Satish Bodakuntla
email: satish.bodakuntla@curie.frjijumon.sn@curie.fr

A S Jijumon

Carsten Janke
email: carsten.janke@curie.fr

Maria M Magiera
email: maria.magiera@curie.fr

Purification of Tubulin with Controlled Posttranslational Modifications and Isotypes from Limited Sources by Polymerization-Depolymerization Cycles

Keywords: Microtubules, in vitro reconstitution, tubulin, tubulin purification, polymerization, microtubule dynamics, tubulin code, tubulin isotypes, suspension culture, tubulin posttranslational modifications

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approach yields pure, polymerization-competent tubulin, as tubulin resistant to polymerization or depolymerization is discarded during the successive purification steps. We describe the purification of tubulin from cell lines, grown either in suspension or as adherent cultures, and from single mouse brains. The method first describes the generation of cell mass in both suspension and adherent settings, the lysis step, followed by the successive stages of tubulin purification by polymerization-depolymerization cycles. Our method yields tubulin that can be used in experiments addressing the impact of the tubulin code on the intrinsic properties of microtubules and microtubule interactions with associated proteins.

INTRODUCTION:

Microtubules play critical roles in many cellular processes. They give cells their shape, build meiotic and mitotic spindles for chromosome segregation, and serve as tracks for intracellular transport. To perform these diverse functions, microtubules organize themselves in different ways. One of the intriguing questions in the field is to understand the molecular mechanisms that allow the structurally and evolutionarily conserved microtubules to adapt to this plethora of organizations and functions. One potential mechanism is the diversification of microtubules, which is defined by the concept known as the 'tubulin code' [START_REF] Verhey | The tubulin code[END_REF][START_REF] Janke | The tubulin code: Molecular components, readout mechanisms, and functions[END_REF][START_REF] Janke | The tubulin code and its role in controlling microtubule properties and functions[END_REF] . The tubulin code includes two principal components: differential incorporation of aand b-tubulin gene products (tubulin isotypes) into the microtubules and tubulin posttranslational modifications (PTMs). Since the 1970s, in vitro reconstitution experiments, combined with evolving light microscopy techniques, have paved the way for important discoveries about the properties of microtubules: dynamic instability [START_REF] Mitchison | Dynamic instability of microtubule growth[END_REF] and treadmilling [START_REF] Margolis | Opposite end assembly and disassembly of microtubules at steady state in vitro[END_REF] , and their other mechanisms and functions [START_REF] Borisy | Nucleated assembly of microtubules in porcine brain extracts[END_REF][START_REF] Kirschner | The mechanism of microtubule assembly in vitro[END_REF][START_REF] Baas | Hooks and comets: The story of microtubule polarity orientation in the neuron[END_REF][START_REF] Stepanova | Visualization of microtubule growth in cultured neurons via the use of EB3-GFP (end-binding protein 3-green fluorescent protein)[END_REF][START_REF] Nedelec | Self-organization of microtubules and motors[END_REF][START_REF] Bieling | A minimal midzone protein module controls formation and length of antiparallel microtubule overlaps[END_REF][START_REF] Roostalu | Directional switching of the kinesin Cin8 through motor coupling[END_REF][START_REF] Schaedel | Microtubules self-repair in response to mechanical stress[END_REF][START_REF] Hendricks | Reconstituting the motility of isolated intracellular cargoes[END_REF][START_REF] Dogterom | Microtubule organization in vitro[END_REF] . Almost all the in vitro experiments performed so far have been based on tubulin purified from brain tissue using repeated cycles of polymerization and depolymerization [START_REF] Vallee | Reversible assembly purification of microtubules without assemblypromoting agents and further purification of tubulin, microtubule-associated proteins, and MAP fragments[END_REF][START_REF] Castoldi | Purification of brain tubulin through two cycles of polymerization-depolymerization in a high-molarity buffer[END_REF] . Although purification from the brain tissue confers the advantage of obtaining high-quality tubulin in large quantities (usually gram amounts), one important drawback is the heterogeneity as tubulin purified from brain tissue is a mixture of different tubulin isotypes and is enriched with many tubulin PTMs. This heterogeneity makes it impossible to delineate the role of a particular tubulin PTM or isotype in the control of microtubule properties and functions. Thus, producing assembly-competent tubulin with controlled tubulin PTMs and homogenous isotype composition is essential to address the molecular mechanisms of the tubulin code. Recently, an approach to purify tubulin by affinity chromatography using the microtubulebinding TOG (tumor-overexpressed gene) domain of yeast Stu2p has been developed [START_REF] Widlund | One-step purification of assembly-competent tubulin from diverse eukaryotic sources[END_REF] . In this method, tubulin in crude lysates of cells or tissue is passed through a column where it binds to the matrix-immobilized TOG domain, which allows the analysis of the whole tubulin pool of a given, even very small, sample. A long-awaited approach to purify recombinant tubulin has also been described in recent years. It is based on the baculovirus system, in which a bicistronic vector containing aand a b-tubulin genes is expressed in insect cells [START_REF] Minoura | Overexpression, purification, and functional analysis of recombinant human tubulin dimer[END_REF] . However, this method is very cumbersome and time-consuming and is therefore mostly used for studying the impact of tubulin mutations [START_REF] Uchimura | A flipped ion pair at the dynein-microtubule interface is critical for dynein motility and ATPase activation[END_REF] and tubulin isotypes [START_REF] Pamula | The structured core of human beta tubulin confers isotype-specific polymerization properties[END_REF][22][START_REF] Ti | Human beta-tubulin isotypes can regulate microtubule protofilament number and stability[END_REF] in vitro.

In the current protocol, we describe a method that uses the well-established and widely used polymerization-depolymerization approach as a blueprint to generate tubulin with different levels of modification either from cell lines or from mouse brain tissue [START_REF] Souphron | Purification of tubulin with controlled post-translational modifications by polymerization-depolymerization cycles[END_REF] . In this procedure, tubulin is cycled between the soluble (tubulin dimer at 4 °C) and polymerized form (microtubule at 30 °C in the presence of guanosine 5'-triphosphate [GTP]). Each form is separated through successive steps of centrifugation: tubulin dimers will remain in the supernatant after a cold (4 °C) spin, whereas microtubules will be pelleted at 30 °C. Furthermore, one polymerization step is carried out at high piperazine-N,Nʹ-bis(2ethanesulfonic acid (PIPES) concentration, which allows the removal of microtubuleassociated proteins from the microtubules and thus, from the finally purified tubulin. Tubulin purified from HeLa S3 cells grown as suspension or adherent cultures is virtually free of any tubulin PTM and has been used in recent in vitro reconstitution experiments [START_REF] Barisic | Microtubule detyrosination guides chromosomes during mitosis[END_REF][START_REF] Nirschl | alpha-Tubulin tyrosination and CLIP-170 phosphorylation regulate the initiation of dynein-driven transport in neurons[END_REF][START_REF] Guedes-Dias | Kinesin-3 responds to local microtubule dynamics to target synaptic cargo delivery to the presynapse[END_REF][START_REF] Luo | Direct observation of dynamic protein interactions involving human microtubules using solid-state NMR spectroscopy[END_REF] . We have further adapted the method to purify tubulin from single mouse brains, which can be used for a large number of mouse models with changes in tubulin isotypes and PTMs. In the protocol, we first describe the generation of the source material (cell mass or brain tissue), its lysis (Figure 1A), followed by the successive steps of tubulin polymerization and depolymerization to purify the tubulin (Figure 1B). We further describe the process to assess the purity (Figure 2A,B) and quantity (Figure 3A,B) of the purified tubulin. The method can be adapted to produce tubulin enriched with a selected PTM by overexpressing a modifying enzyme in cells prior to tubulin purification (Figure 4B). Alternatively, tubulin-modifying enzymes can be added to tubulin during the purification process. Finally, we can purify tubulin lacking specific isotypes or PTMs from the brains of mice deficient in the corresponding tubulin-modifying enzymes (Figure 4B) [START_REF] Even | ATAT1-enriched vesicles promote microtubule acetylation via axonal transport[END_REF] . The method we describe here has two main advantages: (i) it allows the production of sufficiently large amounts of tubulin in a relatively short time, and (ii) it generates high-quality, pure tubulin, with either specific tubulin isotype composition or PTMs. In the associated video of this manuscript, we highlight some of the critical steps involved in this procedure.

