 10 min, 1 mL in a 1.5 mL glass vial). A 4-layer SMIL capillary coating, used as a substructure, was obtained by flushing (at 96.5 kPa) with polycation solution (PDADMAC, 3 gL -1 in HEPES buffer, 7 min, 200 µL in polypropylene vial), HEPES buffer (3 min, 1 mL in a 1.5 mL glass vial) to remove the excess of polycation, polyanion solution (PSS, 3 gL -1 in HEPES buffer, 7 min, 200 µL in polypropylene vial), HEPES buffer (3 min, 1 mL in a 1.5 mL glass vial) to remove the excess of polyanion. To avoid polyelectrolyte contamination, HEPES solutions for PDADMAC and PSS rinsing flushes were stored in different vials. This procedure was repeated once more to obtain the 4-layer SMIL substructure. The fifth layer was deposited by flushing (at 96.5 kPa, 7 min, 200 µL in polypropylene vial) with either PDADMAC, PAH, PEI, εPLL, αPLL (either native or PEGylated), followed by a rinsing step with HEPES buffer (3 min, 1 mL in another 1.5 mL glass vial) and 5 min wait. Finally, the coated capillary was flushed (at 96.5 kPa) with pure water (3 min) and background electrolyte (BGE) (2 M acetic acid, pH 2.2, 10 min) followed by another 5 min wait before analysis.2.8. Separation of intact proteins by capillary zone electrophoresisCapillary zone electrophoresis experiments were performed on a P/ACE MDQ Beckman Coulter (Sciex, Villebon-sur-Yvette, France) piloted by 32 Karat software. Analyses were performed on a 40 cm total length capillary (30 cm effective length), with inner (i.d.) and outer (o.d.) diameters of 50 µm (± 0.3 µm) and 363 µm (± 15 µm), respectively. Electrophoretic separations were performed in a 2 M acetic acid BGE, pH 2.2 (6.4 mM ionic strength). BGE was flushed (at 96.5 kPa) for 3 min between each run. Separations were performed at different voltages between ±10

  studying the impact of the PEGylation of the polycationic last layer on the CE separation performances, the 5-layer SMIL coating terminating with the non-PEGylated PDADMAC, PAH, PEI, εPLL or αPLL were first compared. The electropherograms (five repetitions) obtained for the separation of the protein mixture at different separation voltages (i.e., -20 kV, -17.5kV, -15kV, -12.5kV and -10 kV, with typical electric currents of 18.1 µA, 15.7 µA, 13.4 µA, 11.0 µA and 8.8 µA, respectively) are shown in Supporting Information (see Figure SI 21 to Figure SI 25). An example of electropherograms, repetitions obtained at -10 kV is displayed in Figure 2A. Excellent repeatabilities were generally obtained as demonstrated by the very low RSDtm (typically lower

Figure SI 26

 26 FigureSI 26 for the experimental plots on TI, RNAse A and Lyz). Using equation (1), all the H=f(u) plots were fitted taking k and A as variable parameters and by neglecting the third term in the right-hand side of the equation which is related to the resistance in mass transfer in stationary phase. Table1gathers the numerical adjusted data for k and A for the 4 proteins. From Figure2A,

Figure 3 9 Hz and 2

 392 shows the dissipation and frequency shifts of the 3 rd harmonic during the SMIL construction. Since films terminated by PEI and αPLL presented similar behavior as PAH, but with lower analytical efficiency, only films terminated by PAH were investigated by QCM-D. Frequency shifts were measured at each injection step of oppositely charged polyelectrolyte (see Section 3 in Supporting Information), signaling the adsorption of the next polyelectrolyte layer by charge overcompensation promoting[START_REF] Decher | Buildup of ultrathin multilayer films by a self-assembly process: II. Consecutive adsorption of anionic and cationic bipolar amphiphiles and polyelectrolytes on charged surfaces[END_REF]. The first increment corresponds to PDADMAC (at 3 gL -1 in 20 mM HEPES) adsorption onto the bare sensor, followed by rinsing with 20 mM HEPES. Frequency and dissipation shift values of the first step were larger than those of following steps since the adsorption onto the bare QCM sensor is different than onto polyelectrolytes. Average PDADMAC and PSS shift in frequency were Hz per step, respectively, showing growth of very thin film. As observed in FigureSI32, low final frequency (-27 Hz) and dissipation shift (3.4 × 10 -7 ) for the (PDADMAC/PSS)2 SMIL indicated thin films, while overlapping overtones indicated rigid properties. Due to their high linear charge density, PDADMAC and PSS multilayered films are known to form thin and rigid films in low ionic strength conditions (typically 15 mM NaCl[START_REF] Elżbieciak-Wodka | Comparison of permeability of poly(allylamine hydrochloride)/and poly(diallyldimethylammonium chloride)/poly(4styrenesulfonate) multilayer films: Linear vs. exponential growth[END_REF]

Figure 7

 7 Figure 7 compares the (PDADMAC/PSS)2-(PAH)1 SMIL building to two other films obtained with PEGylated PAH as the last layer. The PAH backbone interacts electrostatically with the last PSS layer, while neutral mPEG pendant chains in PEGylated PAH could extend at the SMIL/BGE interface. As the last layer, PAH (unmodified or PEGylated) led to films with slightly less rigid behavior and frequency overtone spreading (Figures 3 and SI 33) compared to PDADMAC last layered SMIL. When adsorbing PEGylated PAH chains on SMIL, two adverse effects can beinferred: on the one hand the molar mass of each polycationic chain is significantly increased by grafted neutral pendant chains, leading to thicker films and larger mass deposition. On the other hand, the charge density of grafted PAH is decreased while steric hindrance increases, which can limit polymer chain adsorption and lead to thinner films. When comparing SMILs capped by either unmodified PAH or 5kDa PEGylated PAH (11% grafting ratio) as the last layer, the latter exhibited a larger frequency shift concomitant with a larger overtone spread (FigureSI35); however similar dissipation values were observed in both cases. In contrast, the frequency shift, overtone spreading (FigureSI 36) and dissipation values, were similar or inferior to that of PAH when 1 kDa PEGylated PAH (20% grafting rate) were adsorbed as a capping layer. These results

Figure 1 . 2 substructure.

 12 Figure 1. Schematic representation of SMIL coatings used in this work based on (PDADMAC/PSS)2 substructure. The last polycationic layer can be composed of unmodified PDADMAC, PAH, PEI, ePLL and aPLL, or their respective PEGylated chains with different chain lengths.

Figure 2 .

