Shailaja Seetharaman

Benoit Vianay

Vanessa Roca

Aaron Farrugia

Chiara De Pascalis

Batiste Boëda

Florent Dingli

Damarys Loew

Stephane Vassilopoulos

Alexander Bershadsky

Stéphane Vassilopoulos

Manuel Théry

Sandrine Etienne-Manneville

Microtubules tune mechanosensitive cell responses

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however, very little is known about microtubules in this context. In this study, we used primary rat astrocytes and primary human umbilical vein endothelial cells (HUVECs) (both types of cells are dependent on efficient microtubule dynamics for migration or invasion [START_REF] Lyle | Microtubule dynamics regulation contributes to endothelial morphogenesis[END_REF][START_REF] Myers | Distinct ECM mechanosensing pathways regulate microtubule dynamics to control endothelial cell branching morphogenesis[END_REF][START_REF] Etienne-Manneville | Integrin-mediated activation of Cdc42 controls cell polarity in migrating astrocytes through PKCzeta[END_REF][START_REF] Etienne-Manneville | In vitro assay of primary astrocyte migration as a tool to study Rho GTPase function in cell polarization[END_REF][START_REF] Bance | Microtubule acetylation but not detyrosination promotes focal adhesion dynamics and astrocyte migration[END_REF] ), to address the role of microtubules in rigidity sensing and mechanosensitive migration.

The crucial factors affecting the functions of the microtubule network are post-translational modifications (PTMs) of tubulin such as acetylation, which occurs at the K40 residue of α-tubulin. The enzyme responsible for microtubule acetylation, αTAT1 (α-tubulin acetyltransferase 1, also called as MEC-17), is present in the lumen of microtubules [START_REF] Coombes | Mechanism of microtubule lumen entry for the α-tubulin acetyltransferase enzyme αTAT1[END_REF] and is highly specific to α-tubulin K40. On the other hand, the enzymes involved in deacetylation at K40 are histone deacetylase family member 6 (HDAC6) and sirtuin type 2 (Sirt2), both of which target other substrates as well [START_REF] Valenzuela-Fernandez | HDAC6: a key regulator of cytoskeleton, cell migration and cell-cell interactions[END_REF] . In this study, we used previously characterized siRNAs targeting αTAT1 [START_REF] Bance | Microtubule acetylation but not detyrosination promotes focal adhesion dynamics and astrocyte migration[END_REF] (set 1: siαTAT1-1 and set 2: siαTAT1-2; Extended Data Fig. 1a, 1b and 1c) to decrease acetylation, and Tubacin (Extended Data Fig. 1a and1c), a drug which increases microtubule acetylation by inhibiting HDAC6 without modifying the acetylation of other HDAC6 substrates such as histones [START_REF] Haggarty | Multidimensional Chemical Genetic Analysis of Diversity-Oriented Synthesis-Derived Deacetylase Inhibitors Using Cell-Based Assays[END_REF][START_REF] Haggarty | Domain-selective small-molecule inhibitor of histone deacetylase 6 (HDAC6)-mediated tubulin deacetylation[END_REF] .

Substrate rigidity promotes microtubule acetylation through integrin β 1 signalling

We have previously shown that microtubules come in close proximity to FAs, and that microtubule acetylation promotes FA turnover during migration [START_REF] Bance | Microtubule acetylation but not detyrosination promotes focal adhesion dynamics and astrocyte migration[END_REF] . Since FAs are major mechanosensitive structures [START_REF] De Pascalis | Single and collective cell migration: the mechanics of adhesions[END_REF] , we investigated whether the extracellular matrix rigidity affects microtubule acetylation.

Astrocytes were plated sparsely on polyacrylamide hydrogels of different rigidities: 1.26 kPa, 2 kPa, 9 kPa and 48 kPa (Fig. 1a and1b). Astrocytes on soft substrates exhibited lower levels of acetylated tubulin than cells on stiff substrates (Fig. 1a and1b). Tubulin acetylation levels showed a two fold increase with increasing rigidity (1.26 kPa to 48 kPa), a strongly significant change in comparison to the maximal 3.8 fold increase obtained after Tubacin treatment (Extended Data Fig. 1d). Subsequently, to determine whether microtubule acetylation may be indirectly caused by a mechanism involving differential cell spreading, we plated astrocytes on adhesive micropatterns (area 2500 μm 2 ) printed on 2 kPa and 40 kPa hydrogels (Extended Data Fig. 1e). Similar to stiff substrates, astrocytes on soft substrates adopted a crossbow shape and identical spread area, and yet, microtubule acetylation was increased on stiff substrates as compared to softer substrates (Extended Data Fig. 1e). In contrast to tubulin acetylation, tubulin detyrosination (another tubulin PTM), remained unaffected by increased substrate rigidity in astrocytes (Extended Data Fig. 1f and1g), and had no effect on astrocyte adhesion and migration [START_REF] Bance | Microtubule acetylation but not detyrosination promotes focal adhesion dynamics and astrocyte migration[END_REF] , suggesting that rigidity sensing specifically affects microtubule acetylation.