PROTOCOL:

Animal care and use for this study were performed in accordance with the recommendations of the European Community (2010/63/UE). Experimental procedures were specifically approved by the ethics committee of the Institut Curie CEEA-IC #118 (authorization no. 04395.03 given by National Authority) in compliance with the international guidelines.

1.

Preparation of Reagents for Tubulin Purification NOTE: All the buffers used for tubulin purification should contain potassium salts and NOT sodium salts [START_REF] Wolff | Cation selective promotion of tubulin polymerization by alkali metal chlorides[END_REF] . 1.1. Prepare 1 L of complete medium: Dulbecco's modified Eagle medium (DMEM), with 10% fetal bovine serum (FBS, 100 mL), 200 mM L-glutamine (10 mL of 2 M stock), and 1x penicillin-streptomycin (10 mL of 100x stock). Store at 4 °C. 1.2. Prepare 10 M potassium hydroxide (KOH) by dissolving 140 g of KOH in water, adjust the final volume to 250 mL, and store at room temperature. 1.3. Prepare 0.5 M ethylenediamine tetraacetic acid (EDTA), pH 8, by dissolving 36.5 g of EDTA in water, adjust the pH to 8.0 using KOH (otherwise EDTA will not dissolve) and the final volume to 250 mL, filter-sterilize, and store at room temperature. 1.4. Prepare 5 mM phosphate-buffered saline (PBS)-EDTA by adding 5 mL of 0.5 M EDTA to 500 mL of PBS, filter-sterilize, and store at room temperature. 1.5. Prepare 0.5 M K-PIPES, pH 6.8, by dissolving 75.5 g of PIPES in water, adjust to pH 6.8 with KOH (otherwise PIPES will not dissolve) and the final volume to 500 mL, filter-sterilize, and store at 4 °C. 1.6. Prepare 1 M K-PIPES, pH 6.8, by dissolving 15.1 g of PIPES in water, adjust to pH 6.8 with KOH and the final volume to 50 mL, filter-sterilize, and store at 4 °C. 1.7. Prepare 0.5 M potassium-ethylene glycol-bis(β-aminoethyl ether)-N,N,Nʹ,Nʹtetraacetic acid (K-EGTA, pH 7.7) by dissolving 47.5 g of EGTA in water, adjust to pH 7.7 with KOH and the final volume to 250 mL, filter-sterilize, and store at room temperature. 1.8. Prepare BRB80 (80 mM K-PIPES, pH 6.8; 1 mM K-EGTA; 1 mM magnesium chloride [MgCl2)) solution by mixing 3.2 mL of 0.5 M PIPES, 40 µL of 0.5 M K-EGTA, and 20 µL of 1 M MgCl2 and adjust the final volume to 20 mL. Store at 4 °C. 1.9. Prepare 0.1 M phenylmethanesulfonyl fluoride (PMSF) by dissolving 435 mg of PMSF in isopropanol to obtain a final volume of 25 mL and store at -20 °C. 1.10. Prepare protease inhibitors mix (200x) by dissolving 10 mg of aprotinin, 10 mg of leupeptin, and 10 mg of 4-(2-aminoethyl)-benzenesulfonyl fluoride in water to obtain a final volume of 2.5 mL, make aliquots of 100 µL, and store at -20 °C. 1.11. Prepare 10% Triton X-100 by mixing 5 mL of Triton X-100 in 45 mL of water, filtersterilize, and store at room temperature. 1.12. Prepare Lysis buffer (BRB80 supplemented with 1 mM 2-mercaptoethanol, 1 mM PMSF, 1x protease inhibitors mix and, optionally for HEK-293 cells, 0.2% Triton X-100) on the day of tubulin purification by mixing 20 mL of BRB80 with 1.5 µL of 2-mercaptoethanol, 200 µL of 0.1 M PMSF, 100 µL of the protease inhibitors mix and, optionally for HEK-293 cells, 400 µL of 10% Triton X-100. NOTE: 2-mercaptoethanol is toxic and should be used in the fume-hood. 1.13. Prepare 0.2 M GTP by dissolving 1 g of GTP in 9.5 mL of water, adjust the pH to 7.5 using KOH, make aliquots of 20 µL, and store at -20 °C. Avoid repeated freeze-thaw cycles.

1.14. Prepare 1 M tris(hydroxymethyl) aminomethane-hydrochloride (Tris-HCl) by dissolving 60.56 g of Tris in water, adjust to pH 6.8 with HCl, complete to a final volume of 500 mL, filtersterilize, and store at room temperature. 1.15. Prepare 5x Laemmli sample buffer (450 mM dithiothreitol (DTT); 10% sodium dodecylsulfate (SDS); 400 µM Tris-HCl, pH 6.8; 50% glycerol; ~0.006% bromophenol blue) by adding 4 g of SDS to 16 mL of preheated 1 M Tris-HCl, pH 6.8, and mix the solution gently. Add 2.6 g of DTT and 20 mL of 100% glycerol to the mix and stir until the solution becomes homogenous. Add the desired amount of bromophenol blue (2.5 mg) to reach the required color intensity. Make 5 mL aliquots and store at -20 °C. Prepare the 2x working solution of Laemmli sample buffer by diluting the 5x stock in distilled water.

2.

Amplification and harvesting sources of tubulin NOTE: In this protocol, three sources of tubulin were used: (i) cells (HeLa S3 and HEK-293) grown as suspension cultures; (ii) cells grown as adherent cultures (HEK-293, HeLa, and U-2 OS); and (iii) mouse brain tissue. This protocol considers the day of tubulin purification as 'day 0' and accordingly, other steps have been described relative to day 0. 2.1. Amplification of Cells 2.1.1. Cells Grown as Suspension Cultures NOTE: To successfully purify tubulin from suspension cultures, use at least 2 L of suspension culture. 2.1.1.1.

For 2 L of suspension culture, revive and grow the preferred cell type to obtain 6 ´ 10 7 cells 10 days before the day of preparation. On day -10, plate cells on six 15 cm diameter dishes at 10 7 cells per plate. 2.1.1.2.

On day -8, preheat the required amount of complete medium to 37 °C. Add 1 L of pre-heated medium to each spinner bottle under sterile conditions. Place the spinners on a stirrer table set at 20-25 rpm inside the cell culture incubator, slightly open the lateral spinner caps to allow the medium to equilibrate to the incubator's atmosphere. NOTE: To avoid any contamination, thoroughly clean the outer surface of the media and spinner bottles using 70% ethanol. 2.1.1.3.

On day -7, trypsinize and collect the cells grown to 80-90% confluence (approximately 1.8 ´ 10 8 cells). Collect cells from 3 dishes at a time, spin down (200 ´ g, 5 min, room temperature), and re-suspend all cells in 10 mL of DMEM. NOTE: Thorough dissociation of the cells at this point is very important to avoid the formation of larger aggregates in the spinner bottles, which affects cell survival and results in low tubulin yield.

2.1.1.4.