 2 Figure 2. Impact of the nature of the last layer on protein separation at -10kV (A), on the H=f(u) curve in the case of Myo (B) and on the resolution of the Myo / RNAse A pair (C). Experimental conditions: (PDADMAC/PSS)2-(Polycation)1 SMIL coated capillary, 40 cm total length (30 cm to detector) × 50 µm i.d. BGE: 2 M acetic acid, pH 2.2. Hydrodynamic injection of proteins at 0.25 gL -1 each in BGE: 2.7 kPa, 4s (0.61% Vtot). Co-injection of 0.001% DMF in BGE (w/w): 2.7 kPa, 0.4s. Protein mixture: (1) TI, (2) Myo, (3) RNAse A and (4) Lyz. Applied voltages: -10kV, -12.5kV, -15kV, -17.5kV and -20kV. Temperature: 25°C. See Section 2.7 for the coating procedure.

Figure 3 .

 3 Figure 3. Evolution of frequency (continuous lines) and dissipation (dashed lines) shifts for the 3 rd harmonic during SMIL construction procedure in QCMD. Experimental conditions: (PDADMAC/PSS)2-(Polycation)1 SMIL deposited on a silica-coated QCM sensor. Hydrodynamic flow: 0.05 mLmin -1 . Temperature: 23°C. Average (PDADMAC/PSS)2 layers construction in grey. See Section 2.9 for the coating procedure.

Figure 4 .

 4 Figure 4. Impact of the PAH and PEI PEGylated polycation last layer on the electroosmotic flow. Experimental conditions: (PDADMAC/PSS)2-(Polycation)1 SMIL capillary coated, 40 cm total length capillary (30 cm to detector) × 50 µm i.d. BGE: 2 M acetic acid (pH 2.2). Hydrodynamic injection: 2.7 kPa, 4s. Co-injection of 0.002% DMF water (w/w). Protein mixture: (1) TI, (2) Myo, (3) RNAse A and (4) Lyz, at 0.25 gL -1 each in BGE. Temperature: 25°C. See Section 2.7 for the coating procedure.

Figure 5 .

 5 Figure 5. Impact of PEGylation of the last PAH layer with 1kDa PEG (A) and 5kDa PEG (B). Same experimental conditions as in Figure 2. Reversed polarity was used for (PDADMAC/PSS)2-(PAH-g-(mPEG5kDa)0.11)1.

Figure 6 .

 6 Figure 6. Impact of the PEGylation for the last PAH layer onMyo separation efficiency (A), Lyz separation efficiency (B) and peak resolution for the Myo / RNAse A pair (C). Same experimental conditions as in Figure 2. Reversed polarity was used for (PDADMAC/PSS)2-(PAH-g-(mPEG5kDa)0.11)1. See Figure SI 31 for impact on TI and RNAse A.

Figure 7 .

 7 Figure 7. Evolution of shift frequency (continuous lines) and shift dissipation (dashed lines) for the 3 rd harmonic during SMIL construction procedure in QCMD. Experimental conditions: see Figure 3. See detailed construction in supporting information Figure SI 32 to SI 34.

Figure 2 .

 2 Figure 2. Schematic representation of SMIL coatings used in this work based on (PDADMAC/PSS)2 substructure. The last polycationic layer can be composed of unmodified PDADMAC, PAH, PEI, ePLL and aPLL, or their respective PEGylated chains with different chain lengths.

Figure 2 .

 2 Figure 2. Impact of the nature of the last layer on protein separation at -10kV (A), on the H=f(u) curve in the case of Myo (B) and on the resolution of the Myo / RNAse A pair (C). Experimental conditions: (PDADMAC/PSS)2-(Polycation)1 SMIL coated capillary, 40 cm total length (30 cm to detector) × 50 µm i.d. BGE: 2 M acetic acid, pH 2.2. Hydrodynamic injection of proteins at 0.25 gL -1 each in BGE: 2.7 kPa, 4s (0.61% Vtot). Coinjection of 0.001% DMF in BGE (w/w): 2.7 kPa, 0.4s. Protein mixture: (1) TI, (2) Myo, (3) RNAse A and (4) Lyz. Applied voltages: -10kV, -12.5kV, -15kV, -17.5kV and -20kV. Temperature: 25°C. See Section 2.7 for the coating procedure.

Figure 3 .

 3 Figure 3. Evolution of frequency (continuous lines) and dissipation (dashed lines) shifts for the 3 rd harmonic during SMIL construction procedure in QCMD. Experimental conditions: (PDADMAC/PSS)2-(Polycation)1 SMIL deposited on a silica-coated QCM sensor. Hydrodynamic flow: 0.05 mLmin -1 . Temperature: 23°C. Average (PDADMAC/PSS)2 layers construction in grey. See Section 2.9 for the coating procedure.

Figure 4 .

 4 Figure 4. Impact of the PAH and PEI PEGylated polycation last layer on the electroosmotic flow. Experimental conditions: (PDADMAC/PSS)2-(Polycation)1 SMIL capillary coated, 40 cm total length capillary (30 cm to detector) × 50 µm i.d. BGE: 2 M acetic acid (pH 2.2). Hydrodynamic injection: 2.7 kPa, 4s. Co-injection of 0.002% DMF water (w/w). Protein mixture: (1) TI, (2) Myo, (3) RNAse A and (4) Lyz, at 0.25 gL -1 each in BGE. Temperature: 25°C. See Section 2.7 for the coating procedure.

Figure 5 .

 5 Figure 5. Impact of PEGylation of the last PAH layer with 1kDa PEG (A) and 5kDa PEG (B). Same experimental conditions as in Figure 2. Reversed polarity was used for (PDADMAC/PSS)2-(PAH-g-(mPEG5kDa)0.11)1.

Figure 7 .

 7 Figure 7. Evolution of shift frequency (continuous lines) and shift dissipation (dashed lines) for the 3 rd harmonic during SMIL construction procedure in QCMD. Experimental conditions: see Figure 3. See detailed construction in supporting information Figure SI 32 to SI 34.

  -1 until stabilization of QCM signals, followed by flushing with the construction buffer (HEPES 20 mM, NaOH 10 mM, pH 7.4) for 20 to 30 min. The film was assembled by pure water with a 0.05 mLmin -1 flow rate. Both the normalized resonance frequency shift (Δfn/n in Hz; related to the mass deposited on the sensor), and the normalized dissipation shift (ΔDn/n; related to the energy dissipated by the resonator in its direct surrounding), were discussed. High ΔDn/n values and Δfn/n spreading are characteristics of a viscoelastic coating, such as hydrated SMIL film structures. Overtones frequency shifts overlap and low dissipation shift values (i.e., when the final ΔDn/n divided by the final Δfn/n is equal or lower to 4 × 10 -7

	sequential injection of PDADMAC and PSS solutions (at 3 gL -1 in construction buffer) for 30 min
	at 0.05 mLmin -1 , starting with PDADMAC as first layer. Rinsing with the construction buffer was
	performed for 30 min between each polyelectrolyte adsorption step. Once a 4-layer SMIL

(PDADMAC/PSS)2 was obtained, a differentiated capping layer based either on εPLL or PAH (at 3 gL -1 in construction buffer) with PEG grafting ratios of 0, 11 and 20 % was deposited for 30 min at 0.05 mLmin -1 . After rinsing with the construction buffer, the SMIL-coated QCM sensor was stabilized in

Table 1

 1

gathers the numerical adjusted data for k and A for the 4 proteins. From Figure

2A

, it clearly appears that PAH, and to a lower extend PDAMAC are the best coatings leading to the lower H values (see Figure

SI 26)

. This conclusion is also confirmed by the ranking based on the k values (see Table

Table 3 . Impact of PEGylated polycation in the last layer on protein retention factor (k) and RSDtm at -10kV and -20kV.