Integrin-based FAs sense the matrix rigidity and trigger mechanosensitive signalling pathways [START_REF] Seetharaman | Integrin diversity brings specificity in mechanotransduction[END_REF] . To determine the role of integrin signalling in controlling microtubule acetylation, we first used a scratchinduced migration assay to trigger integrin activation at the wound edge [START_REF] Etienne-Manneville | Integrin-mediated activation of Cdc42 controls cell polarity in migrating astrocytes through PKCzeta[END_REF] . Addition of cyclic RGD (cRGD) peptide, to prevent the binding of integrins to the RGD motif of extracellular matrix proteins, reduced tubulin acetylation (Extended Data Fig. 2a). Furthermore, integrin activation using MnCl 2 [START_REF] Cluzel | The mechanisms and dynamics of αvβ3 integrin clustering in living cells[END_REF] led to an increase in microtubule acetylation as compared to control cells (Extended Data Fig. 2b). Finally, depletion of β 1 integrin using a siRNA (Extended Data Fig. 2c) also resulted in a significant decrease in tubulin acetylation as compared to control cells (Extended Data Fig. 2c and2d), suggesting that β 1 integrins are major players in the control of microtubule acetylation. We then investigated the involvement of two central players in integrin signalling; the proto-oncogene c-Src and the Focal Adhesion Kinase (FAK). In migrating astrocytes, inhibition of Src (using Src kinase inhibitor 1 -referred to as Src kin 1) and FAK (using PF-562271) drastically reduced the levels of acetylated tubulin (Extended Data Fig. 2e). In addition, following Src and FAK inhibition, microtubule acetylation induced by cell spreading on 48 kPa substrates was decreased to a level similar to that observed on soft substrates (Fig. 1C and Extended Data Fig. 2f), indicating that Src and FAK are required for the rigidity-dependent increase in acetylation. Altogether, these results strongly suggest that β 1 integrin-mediated signalling promotes substrate rigidity-induced acetylation of microtubules.

αTAT1 interacts with Talin in mechanosensitive manner

Microtubule acetylation is mediated by αTAT1; however, the recruitment and/or activation of αTAT1 remains unknown [START_REF] Janke | Causes and Consequences of Microtubule Acetylation[END_REF] . Therefore, we carried out a quantitative mass spectrometry screen to identify interacting partners of αTAT1 using HEK293 cells. The mass spectrometry data (data available via ProteomeXchange with identifier PXD015871; check methods for details; Extended Data Fig. 2g) revealed interesting potential interactors, amongst which the proteins enriched in the gene ontology for focal adhesions depicted in red in Extended Data Fig. 2g. One of the significant interactors on the mass spectrometry screen was Talin (Extended Data Fig. 2g), a mechanosensitive partner of integrins [START_REF] Seetharaman | Integrin diversity brings specificity in mechanotransduction[END_REF] . We confirmed the interaction of αTAT1 with Talin using pulldown experiments. GST-αTAT1, but not GST alone, associated with endogenous Talin from astrocyte lysates (Fig. 1d and Extended Data Fig. 2h). The interaction was further confirmed by immunoprecipitation using GFP-αTAT1-transfected astrocytes.

Talin co-immunoprecipitated with GFP-αTAT1 but not with GFP alone (Fig. 1e and Extended Data Fig. 2i).

In addition, using TIRF microscopy, we observed that GFP-αTAT1 strongly localised at mCherry-vinculinpositive FAs (Fig. 1f) [START_REF] Bance | Microtubule acetylation but not detyrosination promotes focal adhesion dynamics and astrocyte migration[END_REF] . We then investigated whether Talin played a role in the recruitment of αTAT1 to FAs and the acetylation of microtubules. Therefore, we used specific siRNAs to reduce Talin expression in primary astrocytes (Extended Data Fig. 2j). Talin depletion reduced GFP-αTAT1 recruitment to FAs (Fig. 1f). In addition, Talin-depleted astrocytes plated on 48 kPa substrates exhibited decreased levels of tubulin acetylation as compared to the control cells (Fig. 1g).