Add 5 mL of the cell suspension to each spinner bottle containing 1 L of DMEM, return the spinners to the stirrer table in the cell culture incubator, and allow cells to grow for one week. 2.1.2. Cells Grown as Adherent Cultures NOTE: To successfully purify tubulin from adherent cells, use a minimum of 10 dishes of 80-90% confluence. 2.1.2.1.

Revive and amplify the desired cell type to obtain 1 ´ 10 8 cells three days before the day of the tubulin preparation. 2.1.2.2.

On day -3, plate these cells on ten 15 cm dishes at 1 ´ 10 7 cells per dish and allow them to grow 80-90% confluence. 2.1.2.3.

On day -1, if required, transfect cells with a plasmid to express a tubulinmodifying enzyme or a particular tubulin isotype. 2.2. Harvesting the Cells/Brain Tissue 2.2.1. Cells Grown as Suspension Cultures 2.2.1.1.

Transfer the cell suspension from spinners into 1 L centrifuge bottles (Table of Materials) and pellet cells at 250 ´ g, 15 min, room temperature. For immediately starting another culture of HeLa S3 cells in the spinner bottles, leave 100 mL of cell suspension in the spinners, and add 1 L of complete, pre-heated DMEM to the spinner bottle. NOTE: Carefully check for bacterial contamination before proceeding for tubulin purification.

2.2.1.2.

Resuspend pelleted cells from each centrifuge bottle in 10 mL of ice-cold PBS, and transfer all the cells into 50 mL screw-cap tubes. During re-suspension, keep the cells on ice. Pellet the cells at 250 ´ g, 15 min, 4 °C. NOTE: Follow recommendations for spinner bottle cleaning and storage (see Table of materials).

2.2.1.3.

Discard the supernatant and determine the volume of the cell pellet. From 2 L of suspension culture (two spinner bottles), expect a cell pellet of 5-6 mL. NOTE: In the protocol described below, the cell pellet volume is assumed to be 10 mL. Adjust the experiment according to the pellet volumes. 2.2.1.4.

Add 1 volume (10 mL) of lysis buffer and re-suspend the cell pellet. NOTE: The ratio of cell pellet volume to lysis buffer volume is very important for successful tubulin purification. Adding more lysis buffer decreases tubulin concentration, which then fails to reach the critical concentration needed for polymerization, thus greatly reducing the tubulin yield. NOTE: Cells resuspended in lysis buffer can be snap-frozen in liquid nitrogen and stored at -80 °C for two months. 2.2.2. Cells Grown as Adherent Cultures NOTE: Cells from adherent cultures must be harvested very quickly for successful tubulin purification (approximately 15 mins for harvesting ten 15 cm dishes). Three people participated in this step of the protocol. 2.2.2.1.

Remove the medium from 15 cm dishes by inclining the dishes, and then gently wash the cells with 7 mL of PBS-EDTA at room temperature (person 1). Work only with three 15 cm dishes at a time to avoid leaving the cells without medium or buffer. 2.2.2.2.

Add 5 mL of PBS-EDTA to the cells and incubate them for 5 min at room temperature.

2.2.2.3.

Use a cell lifter to gently detach the cells by shoveling them to one edge of the dish (person 2), and collect all the cells in a 50 mL screw-cap tube (person 3). Rinse each plate with an additional 2 mL of PBS-EDTA to collect any remaining cells from the dishes. During this step, keep the 50 mL screw-cap tube containing the cell suspension on ice.

2.2.2.4.

Pellet the cells at 250 ´ g, 10 min, 4 °C. Discard the supernatant and determine the volume of the cell pellet. Expect a volume of ~1 mL from ten 15 cm dishes. NOTE: In the protocol described below, the cell pellet volume is assumed to be 10 mL. Adjust the experiments according to the pellet volumes. 2.2.2.5.

Resuspend the cells in 1 volume (10 mL) of lysis buffer. NOTE: Cells resuspended in lysis buffer can be stored at -80 °C for up to two months. 2.2.3. Brain Tissue NOTE: Mice of any age, sex, or genetic background can be used. The choice of the transgenic mouse strain will depend on the scientific question to be addressed. In this manuscript, we show the example of tubulin purified from the ttll1 -/-mouse, lacking a major brain glutamylating enzyme, the tubulin tyrosine ligase-like 1 (TTLL1) protein [START_REF] Magiera | Excessive tubulin polyglutamylation causes neurodegeneration and perturbs neuronal transport[END_REF] .

2.2.3.1.

Sacrifice the mouse by cervical dislocation, quickly decapitate, and collect the brain into a round-bottom tube. If there is excess blood on the brain, quickly wash with lysis buffer. Collect the brain as soon as the mouse is sacrificed as a post-mortem delay can affect the success of tubulin purification. Use round-bottom tubes to accommodate the width of the probe used for homogenization. NOTE: Collected mouse brains can be snap-frozen in liquid nitrogen and stored at -80 °C for up to 3 years.

2.2.3.2.

Add 500 µL of lysis buffer to a single brain extracted from an adult mouse. For the rest of the protocol, the volume of lysis buffer added is assumed to be 10 mL. Adjust for ytheour experiment according to the number of brains to be used.

3.

Lysis of Cells or Brain Tissue 3.1. Cells Grown as Suspension Cultures 3.1.1. For HEK-293, lyse the cells on ice by repetitively pipetting up and down using pipette tips of different widths. First, attach a p1000 tip to a 10 mL pipette, and pipette the cell suspension up and down every 5 min, for 10 min (three cycles of pipetting). Second, attach a p200 tip to a p1000 tip and further pipette every 5 min, for 20 min (five cycles of pipetting). 3.1.2. For HeLa S3, lyse the cells using a French press (see Table of Materials for settings). 3.1.3. Take 1/100 th volume of the lysis mix (L) (200 µL for 20 mL of L), and add the same volume of 2x Laemmli buffer, boil for 5 min, and store at -20 °C for further analysis. 3.2. Cells Grown as Adherent Cultures 3.2.1. Transfer the cells into a 14 mL round-bottom tube whose height has been reduced to accommodate the sonicator probe (see Table of Materials for settings). Sonicate the cells for ~45 pulses, and confirm cell lysis by sampling a drop of the lysis mix under a microscope. NOTE: The number of pulses could vary according to the cell type used for tubulin purification. Sonicating cells too much could cause tubulin to precipitate and will negatively affect the purification yield. 3.2.2. Pipette the cells up and down on ice every 5 min for 20 min (five cycles of pipetting), using a p200 tip. 3.2.3. Take 1/100 th volume of lysis mix (L) (200 µL for 20 mL of L) and add the same volume of 2x Laemmli buffer, boil for 5 min and store at -20 °C for further analysis. 3.3. Brain Tissue 3.3.1. Lyse the brain tissue using a tissue blender (see Table of Materials for settings). Alternatively, lyse the tissue using a microtube pestle or an equivalent equipment and pipette up and down on ice with a 1 mL syringe with an 18 G needle. 3.3.2. Take 1/100 th volume of lysis mix (L) (200 µL for 20 mL of L), and add the same volume of 2x Laemmli buffer, boil for 5 min, and store at -20 °C for further analysis.

4.

Purification of Tubulin

REPRESENTATIVE RESULTS:

The main goal of this method is to produce high-quality, assembly-competent tubulin in quantities sufficient to perform repeated in vitro experiments with the purified components. Microtubules assembled from this tubulin can be used in reconstitution assays based on the total internal reflection fluorescence (TIRF) microscopy technique with either dynamic or stable microtubules, in experiments testing microtubule dynamics, interactions with MAPs or molecular motors, and force generation by the motors [START_REF] Barisic | Microtubule detyrosination guides chromosomes during mitosis[END_REF] . They can also be used in microtubule-MAP co-pelleting assays and solid-state NMR spectroscopy [START_REF] Luo | Direct observation of dynamic protein interactions involving human microtubules using solid-state NMR spectroscopy[END_REF] .