 3 Corresponding electropherograms and H=f(u) graphs are available in Supporting Information, from Figures SI 27 toFigure SI 30.

Table 4 . Impact of the last polycationic layer on SMIL construction

 4

. Experimental conditions: see Figure

3

.

Table 1

 1

Table 1 . Impact of the chemical nature of the last polycationic layer on protein retention factor (k) and on RSD on migration times at -10kV and -20kV.

 1 Experimental conditions: see Figure2.

	Last layer	Parameters			Proteins
	polymer (µeo)		TI	Myo	RNAse A	Lyz	Average
	PDADMAC	k × 10 -2	6.8	5.1	6.8	7.3	6.5
	(-45.5 TU)	A [µm]	4.8	1.6	3.0	1.5	2.7
		RSDtm (-10kV)	0.41	0.49	0.48	0.61	0.50
		RSDtm (-20kV)	0.05	0.06	0.06	0.10	0.07
	PAH	k × 10 -2	7.3	5.7	6.7	8.0	6.9
	(-45.2 TU)	A [µm]	4.3	0.3	2.4	0.7	1.9
		RSDtm (-10kV)	0.34	0.41	0.44	0.54	0.43
		RSDtm (-20kV)	0.06	0.05	0.10	0.15	0.09
	PEI	k × 10 -2	8.3	6.2	7.2	7.6	7.3
	(-44.4 TU)	A [µm]	4.4	2.2	4.2	2.5	3.3
		RSDtm (-10kV)	0.03	0.04	0.02	0.04	0.03
		RSDtm (-20kV)	0.05	0.05	0.04	0.06	0.05
	εPLL	k × 10 -2	3.1	3.7	4.6	3.9	3.8
	(-40.7 TU)	A [µm]	10.0	5.6	7.0	5.1	6.9
		RSDtm (-10kV)	1.73	1.98	2.16	2.75	2.15
		RSDtm (-20kV)	0.17	0.14	0.13	0.17	0.15
	αPLL	k × 10 -2	8.7	6.3	7.3	7.9	7.5
	(-45.3 TU)	A [µm]	3.1	1.3	4.0	1.6	2.5
		RSDtm (-10kV)	0.98	1.12	1.13	1.37	1.15
		RSDtm (-20kV)	0.28	0.29	0.36	0.40	0.33

Table 3 Table 3 . Impact of PEGylated polycation in the last layer on protein retention factor (k) and RSDtm at -10kV and -20kV.

 33 Corresponding electropherograms and H=f(u) graphs are available in Supporting Information, from Figures SI 27 toFigure SI 30.

				Proteins
	Last layer polymer (µeo)	Parameters	TI	Myo	RNAse A	Lyz	Average
		k × 10 -2	7.3	5.7	6.7	8.0	6.9
	PAH	A [µm]	4.3	0.3	2.4	0.7	1.9
	(-45.2 TU)	RSDtm (-10kV)	0.34	0.41	0.44	0.54	0.43
		RSDtm (-20kV)	0.06	0.05	0.10	0.15	0.09
		k × 10 -2	5.8	3.0	3.7	5.1	4.4
	PAH 13.3%	A [µm]	6.5	3.6	4.7	2.9	4.4
	(-41.7 TU)	RSDtm (-10kV)	1.34	0.19	0.20	0.22	0.49
	1kDa	RSDtm (-20kV) k × 10 -2	0.53 8.3	0.59 4.1	0.64 5.5	0.83 6.3	0.65 6.1
	PAH 23.5%	A [µm]	4.5	2.6	3.6	2.3	3.3
	(-41.3 TU)	RSDtm (-10kV)	1.36	1.54	1.62	2.03	1.64
		RSDtm (-20kV)	0.41	0.46	0.48	0.61	0.49
		k × 10 -2	9.4	7.8	9.5	10.8	9.4
	PAH 2.1%	A [µm]	7.7	3.9	3.9	3.1	4.6
	(-40.2 TU)	RSDtm (-10kV)	0.08	0.08	0.09	0.11	0.09
	5kDa	RSDtm (-20kV) k × 10 -2	0.11 7.6	0.14 4.5	0.14 3.1	0.20 3.9	0.15 4.8
	PAH 11%	A [µm]	8.3	3.8	5.5	3.5	5.3
	(-5.5 TU)	RSDtm (-10kV)	0.02	0.02	0.01	0.02	0.02
		RSDtm (-20kV)	0.04	0.05	0.05	0.01	0.04

Acknowledgement

This work was supported by the CNRS, the french Ministry of Research (MESR) and the ANR (grant ANR-20-CE92 SMIL E, for L.D.). The authors thank the Technological Center in Micro and nanoelectronics (CTM) in Montpellier for AFM experiments.


Nabil Madali
email: nabil.madali@b-com.com

Antonin Gilles

Patrick Gioia

Luce Morin

Automatic depth map retrieval from digital holograms using a deep learning approach

Information extraction from computer-generated holograms using learning-based methods is a topic that has not received much research attention. In this article, we propose and study two learning-based methods to extract the depth information from a hologram and compare their performance with that of classical depth from focus (DFF) methods. We discuss the main characteristics of a hologram and how these characteristics can affect model training. The obtained results show that it is possible to extract depth information from a hologram if the problem formulation is well-posed. The proposed methods are faster and more accurate than state-of-the-art DFF methods.

Introduction

Holography [START_REF] Gabor | A new microscopic principle[END_REF] is an immersive technology that records and reproduces the emitted wavefronts of an illuminated 3D scene, providing all the depth perception cues of the human visual system [START_REF] Goodman | Introduction to fourier optics[END_REF].