The enrichment and purity of tubulin throughout the purification process can be monitored by using a Coomassie-stained SDS-polyacrylamide gel electrophoresis (PAGE) gel, preferably the 'TUB' SDS-PAGE gels, that allow for the separation of aand b-tubulins, which co-migrate as a single band in classical gels [START_REF] Lacroix | Generation of differentially polyglutamylated microtubules[END_REF] . Lysates collected at different steps (except for the very last depolymerization, see protocol) are loaded onto the gel in comparable amounts for assessing the success of tubulin purification (Figure 2A) [START_REF] Souphron | Purification of tubulin with controlled post-translational modifications by polymerization-depolymerization cycles[END_REF] . The final tubulin sample, which is very precious, is only loaded on the gel run for the determination of tubulin concentration. It is normal to lose some tubulin in the process of repeated cycles of polymerization and depolymerization. A lower-than-expected yield of the final purified tubulin can be due to either (i) incomplete depolymerization of microtubules, visualized by the presence of an important amount of tubulin in fractions P3, P5, and P7, or (ii) an inefficient tubulin polymerization into microtubules, in which case a lower amount of tubulin is present in fractions P2, P4, and P6 and higher in fractions SN2, SN4, and SN6 (Figure 2B). If the tubulin is lost during polymerization steps (lower amounts of P2 and P4) (i) ensure sufficient tubulin concentration during polymerization (ii) use a fresh aliquot of GTP, and/or (iii) reconfirm the temperature of the polymerization reaction. If the tubulin is lost during depolymerization steps (lower amounts of SN3 and SN5), increase the time as well as pipetting of the mix on ice.

For the quantification of purified tubulin, run the samples along with the known quantities of bovine serum albumin (BSA, 0.5 µg -1 µg -2 µg -4 µg) (Figure 3A) on SDS-PAGE. Gels are stained with Coomassie brilliant blue, scanned, and the intensities of BSA and tubulin bands are measured by quantitative densitometry (Figure 3B) as described at https://openwetware.org/wiki/Protein_Quantification_Using_ImageJ. Please note that the same analysis can be done in Fiji, an upgraded version of ImageJ [START_REF] Schneider | NIH Image to ImageJ: 25 years of image analysis[END_REF] . Values from the BSA bands were used to determine the linear regression equation, which was used to calculate the amount of protein in the tubulin bands. Only tubulin band intensities within the range of the BSA curve are used to determine tubulin concentration. Based on the calculated tubulin concentration, aliquots of desired volumes of tubulin are prepared, snap-frozen in liquid nitrogen, and stored at -80 °C. We usually obtain about ~2 mg of tubulin from four spinner bottles of HeLa S3 suspension cultures (~15 g of cells), ~250 µg of tubulin from ten 15-cm diameter dishes (~1.2 g of cells), and ~1 mg of tubulin from 1 g of mouse brain tissue.

To confirm the enrichment of a particular tubulin isotype or modification, ~0.1 µg of the purified tubulin can be immunoblotted using respective antibodies [START_REF] Magiera | Methods in Cell Biology[END_REF][START_REF] Hausrat | Alpha-and betatubulin isotypes are differentially expressed during brain development[END_REF] . The control tubulin will vary depending on the tubulin of interest. For tubulin modified in vitro with a modifying enzyme, use non-treated tubulin as control. For tubulin modified in cellulo by the overexpression of a modifying enzyme, use tubulin purified from cells that do not express the enzyme as control (Figure 4A). Control tubulin for tubulin purified from knockout-mouse brains will be tubulin from wild type mice (Figure 4B). In all immunoblot analyses, an equal load of tubulin is verified by using a PTM-independent anti-a-tubulin antibody (12G10).

FIGURE LEGENDS:

Figure 1: Tubulin purification from different sources using polymerization-depolymerization cycles. (A) Different sources of tubulin are lysed using specific strategies. HeLa S3 cells cultured in suspension are lysed using a French press; HEK-293 cells are lysed by repetitive pipetting. Adherent cells were lysed using short pulses of sonication and mouse brain tissue using a tissue homogenizer. (B) Schematic representation of the successive steps of the tubulin purification protocol using cycles of cold-depolymerization and warm-polymerization. After lysis and lysate clarification, microtubules are polymerized and pelleted. Microtubules are then depolymerized and subsequently allowed to polymerize in a high-molarity buffer, preventing microtubule-associated protein (MAP) co-sedimentation with the microtubules. MAP-free microtubules are then depolymerized and can be further subjected to a third cycle of polymerization-depolymerization to remove trace amounts of the high-molarity buffer. (B) In an unsuccessful tubulin purification, the final tubulin yield is low, and tubulin remains either in the pellet after depolymerization or in the supernatant after polymerization (red boxes). In the example shown here, tubulin did not polymerize efficiently in both polymerization steps. Coomassie-stained SDS-PAGE gel with known quantities of bovine serum albumin (BSA; 0.5, 1, 2 and 4 µg, gray gradient line) and different volumes (0.5 and 1 µL, light and dark colors, respectively) of purified tubulin. In the example shown, tyrosinated tubulin (HeLa S3 tubulin, light and dark orange) and detyrosinated tubulin (HeLa S3 tubulin treated with carboxypeptidase A, light and dark blue) were loaded on the gel. (B) BSA bands from (A) were quantified using ImageJ (in arbitrary units, AU) and plotted against the amount of protein loaded (gray to black points). Those points were used to calculate the linear regression line (the gray gradient line) and equation, which were used to calculate the amounts of protein in the tubulin samples (light and dark orange and blue points) loaded on the gel. This facilitated the calculation of the concentration of the tubulin samples. Note that the points that lie beyond the BSA standard curve should not be used to determine concentration (dark orange and blue points).

DISCUSSION:

The method described here provides a platform to rapidly generate high-quality, assemblycompetent tubulin in medium-large quantities from cell lines and single mouse brains. It is based on the gold-standard protocol of tubulin purification from bovine brains used in the field for many years [START_REF] Vallee | Reversible assembly purification of microtubules without assemblypromoting agents and further purification of tubulin, microtubule-associated proteins, and MAP fragments[END_REF][START_REF] Castoldi | Purification of brain tubulin through two cycles of polymerization-depolymerization in a high-molarity buffer[END_REF] . One particular advantage of the approach is the use of suspension cultures of HeLa S3 cells, which, once established, yields large amounts of cells while requiring little hands-on time. This makes the protocol relatively easy to perform in any cell biology lab, whereas other tubulin purification methods [START_REF] Widlund | One-step purification of assembly-competent tubulin from diverse eukaryotic sources[END_REF][START_REF] Minoura | Overexpression, purification, and functional analysis of recombinant human tubulin dimer[END_REF][START_REF] Lacroix | Generation of differentially polyglutamylated microtubules[END_REF][START_REF] Newton | Intrinsically slow dynamic instability of HeLa cell microtubules in vitro[END_REF] require specific equipment and expertise and are thus mostly used by laboratories with a strong background in protein purification. When producing smaller quantities of tubulin from adherent cell lines, a variety of cell lines can be used. We have successfully purified tubulin from HeLa, U-2 OS, and HEK-293 cells. If a larger-scale purification is needed, harvested cells or brains can be snap-frozen in lysis buffer and stored at -80 °C, and multiple cell pellets or brains can be pooled together to purify larger amounts of tubulin. Tubulin purified from cell lines is virtually free of tubulin PTMs. This Tyr-tubulin can readily be converted to detyrosinated (deTyr-) tubulin in a single straightforward step [START_REF] Barisic | Microtubule detyrosination guides chromosomes during mitosis[END_REF] . To produce tubulin with other PTMs, specific tubulin-modifying enzymes can be overexpressed in cells prior to tubulin purification. Furthermore, using cell lines of human origin as the source of material helps avoid potential cross-species issues when studying interactions between microtubules and human MAPs. Further, tubulin from untransformed (such as HEK293) or transformed (such as HeLa) cells can provide information about the effects of microtubuledirected drugs (e.g., taxanes) on normal-vs. tumor-cell microtubules. Finally, our protocol facilitates the purification of tubulin from single mouse brains. As an increasing number of mouse models of tubulin mutations and modifications are being generated, this protocol allows direct analysis of the properties and interactions of microtubules with altered tubulin isotype composition [START_REF] Belvindrah | Mutation of the alpha-tubulin Tuba1a leads to straighter microtubules and perturbs neuronal migration[END_REF][START_REF] Breuss | Mutations in the murine homologue of TUBB5 cause microcephaly by perturbing cell cycle progression and inducing p53 associated apoptosis[END_REF][START_REF] Latremoliere | Neuronal-specific TUBB3 is not required for normal neuronal function but is essential for timely axon regeneration[END_REF] or tubulin PTMs [START_REF] Magiera | Excessive tubulin polyglutamylation causes neurodegeneration and perturbs neuronal transport[END_REF][START_REF] Morley | Acetylated tubulin is essential for touch sensation in mice[END_REF] .