The recorded hologram contains all the information that describes the acquired 3D scene. However, the extraction of 3D information directly from the hologram is not straightforward, due to the lack of spatial localization of scene objects inside the hologram plane. Recovering the scene geometry is a necessary step for several tasks, such as the segmentation and edition of the different objects contained in the scene. To directly recover the scene geometry from a hologram, researchers often use the Depth From Focus (DFF) [START_REF] Grossmann | Depth from focus[END_REF] method on a sampled reconstruction volume. Such a volume is created from multiple reconstructions computed at different focus distances chosen in a predefined interval with a fixed depth sampling rate. Given the computed reconstruction volume, the acquisition depth can be estimated by applying focus measure operators [START_REF] Muller | Real-time correction of atmospherically degraded telescope images through image sharpening[END_REF] on each reconstruction and then selecting for each pixel the depth for which the focus is optimal. Numerous focus measure operators such as Entropy [START_REF] Gillespie | The use of self-entropy as a focus measure in digital holography[END_REF], Gray-level variance [START_REF] Ma | Numerical reconstruction of digital holograms for three-dimensional shape measurement[END_REF], Gini index [START_REF] Zonoobi | Gini index as sparsity measure for signal reconstruction from compressive samples[END_REF], Tamura coefficient [START_REF] Memmolo | Automatic focusing in digital holography and its application to stretched holograms[END_REF] and Wavelet transform [START_REF] Liebling | Autofocus for digital fresnel holograms by use of a fresnelet-sparsity criterion[END_REF] have been specifically designed for their usage on holographic reconstructions. The DFF method leads to relevant results when the focus measure operator, the reconstruction interval and the sampling rate are judiciously chosen. However, the large number of numerical reconstructions leads to a high computational cost.

With the emergence of deep learning methods, numerous research works -mainly in Holographic Microscopy where the scene is acquired at a single depth -have attempted to estimate the scene information directly from the hologram without the need for any numerical reconstruction. Given a well-designed network architecture and sufficient amount of data, research works on the subject discussed in Section 2 have shown that the trained model gives more accurate and faster results than traditional methods while maintaining to some extent their performance on new samples not seen during the training. In most cases, the trained network represents a one-to-one mapping between the hologram and a single estimated acquisition depth. Using such a network to perform inference on complex scenes with multiple objects located at different depths -such as those used in Computer-Generated Holograms (CGH) -would require the hologram to be first segmented in an unsupervised manner into individual sub-holograms corresponding to each acquisition depth, before feeding them to the network. This is an intractable problem due to the segmentation step. In this work, we first review the possibility of learning a one-to-many mapping between the acquired hologram and a depth map of the scene. Then, a patch-based method that locally segments focused regions from a holographic reconstruction stack is presented. Finally, we compare the direct method and the patch-based in terms of generalization and scalability.

The remaining of the article is organized as follows: Section 3, first introduces the baseline approach based on the DFF method. Next, the direct mapping between the hologram and the scene depth map is introduced. Finally, the proposed patch-based approach and the used network architectures are detailed. In Section 4, a series of experiments are conducted to validate the proposed approach, then its advantages and limitations are discussed in Section 5.

Related work

Depth estimation is a well-studied problem in Digital Holographic Microscopy (DHM). To recover the acquisition depth value from a given hologram, most methods rely on a set of holographic reconstructions computed at different depths from the hologram plane and then estimate a sharpness value for each reconstruction using a Focus Measure (FM) operator [START_REF] Muller | Real-time correction of atmospherically degraded telescope images through image sharpening[END_REF]. The maximum of the sharpness curve indicates the distance at which the microscopic object (cell, molecule, etc.) is located from the hologram plane. Several focus operators designed for DHM have been proposed in the literature; a fairly exhaustive review can be found in [START_REF] Sheridan | Roadmap on holography[END_REF].

With the advent of deep learning, several research works tried to solve this problem directly, by designing end-to-end architectures that directly use the hologram without requiring additional pre-processing, making it possible to obtain more precise and faster results than traditional methods.

One of the pioneer works on the subject was done by Pitkaaho et al. [START_REF] Pitkäaho | Performance of autofocus capability of deep convolutional neural networks in digital holographic microscopy[END_REF]. It uses the AlexNet [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF] network architecture to estimate the in-focus depth from a set of holographic reconstructions computed at manually selected axial positions. Each recorded hologram is manually preprocessed to remove zero and twin order terms. Then, twenty-one holographic reconstructions are computed with a fixed reconstruction step of 10mm within a ±100mm range around the ground truth in-focus depth. The reconstruction amplitudes are resized, randomly cropped, and augmented using different rotation angles, before feeding them into the network. The network is supervised to predict where the focus is optimal among the twenty-one reconstructions. Experimental results show that the network generalizes well using different cell types when the hologram is recorded using the same optical setup as for the training. However, it fails to generalize under different magnification and lighting conditions during hologram acquisition.

Subsequently, Ren et al. [START_REF] Ren | Autofocusing in digital holography using deep learning[END_REF] proposed to formalize the depth estimation as a classification problem. Given the input hologram, the CNN network is supervised to predict a label which represents the discretized object distance from the hologram plane. The network is trained on a large-scale dataset, composed of 1000 holograms acquired optically from the local area of the negative USAF 1951 at different lateral and axial positions. Each hologram was recorded at one of five possible recording distances, acting as the targeted labels. The experimental results show that through the training, the network learns the relevant characteristics to cluster each input hologram into one of the five possible classes, outperforming the results obtained using kNN and SVM by a large margin for both validation and test, with almost 98% accuracy.

The same authors extended their work in [START_REF] Ren | Learning-based nonparametric autofocusing for digital holography[END_REF] by reformulating the depth estimation as a regression problem. The used CNN architecture consists of five convolution blocks, each block containing a convolution layer followed by a batch normalization and maximum pooling layers. The output of the last convolution block is vectorized and passed through two fully connected layers to produce the estimated focus depth. The whole network is supervised using an ℓ 2 norm between the estimated and ground truth values. The authors reviewed the network performance on amplitude and phase objects using holograms acquired optically from local areas of USAF 1951 and several biological specimens. For amplitude objects, 500 holograms with ten lateral positions were used, and for phase objects, 2000 holograms were used with five different lateral positions and four magnification levels. The experimental results showed that CNN outperforms MLP, k-nn, and SVM models for both amplitude and phase objects. The network shows a reasonable ability to generalize to different optical recording exposures and axial distances inside the training range. However, outside the training range, the network fails to produce the correct focal distance estimate. Overall, the proposed network is faster, more accurate, and with fewer hyper-parameters than conventional auto-focusing methods.

Shimobaba et al. [START_REF] Shimobaba | Convolutional neural network-based regression for depth prediction in digital holography[END_REF] improved the previous work by introducing the hologram power spectrum as an additional input data to the CNN, arguing that it gives more relevant information than the raw interference pattern for predicting focused depth. To verify this assumption the authors trained two identical networks, one for each input type, with holograms computed from Caltech-256 datasets [START_REF] Griffin | Caltech-256 object category dataset[END_REF]. The obtained results demonstrate that the network trained on the raw interference model pattern is unable to give a correct prediction on the validation set. On the other hand, the network trained on the power spectrum not only performs better on the training set but also maintains its performance on the validation set. The proposed approach yields faster and better results than Tamura coefficient [START_REF] Tamamitsu | A robust holographic autofocusing criterion based on edge sparsity: comparison of Gini index and Tamura coefficient for holographic autofocusing based on the edge sparsity of the complex optical wavefront[END_REF] method without the need for a computationally expensive deep search.

In the previously cited works, the network is supervised to predict a single acquisition depth per hologram. Given a pre-trained model, the inference for a complex scene containing multiple objects requires the hologram to be segmented in an unsupervised manner into a set of subholograms, each corresponding to a unique acquisition depth. Then, the inference results of the different sub-holograms must be merged to obtain the depth map of the scene. This is an intractable problem due to the lack of spatial localization in the hologram plane.

In the following section, we describe our deep learning-based methodology to estimate the depth map from a computer-generated hologram of a complex scene with multiple focal distances.

Methodology

In this section, we first recall the DFF method and how to use it to extract multiple depths per hologram. Then, we propose two different deep learning-based approaches for retrieving the scene geometry from a given hologram, illustrated in Figures 2 and3.

Depth From Focus (DFF) Principle

The following section briefly describes the DFF approach which is often applied in holographic microscopy [START_REF] Sheridan | Roadmap on holography[END_REF] where a single depth value is expected per hologram. The DFF approach will serve as a baseline model to evaluate the performances of the proposed approaches.

Single depth extraction per hologram

Given a hologram denoted by 𝐻 ∈ C 𝐿×𝐿 , a reconstruction volume is acquired by computing a set of numerical reconstructions on planes that are parallel to the hologram and located in a predefined depth interval [𝑧 min , 𝑧 max ], where 𝑧 min and 𝑧 max are the minimal and maximal depths of the scene respectively. Each numerical reconstruction is computed by the Angular Spectrum Method [START_REF] Goodman | Introduction to fourier optics[END_REF] defined as

P 𝑧 𝑖 {𝐻} = F -1 F (𝐻)𝑒 𝑗2 𝜋𝑧 𝑖 √ 𝜆 -2 -𝑓 2 𝑥 -𝑓 2 𝑦 , (1)
where 𝑓 𝑥 and 𝑓 𝑦 are the spatial frequencies along the 𝑋 and 𝑌 axis, 𝜆 is the acquisition wavelength, and 𝑧 𝑖 is the reconstruction depth, given by

𝑧 𝑖 = 𝑧 max -𝑧 min 𝑁 𝑖 + 𝑧 min . (2)
The set of numerical reconstruction amplitudes constitutes the reconstruction volume defined as Ω = { P 𝑧 0 {𝐻} , P 𝑧 1 {𝐻} , ..., P 𝑧 𝑁 {𝐻} }.

When the original scene is located at a single depth, this depth value can be recovered by evaluating the depth for which the reconstruction is sharpest with a focus measure operator 𝐹 𝑀 according to

𝑑 = arg max 𝑖 ∈ [1, 𝑁 ] 𝐹 𝑀 ( P 𝑧 𝑖 {𝐻} ) . (4)

Extraction of several depths per hologram

For a complex scene with multiple objects located at different depths, each reconstruction plane in the reconstruction volume must first be decomposed [START_REF] Ma | Numerical reconstruction of digital holograms for three-dimensional shape measurement[END_REF] into either non-overlapping or overlapping patches of predefined resolution (𝑠 × 𝑠). An optimal decomposition ensures that only one depth value is expected per patch 𝑅 𝑚,𝑛,𝑖 given by

𝑅 𝑚,𝑛,𝑖 (𝑢, 𝑣) = P 𝑧 𝑖 {𝐻} (𝑚 + 𝑢 -𝑠/2, 𝑛 + 𝑣 -𝑠/2). (5)
The depth 𝑑 𝑚,𝑛 is therefore estimated independently for each set of patches

{𝑅 𝑚,𝑛,𝑖 | 𝑖 ∈ [1, 𝑁]} as 𝑑 𝑚,𝑛 = arg max 𝑖 ∈ [1, 𝑁 ] 𝐹 𝑀 (𝑅 𝑚,𝑛,𝑖 ) , (6)
yielding a depth map containing for each pixel (𝑚, 𝑛) the depth at which the focus of the corresponding patch centered around the pixel is optimal. The numerical error between the estimated and ground truth depths depends on the chosen reconstruction interval, sampling rate, and focus measure operator. The reconstruction interval and sampling rate control the precision of the committed numerical error. With a reconstruction interval closer to the optimal focal plane and a finer sampling rate, the numerical error is small, since the reconstructions are performed closer to the optimal solution. On the other hand, a larger reconstruction interval or low sampling rate induces a high numerical error, since most reconstructions are far from the optimal focus plane.

Besides the holographic reconstruction parameters, the major drawbacks of the DFF approach are the optimal decomposition of the reconstruction volume and the choice of the patch size. Figure 1, illustrates two cases that can be encountered in practice. In the first case, the extracted patch is made up of four planar sub-regions, each with a different acquisition depth. If this patch is used to estimate the depth of the central pixel, there is a 1/4 chance that the estimate will be accurate; however, this probability can be raised by reducing the patch size. The patch defined by the red dashes is the most optimal and ensures a single depth value per patch. In the second scenario, the patch is divided into several sub-regions, each with a distinct geometry. It is therefore not possible to accurately predict the depth of the central pixel by reducing the size of the patch because it will always contain multiple sub-regions, creating uncertainty in the depth prediction.

To alleviate these drawbacks, alternative methods based on deep learning are described in the next section.

Reducible Case Depth Map

Amplitude Map Irreducible Case 9 × 9

× 23

Depth Map Amplitude Map Fig. 1. Illustration of the main drawbacks of the DFF approach using two cropped patches of size (32×32) extracted at two different positions. The dotted lines correspond to the new spatial boundaries of the cropped patches with a crop size of (23 × 23) and (9 × 9) respectively. In the first case, the depth value of the central pixel can be correctly estimated if the patch size is considerably reduced. In the second case, regardless of the chosen patch size, the depth prediction is subject to uncertainty.

Proposed approaches based on CNN

In this section, two learning-based approaches are discussed. First, a direct approach is illustrated in Figure 2 that directly uses the hologram as input and is supervised to predict the scene depth map. Then, a patch-based method illustrated in Figure 3 alleviates the main drawback of the DFF approach.

E D Hologram Real and Imaginary Part Depth Map U-Net Network

𝐿 × 𝐿 𝐿 × 𝐿 × 2 𝐿 × 𝐿 Fig. 2. Illustrations of the direct mapping approach where the hologram is directly used as the input of the network.

Direct mapping approach from the hologram to the depth map

The first learning-based approach that was considered, is a direct mapping denoted by G 1 from the recorded hologram 𝐻 to the focused depth 𝑑 ∈ R 𝐿×𝐿 , such that

𝑑 = G 1 {𝐻}, 𝐻 ∈ C 𝐿×𝐿 (7)
which is a reasonable assumption since the hologram contains both the intensity and depth information of the acquired 3D scene.