The approach is based on cycles of polymerization and depolymerization. Thus, specific tubulin isotypes or tubulin with particular PTMs that affect the assembly and disassembly properties of microtubules could result in a disproportionate loss or reduction of such tubulin forms during the purification process. Nevertheless, we have shown that major tubulin PTMs, such as acetylation, detyrosination, glutamylation, and glycylation, are retained on the microtubules throughout the tubulin purification process [START_REF] Souphron | Purification of tubulin with controlled post-translational modifications by polymerization-depolymerization cycles[END_REF] . However, it should be noted that for quantitative analyses of the tubulin composition in cells or tissues, the TOG-column-based tubulin purification approach is more appropriate as it would allow an unbiased, polymerization-independent tubulin purification [START_REF] Widlund | One-step purification of assembly-competent tubulin from diverse eukaryotic sources[END_REF] . Despite its limitation, our protocol offers a great advantage in generating large amounts of high-quality tubulin that can be used in meticulous in vitro reconstitution experiments. In particular, it facilitates the use of PTM-rich brain tubulin in routine experiments.

DISCLOSURES:

The authors have nothing to disclose.

ACKNOWLEDGMENTS:

This work was supported by the ANR-10-IDEX-0001-02, the LabEx Cell'n'Scale ANR-11-LBX-0038 and the Institut de convergence Q-life ANR- We thank all members of the Janke lab, in particular J. Souphron, as well as G. Lakisic (Institut MICALIS, AgroParisTech) and A. Gautreau (Ecole Polytechnique) for help during the establishment of the protocol. We would like to thank the animal facility of the Institut Curie for help with mouse breeding and care. The antibody 12G10, developed by J. Frankel and M. Nelson, was obtained from the Developmental Studies Hybridoma Bank developed under the auspices of the NICHD and maintained by the University of Iowa.

Figure 2 :

 2 Evaluating the success of the tubulin purification. Samples collected at different steps of the tubulin purification protocol were run on a 'TUB' sodium dodecylsulfatepolyacrylamide gel electrophoresis (SDS-PAGE) gel (see protocol for details) and stained with Coomassie brilliant blue. (A) In a successful tubulin purification, aand b-tubulins are progressively enriched throughout the process. After the second polymerization, the microtubule pellet (P4) is virtually free of contamination from other proteins or microtubuleassociated proteins (MAPs). Note that it is normal to lose some tubulin during the procedure.

Figure 3 :

 3 Quantification of the purified tubulin using Coomassie-stained sodium dodecylsulfate-polyacrylamide gel electrophoresis (SDS-PAGE) gels and densitometry. (A)

Figure 4 :

 4 Immunoblot analysis of purified tubulin with different PTMs. (A)Tubulins purified from HEK-293 cells: wild type, or cells overexpressing TTLL5 or TTLL7 were analyzed for the specific enrichment of polyglutamylation using the GT335 antibody. While TTLL5 overexpression increases polyglutamylation on aand b-tubulin, TTLL7 overexpression specifically enriches b-tubulin glutamylation. (B) Tubulin purified from brain tissues of wild type and ttll1 -/-mice were analyzed for patterns of glutamylation. Note the strong reduction of polyglutamylation of tubulin from ttll1 -/-mice, which lack the major brain glutamylase TTLL1[START_REF] Janke | Tubulin polyglutamylase enzymes are members of the TTL domain protein family[END_REF] . 'TUB' gels were used to separate aand b-tubulin. An equal amount of tubulin load was confirmed by 12G10, an anti-a-tubulin antibody.

  4.1. Lysate Clarification 4.1.1. Clear the lysate (separating pellet and soluble fraction of the lysis mix) by centrifugation at 150,000 ´ g, 4 °C, 30 min. See Table of Materials for details about ultracentrifuge rotors and tubes. For cell extracts, a white floating layer is often formed after centrifugation. Do not transfer this floating layer along with the supernatant, as it interferes with tubulin polymerization. Use a syringe of appropriate volume attached to a long 20 G or 21 G needle to gently remove the supernatant without disturbing the floating layer. If the supernatant is still cloudy, centrifuge at 5,000 ´ g, 4 °C for 10 min. 4.1.2. Transfer the supernatant (SN1) to an ultracentrifuge tube and note its volume. For a 10 mL cell pellet, expect a volume of ~12 mL for SN1. 4.1.3. Take 1/100 th volume of SN1 (120 µL for 12 mL of SN1), and add the same volume of 2x Laemmli buffer, boil for 5 min, and store at -20 °C for further analysis. 4.1.4. Resuspend the pellet (P1) in BRB80 (Table of Materials) using the same volume as SN1. Take 1/100 th volume of P1 (200 µL for 20 mL of P1), and add the same volume of 2x Laemmli buffer, boil for 5 min, and store at -20 °C for further analysis. : The microtubule pellet after the second polymerization step can be snap-frozen and stored at -80 °C for up to 1 year. 4.4.5. Take 1-4 µL and add 9 volumes of 2x Laemmli buffer, boil for 5 min, and store at -20 °C for further analyses. 4.5. Second Depolymerization 4.5.1. Depolymerize microtubules by adding ice-cold BRB80 to the pellet P4, and leave on ice for 5 min: for tubulin from cells, add 1/100 th (120 µL), and for tubulin from brains, add 1/40 th (300 µL) of the volume of the SN1. 4.5.2. Pipette up and down with a p200 tip every 5 min, for 20 min (five cycles of pipetting). 4.5.3. Transfer the solution to a 1.5 mL ultracentrifuge tube, and spin down at 150,000 ´ g, 4 °C for 20 min. Transfer the SN5 to a new 1.5 mL ultracentrifuge tube. The pellet formed after this centrifugation step (P5) contains non-depolymerized microtubules. The supernatant (SN5) contains the soluble tubulin. Depolymerize microtubules by adding ice-cold BRB80 to the pellet P6, and leave on ice for 5 min: for tubulin from cells, add 1/100 th (120 µL), and for tubulin from brains, add 1/40 th (300 µL) of the volume of the SN1. 4.7.2. Pipette up and down with a p200 tip every 5 min, for 20 min (five cycles of pipetting). 4.7.3. Transfer the solution to the appropriate ultracentrifuge tubes, and spin down at 150,000 ´ g, 4 °C for 20 min. Transfer SN7 to a new 1.5 mL ultracentrifuge tube. The pellet (P7) contains small amounts of non-depolymerized microtubules. The supernatant (SN7) contains exclusively depolymerized microtubules (soluble tubulin). 4.7.4. Take 1-4 µL and add 9 volumes of 2x Laemmli buffer, boil for 5 min, and store at -20 °C for further analyses. 4.7.5. Resuspend the pellet P7 in BRB80 (same volume of SN7), take 1-4 µL, and add 9 volumes of 2x Laemmli buffer, boil for 5 min, and store at -20 °C for further analyses.