In previous works, G 1 is a one-to-one mapping between the recorded hologram 𝐻 and a single acquisition depth 𝑑. In the present work, the scenes are composed of several objects having a different shape and location in 3D space. Therefore, during the training process, the network must first extract from 𝐻 the relevant features to segment the scene parts that are at the same depth and then predict their correct focus distance. Learning G 1 from a scene comprising several objects located at different positions requires a much larger dataset, by varying not only the position but also the shapes of the objects that compose the scene. The slightest change in scene geometry or aberration on the hologram can prevent the network from predicting relevant results. It is therefore necessary to ensure that all input parameters are covered during training, thus guaranteeing relevant results in the testing phase.

Besides the need for a very large number of training samples, a direct mapping approach lacks scalability. Indeed, the computational complexity for network training and inference increases linearly with the input resolution.

In order to design an efficient and scalable direct approach method, it is necessary to maintain a fixed spatial resolution for the network input. Therefore, for a high-resolution input, the network training and inference should be performed locally on different sub-parts of the initial input and then merged together to obtain the final results. Training the network locally greatly reduces the computational cost and GPU usage, and allows for a greater generalization ability, because the structural changes that can be observed in a local area are limited and can therefore easily be learned. However, since there is no spatial localization in the hologram plane, it is not possible to segment the hologram into independent sub-parts for network training and inference. Indeed, each pixel of the targeted depth map depends on every pixel of the hologram.

For this purpose, a second learning-based method that combines the benefits of the classic DFF method and the direct approach while alleviating their drawbacks is presented in the next section.

Patch-based approach

As stated previously, to design an efficient learning-based approach, it is important to perform the network training and inference locally using a fixed and low input/output resolution, allowing a faster and more scalable model with good generalization abilities. To design such an approach, the hologram cannot be directly segmented due to the lack of spatial localization of the scene objects in the hologram plane. Therefore, the hologram is first converted into a reconstruction volume to retrieve the spatial locations of the scene objects in the reconstruction images, then a mapping is learned between the cropped reconstruction volumes and the corresponding targeted depth map as illustrated in Figure 3. The proposed method has some similarities with the DFF method. First, for both approaches the same reconstruction volume Ω given in Eq. 3 is used. Secondly, as for the DFF method, the goal of the neural network is to estimate for each pixel the depth for which the focus is locally optimal. The major difference between the two approaches is the used 𝐹 𝑀 function in Eq. 6, as detailed in the following.

In Eq. 6 the FM function can only evaluate the focus level of the given patch and does not provide which part of the patch is in focus. Therefore, in order to compute a complete depth map, the inference operation must be performed locally for each pixel of the reconstruction volume, using a patch centered around each pixel with an optimal patch size to guarantee a single depth per patch. Lets consider the case where the 𝐹 𝑀 function is replaced with a learned network denoted by G 2 . In that case, the network does not only evaluate the level of focus but also segments the in-focus and out-of-focus regions of the given patch, as illustrated in Figure 4. Consequently, the inference operation can be performed using disjoint patches, and the targeted depth map can be inferred from the reconstruction distance used for each segmented region. Thus, the number of operations required to compute a complete depth map is reduced from 𝐿 × 𝐿 × 𝑁 operations to 𝐿 𝑠 × 𝐿 𝑠 × 𝑁 operations plus additional gains from GPU parallelization. The decomposition into a set of non-overlapping patches can be formulated as:

Θ = Λ 𝑝,𝑞,𝑖 𝑝 ∈ 1, 𝑊 𝑠 , 𝑞 ∈ 1, 𝐻 𝑠 , (8)
Λ 𝑝,𝑞,𝑖 = P 𝑧 𝑖 {𝐻} ( 𝑝𝑠 + 𝑢, 𝑞𝑠 + 𝑣), 𝑢 ∈ [0, 𝑠 -1], 𝑣 ∈ [0, 𝑠 -1], 𝑖 ∈ [1, 𝑁] (9)
where ⌊𝑎⌋ is the greater integer value smaller than or equal to 𝑎, 𝐻 and 𝑊 are respectively the height and width of the reconstruction volume, and 𝑠 in the patch size. Given this new decomposition, Eq. 6 can be reformulated as follows:

Î𝑖 = G 2 {Λ 𝑝,𝑞,𝑖 }, (10)
d 𝑝,𝑞 = 𝑁 ∑ 𝑖=1 𝑖 × [ Î𝑖 ] ( 11
)
where Î𝑖 is the predicted in-focus map corresponding to the holographic reconstruction performed at the distance 𝑧 𝑖 , [ Î𝑖 ] is the rounded version of Î𝑖 , and d 𝑝,𝑞 is the predicted depth map.

In order to train the network G 2 , two problems arise. First, it is necessary to sample a large number of patches with a positive sample that includes focused regions with different textures, positions and shapes, as well as negative samples without focused regions. Secondly, to guarantee the occlusion condition, only one maximum focus value can be allowed per pixel. If the different images that compose the cropped volume {Λ 𝑝,𝑞,𝑖 | 𝑖 ∈ [1, 𝑁]} are used independently, this can lead to several extreme values per pixel and therefore to uncertainty when choosing the optimal depth value.

To alleviate these problems, all the images of the cropped reconstruction volume are fed to the network as unique batch, and the network is supervised using cross-entropy loss to predict a single in-focus depth per pixel:

L = ∑ 𝑢,𝑣 𝑁 ∑ 𝑖=1 1 {𝑑 (𝑢,𝑣)=𝑧 𝑖 } • 𝑙𝑜𝑔 Î𝑖 (𝑢, 𝑣) (12)
After the network training is complete, the predicted depth per pixel can be extracted according to

d 𝑝,𝑞 (𝑢, 𝑣) = arg max 𝑖 ∈ [1, 𝑁 ] Î𝑖 (𝑢, 𝑣) (13)
and the associated color intensity value as, Î 𝑚 𝑝,𝑞 (𝑢, 𝑣) = Λ 𝑝,𝑞, d𝑝,𝑞 (𝑢,𝑣) (𝑢, 𝑣)

In cases of occlusion where the focus has multiple peaks, using the k-th maximum value in Equation 13may not be reliable. This is because the network is designed to identify a single optimal focus plane for each input reconstruction volume, which results in a focus curve with a single peak depth value. To address this issue, the network can be applied to reconstruction volumes computed using a sliding window over the reconstruction interval. This approach allows for the extraction of peak values incrementally, rather than relying on a single peak value from the entire reconstruction volume. It is important to ensure that the extracted peaks pass a certain threshold to ensure reliable results.

The obtained RGB-D image ( Î 𝑚, d) provides a comprehensive representation of the scene geometry through the use of holographic reconstructions. This is useful information for many applications, such as holographic video compression where the patch-based method can be used on consecutive holographic video frames to extract their RGB-D representation. Bi-dimensional motion vectors on the 𝑋𝑌 axis can be estimated using optical flow techniques on the RGB data, and additional motion along the 𝑍 axis can be estimated using the depth information. These motion vectors can be incorporated into a compression framework to improve the efficiency and quality of the encoded holographic video data.