	4.3. First Depolymerization 4.7.1.
	4.3.1. Depolymerize microtubules by adding ice-cold BRB80 to the pellet P2, and leave on ice
	for 5 min: for tubulin from cells, add 1/60 th (200 µL), and for tubulin from brains, add 1/20 th
	(600 µL) of the volume of the SN1.
	NOTE: The volume of ice-cold BRB80 added to the pellet during depolymerization steps is
	always relative to the volume of SN1.
	4.3.2. Resuspend the microtubule pellet gently, avoiding air bubbles, until the solution is
	completely homogeneous. Use a p1000 tip for a couple of cycles of pipetting followed by a
	p200 tip every 5 min, for 20 min (five cycles of pipetting). This is a crucial step for the success
	of the tubulin purification.
	4.3.3. Transfer the solution to appropriate ultracentrifuge tubes, and spin down at 150,000
	´ g, 4 °C for 20 min. Transfer the SN3 to a new 1.5 mL ultracentrifuge tube. The pellet formed
	after this centrifugation step (P3) contains precipitated proteins (microtubule-associated
	proteins or MAPs) and non-depolymerized microtubules. The supernatant (SN3) contains
	soluble components: depolymerized tubulin dimers and MAPs, which have detached from the
	depolymerized microtubules.
	4.3.4. Take 1-4 µL of SN3, and add 9 volumes of 2x Laemmli buffer, boil for 5 min, and store
	at -20 °C for further analyses.
	4.3.5. Resuspend the pellet P3 in BRB80 (in the same volume of SN3), take 1-4 µL, and add 4.6. Third Polymerization (in Low-Molarity Buffer)
	9 volumes of 2x Laemmli buffer, boil for 5 min, and store at -20 °C.
	4.6.1. Prepare the polymerization mix: 1 volume of SN5 (120 µL), 1/200 th volume of 0.2-M
	4.4. Second Polymerization (in High-Molarity Buffer) GTP (0.6 µL, final concentration is 1 mM), and 0.5 volume of pre-heated glycerol (60 µL) in a
	tube of the appropriate volume.
	4.4.1. Prepare the polymerization mix by combining 1 volume of SN3 (200 µL), 1 volume of
	pre-heated 1 M PIPES (200 µL, final concentration 0.5 M), 1/100 th volume of 0.2-M GTP (2 µL, 4.6.2. Pipette the mix up and down, gently avoiding formation of air bubbles, and transfer it
	final concentration 1 mM), and 1 volume of pre-heated glycerol (200 µL) in a tube of the to the appropriate ultracentrifuge tubes.
	appropriate volume.	This
	allows the experimenter to directly proceed to the sedimentation of microtubules after the 4.6.3. Cover the tubes with parafilm, transfer them to a water bath set at 30 °C, and incubate
	polymerization step. Do this for all polymerization steps throughout the protocol. 4.4.2. Pipette the mix up and down, avoiding the formation of air bubbles, and transfer it to for 20 min.
	ultracentrifuge tubes.
	4.2.3. Cover the tubes with parafilm, transfer to a water bath set at 30 °C, and incubate for 4.6.4. Centrifuge the tubes at 150,000 ´ g, 30 °C for 30 min. The pellet (P6) contains
	20 min. 4.4.3. Cover the tubes with parafilm, transfer them to a water bath set at 30 °C, and incubate polymerized microtubules and the supernatant SN6 contains small amounts of non-
	for 20 min. polymerized tubulin.
	4.2.4. Centrifuge the tubes at 150,000 ´ g, 30 °C for 30 min. Remove the supernatant (SN2),
	and keep the pellet of polymerized microtubules (P2). 4.4.4. Centrifuge the tubes at 150,000 ´ g, 30 °C for 30 min. Remove the supernatant (SN4), NOTE: Microtubule pellets can be snap-frozen and stored at -80 °C for up to 1 year.
	and keep the pellet of polymerized microtubules (P4). The pellet P4 contains the polymerized
	microtubules, and the supernatant SN4 contains unpolymerized tubulin, MAPs, and other 4.6.5. Take 1-4 µL and add 9 volumes of 2x Laemmli buffer, boil for 5 min, and store at -20
	soluble proteins. °C for further analyses.
	4.7. Third Depolymerization

4.2

. First Polymerization in Low-Molarity Buffer 4.2.1. Prepare the polymerization mix by combining 1 volume of SN1 (12 mL), 1/200 th volume of 0.2 M GTP (60 µL; final concentration 1 mM), and 0.5 volume of pre-heated glycerol (6 mL) in a screw-cap tube of the appropriate volume. NOTE: Glycerol is used as a crowding agent in the polymerization steps throughout the protocol and thus is not considered in the calculations of other components' concentrations. 4.2.2. Pipette the mix up and down, gently avoiding the formation of air bubbles and transfer it to the appropriate ultracentrifuge tubes. NOTE: While transferring the mix to the tubes, adjust the weight of the tubes (in pairs). NOTE: Microtubule pellet can be snap-frozen and stored at -80 °C for up to 1 year. 4.2.5. Take 1/200 th volume of SN2 (90 µL for 18 mL SN2) and add the same volume of 2x Laemmli buffer, boil for 5 min, and store at -20 °C for further analysis. NOTE4.5.4. Take 1-4 µL and add 9 volumes of 2x Laemmli buffer, boil for 5 min, and store at -20 °C for further analyses. 4.5.5. Resuspend the pellet P5 in BRB80 (same volume of SN5), take 1-4 µL, and add 9 volumes of 2x Laemmli buffer, boil for 5 min, and store at -20 °C for further analyses. 4.7.6. Quantify the amount of tubulin (see Representative Results) and aliquot SN7 into small volumes, snap-freeze, and store at -80 °C.

  17-CONV-0005. CJ is supported by the Institut Curie, the French National Research Agency (ANR) awards ANR-12-BSV2-0007 and ANR-17-CE13-0021, the Institut National du Cancer (INCA) grant 2014-PL BIO-11-ICR-1, and the Fondation pour la Recherche Medicale (FRM) grant DEQ20170336756. MMM is supported by the Fondation Vaincre Alzheimer grant FR-16055p, and by the France Alzheimer grant AAP SM 2019 n°2023. JAS was supported by the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 675737, and the FRM grant FDT201904008210. SB was supported by the FRM grant FDT201805005465.