Requiring the full cropped volume for efficient network training may limit the used patch size and the number of reconstructions. It is, therefore, necessary to have a trade-off between the used patch size and the number of holographic reconstructions in order to avoid high GPU usage. In this new formulation, the problem is well-posed due to the out-of-focus behavior of CGH. When modifying the reconstruction distance on CGH, only the parts with a depth at the reconstruction distance are sharp while the remaining regions are contaminated by speckle noise as shown in Figure 5. Therefore, the detection of focused regions is similar to the removal of the noisiest regions.

Network Architecture

The functions G 1,2 are implemented as CNN network, based on the U-Net architecture [START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF] and is shown in Figure 6. The network is composed of two sub-networks, a contracting path and an expansive path. The network can be mathematically described as

𝑋 𝑖, 𝑗 =        D (H (𝑋 𝑖-1, 𝑗 )), 𝑗 = 0 H ( 𝑋 𝑖,0 , U (𝑋 𝑖+1, 𝑗 -1 ) ), 𝑗 = 1 , ( 15
)
where 𝑋 𝑖, 𝑗 is the initial network input, H is a convolution block shown on the right side of concatenation operator. The depth resolution for the contraction path is [32,64,128,256,512] and [256,128,64,32,[START_REF] Gabor | A new microscopic principle[END_REF] for the expansive path. The segmentation network shares the same architecture but uses half the depth resolution in order to avoid over-fitting. The skip-link connections play an important role, especially on holograms that contain a lot of redundant information, since each light wave scattered by each scene point contributes to every pixels during hologram recording. With the skip-link connection, at each stage of the decoder, the network can use both the local and global features extracted from the hologram, therefore, the depth value at each pixel is learnt using a greater amount of information than using a standard auto-encoder.

Experimental results

Experimental Setup

To train the neural networks, a large-scale dataset composed of 4200 RGB holograms obtained from three different scenes (Piano, Table, Woods shown in Figure 10) has been acquired using a layer-based method [START_REF] Gilles | Hybrid approach for fast occlusion processing in computer-generated hologram calculation[END_REF], with a resolution of 1024 × 1024, a pixel pitch of 6𝜇𝑚 and 1400 acquisition angles. For the evaluation phase, two sets are used: a test set composed of 300 holograms computed using 100 additional acquisition angles for each training scene (Piano, Table, Woods), and a validation set composed of 200 holograms recorded from two unseen scenes (Cars, Dices) during the training phase.

The two networks for the direct mapping and patch-based approaches were trained and tested on the same training and validation sets. For direct mapping, the real and imaginary parts of the RGB channels are stacked together and fed directly to the network. The network training is supervised using the ℓ 1 norm with an additional spatial gradient loss to alleviate the blurred effect produced by the CNN. More formally

L = 1 𝑁 𝑁 ∑ 𝑖=1 𝑑 -d 1 + 𝑁 ∑ 𝑖=1 (∇ 𝑥 𝑑 -d 1 + ∇ 𝑦 𝑑 -d 1 ) , (16)
where 𝑁 𝑥 , 𝑁 𝑦 are the height and width of the depth map, and 𝑑 and d are the ground truth and estimated depth maps respectively. In the patch-based approach, a set of holographic reconstructions have been computed at unique depths of the ground truth depth map for each hologram. The resulting reconstruction volume is decomposed spatially into non-overlapping patches. All the patches extracted at the same spatial position are fed into the network, which is supervised using the cross-entropy loss to predict the correct in-focus maps.

The two networks have been trained for 200 epochs using the Adam optimizer and learning decay with a gamma factor of 0.8 every 10 epochs and a starting learning rate of 0.1. The networks are compared to nine commonly used focus operators listed in Table 1

Results

Table 2 reports the obtained results on the validation and test sets. First, we observe that for learning-based approaches, only the patch-based model maintains its performance on both the validation set (Piano, and tends to smooth the final results given an average estimated value of all pixels belonging to the scene. The patch-based approach maintains its performance on the test set, due to the fact that the different patterns that are observed during the test phase are similar to those observed during the training phase. In the case where the observed patterns differ greatly from the training phase, the problem being well-posed thanks to the property of the CGH, the network gives fairly consistent results.

For the DFF approach, there is no operator that maintains its performances for the different scenes; the performance of the operators will indeed depend on the scene geometry. For scenes with occlusions, some objects will be positioned one after the other, resulting in several levels of focus on the patches extracted at the objects' borders; in this case, predicting a single depth for the central pixel results in poor performance. The ratio between the textured and non-textured patches also plays a role on the final performance, some operators will perform better on highly textured areas but will fail on flat areas without texture. Overall, the Laplacian variance and Wavelet-based approaches are the best-performing operators. Figures 7,[START_REF] Memmolo | Automatic focusing in digital holography and its application to stretched holograms[END_REF][START_REF] Sheridan | Roadmap on holography[END_REF][START_REF] Pitkäaho | Performance of autofocus capability of deep convolutional neural networks in digital holographic microscopy[END_REF] give some visual results. From these figures, we observe that the obtained results with the direct mapping are smooth and fail to deal with the occlusions correctly. Indeed, when two objects are one behind the other, their borders are smooth, resulting in the integration of the first object into the second. In addition, while the LAPV operator yields a reasonable performance, it fails in regions with high structural variability. When a patch is extracted over a region where the depth changes rapidly, the patch has a high probability of containing multiple levels of focus, so predicting a single depth for the entire patch will lead to poor performance. The patch-based approach is well suited to this type of scenario. Segmenting locally for each holographic reconstruction only the parts that are in focus, allows multi-level focus processing within the same patch, while maintaining consistent edges and contours between the different extracted regions.

In summary, the proposed patch-based approach is more efficient than the classic DFF method as it can use a single batch inference to determine the depth map for each crop of the reconstruction volume. Additionally, it is more accurate in segmenting the different sub-regions of the cropped reconstruction volume, and does not require further post-processing or adaptive per-pixel patch sizes to maintain depth continuity in the final depth map. The network is trained to identify the relevant characteristics of an in-focus image using fixed-size input patches and properly segment it. Furthermore, the network ensures that the focus level of each pixel follows a perfect Gaussian distribution, with peaks corresponding to the optimal focus, and is therefore not subject to polarity changes when the patch size or reconstruction range is too large. The direct CNN approach has a faster computation time (5.64 seconds per hologram), but it lacks the ability to generalize well compared to the patch-based approach (1.31 seconds per holographic reconstruction and 26.05 seconds for processing a reconstruction volume composed of 256 holographic reconstruction). Both approaches were trained and tested using the same dataset, so this difference in generalization ability cannot be attributed to differences in the quality or quantity of the data. Instead, it is likely due to the different approaches to feature extraction and problem formulation. The direct CNN approach is ill-posed, which causes the network to overfit the data and extract features that are specific to the training set, rather than general features that can be applied to new, unseen data. As a result, the network performs well on the training set but struggles to make accurate predictions on new data because it has not learned to generalize effectively. In contrast, the patch-based approach is able to extract more generalizable features, resulting in better generalization ability and the ability to make accurate predictions on new data.

In addition to generated data, the patch-based approach was applied to some of the IRT-bcom dataset [START_REF] Gilles | Computer generated hologram from multiview-plus-depth data considering specular reflections[END_REF], which was generated using alternative hologram generation techniques (as shown in the Figure 9). The obtained results indicate that the network was able to accurately extract the correct depth value, despite some parts of the depth map being poorly estimated. This is due to the smaller number of reconstruction distances samples leading to the sampled reconstruction distances being too far from the optimal focus planes.

Conclusion

In this paper, we propose two different learning-based approaches for depth estimation from CGH and compare their performances to classical DFF methods. In the first direct mapping approach, the network takes the hologram as input and it is supervised to predict the correct scene depth map. In the second more localized approach, the network takes as input a cropped reconstruction volume and is supervised to predict a local depth map.

Overall, the patch-based approach gives the most accurate depth prediction results. It enables to segment only the parts that are in focus on each holographic reconstruction, allowing multi-level focus processing within the same patch, while maintaining consistent edges and contours between the different extracted regions.

Although the proposed approach is more efficient than the classic DFF method and the direct mapping approach, it has some limitations. Firstly, the GPU consumption increases linearly with the number of holographic reconstructions and the patch size. As the accuracy of the estimated depth is directly related to the number of holographic reconstructions, a trade-off between accuracy and computational complexity must be considered. Similarly, the robustness of the depth map is directly related to the patch size: a larger patch size will provide a more The network was used on a holographic reconstruction volume consisting of 256 holographic reconstructions performed at the uniformly sampled distance between the maximum and minimum depth values provided in each hologram metadata. The results indicated that the network was able to accurately extract depth information, but some depth map regions were estimated poorly. These areas could potentially be improved by implementing more precise holographic distance sampling.

accurate estimation, but at the cost of increased computational complexity. Secondly, the network performance is weaker along the edges of the patches, which can lead to discontinuities in the final depth map. Additionally, like DFF methods, the resulting depth map requires the calculation of a binary mask to distinguish between foreground and background objects, which adds additional computational costs. Future research will focus on investigating novel learning-based techniques that directly learn a transformation in an intermediate space from the hologram where the scene objects are spatially localized and then, in a subsequent step, predict their depths.

Funding. Agence Nationale de la Recherche (ANR-A0-AIRT-07).

Disclosures. The authors declare no conflicts of interest.

Fig. 3 .

 3 Fig.3. Illustrations of the patch-based approach where a reconstruction volume is first constructed from the hologram, then the network is fed with a cropped reconstruction volume to predict the regional depth map.

Fig. 4 .

 4 Fig. 4. First row: input reconstruction volume {Λ 𝑝,𝑞,𝑖 | 𝑖 ∈ [1, 𝑁]}. Second row: ground truth focus maps. Third row: predicted focus maps { Î𝑖 | 𝑖 ∈ [1, 𝑁]}.

Figure 6 ,Fig. 5 . 1 Fig. 6 .

 6516 Fig. 5. Illustration of the predicted in-focus regions on validation sample. (a) Amplitude of holographic reconstruction at a given distance. (b) The ground truth in-focus regions. (c) The predicted in-focus scores.

Fig. 7 .

 7 Fig. 7. Each row contains a sample taken from our datasets. (a) The ground truth all in-focus image. (b) The depth map estimated using the LAPV operator. (c) The depth map estimated using direct mapping. (d) The depth map estimated using the proposed approach.

Fig. 8 .

 8 Fig.8. Illustration of pointclouds predicted using the patch-based method on samples from the test set. On the right the ground truth points cloud and on the left the prediction using the proposed approach.

Fig. 9 .

 9 Fig. 9. Depth ((b) and (d)) and focus maps ((a) and (c)) extracted using the patch-based method on samples (Breakdancers1080p and Ballet1080p) from the IRT-bcom dataset.The network was used on a holographic reconstruction volume consisting of 256 holographic reconstructions performed at the uniformly sampled distance between the maximum and minimum depth values provided in each hologram metadata. The results indicated that the network was able to accurately extract depth information, but some depth map regions were estimated poorly. These areas could potentially be improved by implementing more precise holographic distance sampling.

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Each row contains a sample taken from our datasets. (a) The ground truth all in-focus image. (b) The ground truth depth map. (c) The depth map estimated using direct mapping. (d) The depth map estimated using the proposed approach.

Table 1 .

 1 . The operators are applied on windows of size 17 × 17 which gave the best results on previous experiments. Abbreviations of focus measure operators used in experiments.

	Focus operator	Abbr.	Focus operator	Abbr.
	Energy of Laplacian [21]	LAPE	Modified Laplacian [22]	LAPM
	Variance of Laplacian [23]	LAPV	Diagonal Laplacian [24]	LAPD
	Variance of wavelet coefficients [25] WAVV Ratio of the wavelet coefficients [25] WAVR
	Graylevel variance [26]	GLVA	Normalized Graylevel variance [27] GLVN
	Image contrast [28]	CONT

Table 2 .

 2 Table, Cars) and test set (Dices, Cars). The direct mapping approach results in close performance between the training set and the validation set but fails to generalize on the test set. The lack of generalization can be explained by the problem formulation which is not localized. During training, the network learns from a set of training data that covers only a part of all possible scene shapes and positions. Therefore, for test scenes with sizes, shapes, and spatial positions unseen during training, the direct mapping network fails to give relevant results Piano Table Woods Dices Cars L1 error using different approaches for validation and test sets.

	Direct approach
	U-Net	6.94	8.48	8.62	19.93 23.92
	Patch-based approach
	U-Net	0.66	1.12	0.73	1.21	3.48
	Laplacian-based operators
	LAPE	11.58 25.81 16.35	5.83 32.56
	LAPM	18.89 16.48 23.158 27.0 51.26
	LAPV	4.12	9.28	3.82	2.84 17.07
	LAPD	23.88 19.93 26.81 26.81 54.16
	Wavelet-based operators
	WAVV	9.95 11.24	7.19	13.26 23.88
	WAVR	8.16	8.98	7.45	8.76 14.44
	Miscellaneous operators
	GLVN	17.90 15.10 20.70 25.75 51.65
	GLVA	21.40 17.10 19.38 24.12 51.62
	CONT	16.33 14.89 19.51 21.30 47.13

Data availability.

No data were generated or analyzed in the presented research.