Guanghui Fu

Rosana El Jurdi

Lydia Chougar

Didier Dormont

Romain Valabregue

Stéphane Lehéricy

Olivier Colliot

Introducing Soft Topology Constraints in Deep Learning-based Segmentation using Projected Pooling Loss

Keywords: Segmentation, Loss function, Anatomical priors, Topology, Connected components, Deep Learning

Deep learning methods have achieved impressive results for 3D medical image segmentation. However, when the network is only guided by voxel-level information, it may provide anatomically aberrant segmentations. When performing manual segmentations, experts heavily rely on prior anatomical knowledge. Topology is an important prior information due to its stability across patients. Recently, several losses based on persistent homology were proposed to constrain topology. Persistent homology offers a principled way to control topology. However, it is computationally expensive and complex to implement, in particular in 3D. In this paper, we propose a novel loss function to introduce topological priors in deep learning-based segmentation, which is fast to compute and easy to implement. The loss performs a projected pooling within two steps. We first focus on errors from a global perspective by using 3D MaxPooling to obtain projections of 3D data onto three planes: axial, coronal and sagittal. Then, 2D MaxPooling layers with different kernel sizes are used to extract topological features from the multi-view projections. These two steps are combined using only MaxPooling, thus ensuring the efficiency of the loss function. Our approach was evaluated in several medical image datasets (spleen, heart, hippocampus, red nucleus). It allowed reducing topological errors and, in some cases, improving voxel-level accuracy.

INTRODUCTION

Deep learning methods are the state of the art for medical image segmentation. [START_REF] Hesamian | Deep learning techniques for medical image segmentation: Achievements and challenges[END_REF] The standard approach is to supervise at the voxel level, using a voxel-wise loss (typically cross-entropy 2 and Dice loss [START_REF] Milletari | V-net: Fully convolutional neural networks for volumetric medical image segmentation[END_REF] ). However, this may generate anatomically aberrant segmentations. [START_REF] Bernard | Deep learning techniques for automatic MRI cardiac multi-structures segmentation and diagnosis: is the problem solved?[END_REF] To overcome such problems, one can introduce prior anatomical knowledge in the segmentation procedure. Indeed, medical experts extensively use their anatomical knowledge to perform manual segmentations. In particular, prior knowledge can be introduced through new loss functions. Many loss functions have been designed and we refer the reader to systematic surveys. [START_REF] Ma | Loss odyssey in medical image segmentation[END_REF][START_REF] El Jurdi | High-level prior-based loss functions for medical image segmentation: A survey[END_REF][START_REF] Liu | Anatomy-aided deep learning for medical image segmentation: a review[END_REF] The purpose of such losses is very diverse but some aim at introducing prior information such as for instance shape 8 , 9 position 10 or size [START_REF] Chen | Learning active contour models for medical image segmentation[END_REF][START_REF] Jurdi | A surprisingly effective perimeter-based loss for medical image segmentation[END_REF] of the target region. Topological priors are important because they are usually very robust across patients and because a topologically incorrect segmentation is generally anatomically aberrant. BennTaieb et al. [START_REF] Bentaieb | Topology aware fully convolutional networks for histology gland segmentation[END_REF] proposed a loss combining topological and geometric priors. However, it is not clear if it is applicable to 3D data. He et al. [START_REF] He | Towards topological correct segmentation of macular oct from cascaded fcns[END_REF] introduced a cascaded network which preserves the topological relationships between the layers of the retina. The most general approaches to constrain topology [START_REF] Hu | Topology-preserving deep image segmentation[END_REF][START_REF] Clough | A topological loss function for deep-learning based image segmentation using persistent homology[END_REF][START_REF] Byrne | A persistent homology-based topological loss function for multi-class cnn segmentation of cardiac MRI[END_REF] are based on persistent homology, [START_REF] Edelsbrunner | Persistent homology -a survey[END_REF] an algebraic method for characterizing the topology of shapes and functions. Persistent homology offers a principled way to control topology, such as connected components, holes, or voids. However, it is complex to implement and computationally expensive in particular in 3D.

In this paper, we propose a fast loss function to constrain the topology in 3D medical image segmentation. Unlike approaches based on persistent homology, it only provides an approximate control of topology but has the advantage of being computationally efficient and easy to implement.

PROPOSED METHOD

Our approach has two components: we first project the 3D volume into 2D space according to three views; we then characterize topological features using pooling layers with different kernel sizes (Figure 1). The topological loss corresponds to the difference in topological features between the ground truth and the prediction. Overview of the approach. The method contains two steps: 3D MaxPooling for projection and 2D MaxPooling with different kernel sizes for topological features extraction. Due to space limitation, we do not show the 2D MaxPooling step for sagittal and coronal planes but the specific steps remain the same as for axial plane.

Volume projection by 3D-MaxPooling

We propose to use 3D MaxPooling to project the volume onto three planes (axial, coronal, and sagittal) to create a 2D representation of the 3D object. The projection can be easily obtained by using a MaxPooling operation. The volume projections onto the axial, sagittal and coronal planes are denoted respectively as P axial , P sagittal and P coronal and computed as follows:

P axial = M P 3 kaxi (V w,h,s ); P sagittal = M P 3 ksag (V w,h,s ); P coronal = M P 3 kcor (V w,h,s ) (1)
where V w,h,s denotes the 3D volume, w, h and s are the width, height and number of slices of the volume respectively, M P k 3 denotes the 3D-MaxPooling with kernel k and k axi = (1, 1, s), k sag = (w, 1, 1) and k cor = (1, h, 1) are the kernels for projecting onto the axial, sagittal and coronal planes respectively. Note that P axial is of dimension (w, h), P sagittal is of dimension (h, s) and P coronal is of dimension (w, s).

The above procedure concentrates the target objects along each axis. The idea is to be able to magnify the effect of small errors on the overall loss. Indeed, a small segmentation error (such as for example a small extra connected component) has little effect on the overall loss. For example, the proportion of a voxel mispredicted in the volume V w,h,s is 1 w×h×s , while if we project it onto the axial plane, the error proportion is [START_REF] Hesamian | Deep learning techniques for medical image segmentation: Achievements and challenges[END_REF] w×h . Thus, projecting the prediction into axial planes augments the weight of this error by s times.

Topology feature extraction by 2D-MaxPooling

In a second step, we characterize the topology of the projected view by using two-dimensional MaxPooling with different kernel sizes. The output can be roughly treated as connected components. Some small areas such as noise in predictions are very common, and we don't want to ignore them. We achieve this by using a set of 2D-MaxPooling with different kernel sizes. The topological characterization of the prediction and the ground truth is then defined as follows:

P k topo = M P k 2 (P axial ) + M P k 2 (P sagittal ) + M P k 2 (P coronal ) G k topo = M P k 2 (G axial ) + M P k 2 (G sagittal ) + M P k 2 (G coronal ) (2)
where M P k 2 denotes 3D-MaxPooling with square kernel of size k, G plane (resp. P plane ) is the projection of the ground truth (resp. of the prediction) onto a given plane. After 2D MaxPooling (M P k

2 ) under different kernels k, the correct part of the prediction will occupy the same position as the label.

We then compute the absolute difference between P k topo and P k topo for each kernel size. The topology loss L topo is obtained by averaging over all kernels and over the three projection planes:

L topo = | k∈K G k topo -P k topo | 3 × |K| . ( 3
)
where K is the set of kernels and |K| is the number of kernels. regarded as geometric appreciation.

Final loss

The final loss function is a combination of the voxel-wise Dice loss and the proposed topological loss: L total = L topo + λL dice where λ is the trade-off parameter. In our experiments, we used a fixed value of λ = 1.

EXPERIMENTS AND RESULTS

Implementation

Our loss function is compatible with any convolutional neural network. In our experiments, we used the standard 3D-UNet. [START_REF] Abdulkadir | 3D U-net: Learning dense volumetric segmentation from sparse annotation[END_REF] In some experiments, we used early stopping to avoid over-fitting. The early stopping was done using only the validation set in order to obtain an unbiased performance on the test set. In all experiments, the hyperparameters were identical. Because our approach needs to be pre-trained, each training involved two steps, both using Adam: [START_REF] Kingma | Adam: A method for stochastic optimization[END_REF] (1) pre-train for 150 epochs with learning rate 10 -3 ; (2) fine tuning for the same number of epochs with learning rate 10 -4 . Finally, we used the following sequence of kernels sizes: 4, 5, 8, 10, 20. We use the open-source Python library TorchIO 21 * for medical dataset preprocessing that include: reshaping images to the same size for each task, min-max normalization. Our code is developed based on the PyTorch framework.

Datasets and evaluation framework

Our method was assessed using three public datasets from the Medical Segmentation Decathlon (MSD) [START_REF] Antonelli | The medical segmentation decathlon[END_REF] and one local dataset. The three tasks from MSD were segmentation of spleen, heart and hippocampus (see [START_REF] Antonelli | The medical segmentation decathlon[END_REF] for details). For the local dataset, the task is to segment the red nucleus from Quantitative Susceptibility Mapping (QSM) brain images. The cohort included 18 healthy subjects, 46 patients with early Parkinson's disease, and 16 patients with prodromal parkinsonism (isolated rapid eye movement sleep behavior disorders) from the ICEBERG study (NCT02305147). The institutional ethical committee approved the study (CPP Paris VI/RCB: 2014-A00725-42). All participants gave written informed consent. QSM images were manually segmented by a trained neuroradiologist (L.C.).

Each dataset was split into training, validation and test sets. The splits were done at the participant level to avoid any data leakage. [START_REF] Thibeau-Sutre | Clinicadl: an open-source deep learning software for reproducible neuroimaging processing[END_REF] The dataset distribution can be seen in the Table 1. For the local dataset, we also explored the performance with a reduced training set size by randomly sampling 30% of the training set (15 MRIs).

We report performance at the voxel level and the topological level. We used the following seven classical voxel-level evaluation metrics: Dice score, Precision, Recall, Volume error rate (VER), Absolute volume error rate (AVER), Pearson's r between predicted and ground-truth volume, 95 percentile Hausdorff distance (95HD). Topological errors may have only a small impact on voxel-level metrics if they are of limited size. However, topological errors make the predictions anatomically inconsistent. We thus used additional metrics for evaluation of the topology, computing the 3D connected component error as well as the average 2D connected component error. Topological metrics were the 3D connected component error (CCE) and the average 2D CCE.

Results

Results are displayed in Table 2. The proposed loss performs comparably with respect to the Dice loss baseline in terms of voxel-level metrics. However, the proposed loss substantially decreases the topological errors both in 2D and in 3D for the spleen, heart and red nucleus datasets. For the hippocampus dataset, results were comparable. Furthermore, in the case of the red nucleus, when training on a smaller subset, the Dice score and other voxel-level metrics were also improved. Figure 2 shows an example of result on the red nucleus task. One can observe that only a few iterations of the topology loss are needed to improve topological correctness and Dice score.

DISCUSSION AND CONCLUSION

We proposed a new loss function that integrates soft topological priors for 3D medical image segmentation. We demonstrated that it allows reducing topological errors, without the need for subsequent postprocessing. In some cases, it also allowed to improve voxel-level metrics. A limitation of our approach is that, unlike those based on persistent homology, [START_REF] Hu | Topology-preserving deep image segmentation[END_REF][START_REF] Clough | A topological loss function for deep-learning based image segmentation using persistent homology[END_REF][START_REF] Byrne | A persistent homology-based topological loss function for multi-class cnn segmentation of cardiac MRI[END_REF] it does not strictly guarantee topological correctness. However, it is computationally efficient and easy to implement. Another limitation is that the sequence of kernel sizes needs to be chosen. However, we obtained suitable results for various anatomical structures without changing the kernel size.

Figure 1 .

 1 Figure 1. Overview of the approach. The method contains two steps: 3D MaxPooling for projection and 2D MaxPooling with different kernel sizes for topological features extraction. Due to space limitation, we do not show the 2D MaxPooling step for sagittal and coronal planes but the specific steps remain the same as for axial plane.

Figure 2 .

 2 Figure 2. An illustrative example of red nucleus segmentation. Dice is the result of the Dice loss. Topological ne denotes correspond to n epochs using the proposed loss.

22 Table 1 .

 221 Characteristics of the imaging datasets.

	Dataset Task	Train+val Test Image Size
		Spleen	25+7	9	160,160,128
	Public	Heart	12+4	4	160,160,128
		Hippocampus 166+42	52	64, 64, 48
	Local	Red Nucleus	51+13	16	160,160,128

Table 2 .

 2 Results with our loss and with the standard Dice loss (trained with either 150 or 150+150 epochs).

	Task	Setting Loss Epoch	Dice	Voxel level 95HD Precision Recall MVER MAVER Pearson's r 2D CCE 3D CCE Topology level
		Dice	150	0.887 9.811	0.849	0.938	0.118	0.164	0.890	0.524	2
	Spleen	Dice	150+150 0.894 21.26	0.853	0.948	0.124	0.163	0.897	0.506	1.667
		Ours 150+150 0.895 8.405	0.902	0.898	-0.001	0.098	0.894	0.289	0.222
		Dice	150	0.872 6.923	0.821	0.930	0.136	0.136	0.873	0.543	2.75
	Heart	Dice	150+150 0.872 5.722	0.844	0.902	0.072	0.074	0.872	0.403	0.75
		Ours 150+150 0.864 7.104	0.861	0.870	-0.086	0.087	0.865	0.373	0.25
		Dice	150	0.861 1.368	0.864	0.861	0.001	0.053	0.877	0.2	0.019
	Hippocampus	Dice	150+150 0.861 1.339	0.869	0.858	-0.006	0.054	0.879	0.192	0.019
		Ours 150+150 0.853 1.375	0.877	0.833	-0.048	0.067	0.872	0.203	0
	Red Nucleus (30%: 15 QSM)	Dice Dice Ours 150+150 0.808 3.612 150 0.725 7.646 150+150 0.738 5.493	0.595 0.610 0.783	0.948 0.954 0.849	0.645 0.616 0.107	0.645 0.616 0.184	0.748 0.76 0.812	0.601 0.568 0.349	1.025 1.125 0.375
	Red Nucleus (full: 51 QSM)	Dice Dice Ours 150+150 0.905 1.005 150 0.896 1.005 150+150 0.900 1.005	0.862 0.872 0.896	0.938 0.933 0.918	0.095 0.087 0.03	0.129 0.115 0.09	0.898 0.902 0.906	0.232 0.211 0.144	0.662 0.525 0.05

* https://torchio.readthedocs.io/

ACKNOWLEDGMENTS

The research leading to these results has received funding from the French government under management of Agence Nationale de la Recherche as part of the "Investissements d'avenir" program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute) and reference ANR-10-IAIHU-06 (Agence Nationale de la Recherche-10-IA Institut Hospitalo-Universitaire-6). The ICEBERG study is supported by the European Research Council (ERC) under grant agreement No. 678304, the European Union's Horizon 2020 research and innovation program under grant agreement No. 826421 (TVB-Cloud), Agence Nationale de la Recherche (ANR) under grant agreements ANR-10-IAIHU-06 (IHU ICM), ANR-11-INBS-0006, and ANR-19-JPW2-000 (JPND E-DADS), association France Parkinson (PRECISE-PD project), the Fondation d'Entreprise EDF, Biogen Inc., Fondation Thérèse and René Planiol, Fondation Saint Michel. It received unrestricted support for Research on Parkinson's disease from Energipole (M. Mallart), M. Villain and the Société Française de Médecine Esthétique (M. Legrand). Guanghui Fu is supported by the Chinese Government Scholarship provided by China Scholarship Council (CSC). Lydia Chougar is supported by a Poste d'accueil Inria/AP-HP.