Since αTAT1 is also present within microtubules (Extended Data Fig. 2k) [START_REF] Bance | Microtubule acetylation but not detyrosination promotes focal adhesion dynamics and astrocyte migration[END_REF][START_REF] Coombes | Mechanism of microtubule lumen entry for the α-tubulin acetyltransferase enzyme αTAT1[END_REF] , we tested whether the recruitment of αTAT1 to FAs was dependent on microtubules. To address this, GFP-αTAT1 and mCherryvinculin expressing astrocytes were treated with Nocodazole to depolymerize all microtubules.

Nocodazole-treated astrocytes displayed larger adhesions and increased stress fibres as observed in other cell types (Extended Data Fig. 2l, Supplementary Video 1) [START_REF] Enomoto | Microtubule disruption induces the formation of actin stress fibers and focal adhesions in cultured cells: possible involvement of the rho signal cascade[END_REF][START_REF] Bershadsky | Involvement of microtubules in the control of adhesion-dependent signal transduction[END_REF] , and increased αTAT1 at FAs (Extended Data Fig. 2l, Supplementary Video 1), suggesting that microtubules are not required to localise αTAT1 at FAs, and that increased actomyosin contractility might facilitate αTAT1 concentration at FAs. Since actomyosin contractility is required for Talin-mediated mechanosensing [START_REF] Choquet | Extracellular matrix rigidity causes strengthening of integrin-cytoskeleton linkages[END_REF][START_REF] Wolfenson | Tropomyosin controls sarcomere-like contractions for rigidity sensing and suppressing growth on soft matrices[END_REF] , we used the ROCK inhibitor, Y-27632 (which strongly reduces actomyosin contractility [START_REF] Peglion | Adherens junction treadmilling during collective migration[END_REF] ) to assess its role in the mechanosensitive acetylation of microtubules. By GST-αTAT1 pulldown as well as TIRF microscopy, we observed that Y-27632 treatment decreased αTAT1 interaction with Talin (Fig. 1h and 1i, Supplementary

Video 2) and tubulin acetylation (Fig. 1j and1k) confirming that actomyosin contractility and mechanosensing at FAs influence microtubule acetylation. Altogether, these results suggest that Talinand actomyosin-dependent mechanosensing triggers the recruitment of αTAT1 to FAs, and is required for rigidity-dependent microtubule acetylation.

Microtubule acetylation influences mechanotransduction at focal adhesions

We then tested whether microtubule acetylation may be involved in mechanosensitive cell functions. Astrocytes were sparsely plated on soft (2 kPa) and stiff (48 kPa) polyacrylamide hydrogels and stained for Yes-associated protein (YAP), an actomyosin-and RhoGTPase-dependent transcriptional activator whose nuclear translocation is mechanosensitive [START_REF] Dupont | Role of YAP/TAZ in mechanotransduction[END_REF] . Like in several cell types [START_REF] Panciera | Mechanobiology of YAP and TAZ in physiology and disease[END_REF] , quantification of nuclear-to-cytoplasmic ratio of YAP indicated that YAP nuclear localisation increased when astrocytes were plated on stiff substrates compared to soft substrates (Fig. 2a and2b). In αTAT1-depleted astrocytes, the nuclear recruitment of YAP in cells plated on 48 kPa substrates was strongly inhibited and YAP localisation was similar in cells plated on soft and rigid substrates (Fig. 2a). Conversely, increasing microtubule acetylation by Tubacin treatment led to a strong nuclear recruitment of YAP in cells plated on 2 kPa substrates, recapitulating a phenotype of cells on stiff substrates (Fig. 2b). To determine whether the effects of microtubule acetylation on YAP were specific to astrocytes, we performed similar experiments on HUVECs. HUVECs exhibit low level of acetylated tubulin, even when plated on stiff substrates (Extended Data Fig. 3b,3c). Hence, to test the effects of microtubule acetylation on YAP translocation in HUVECs, cells were treated with Tubacin to increase microtubule acetylation (Extended Data Fig. 3c). Similar to astrocytes, increasing microtubule acetylation in HUVECs plated on 2 kPa substrates resulted in an increased nuclear recruitment of YAP, mimicking the phenotype of cells plated on stiff substrates (Fig. 2c). Altogether, this demonstrates that the manipulation of microtubule acetylation tunes YAP mechanosensitive nuclear-cytoplasmic shuttling (summary cartoon shown in Fig. 2g).

FAs are crucial mechanosensitive structures which sense and adapt to the matrix rigidity [START_REF] Jansen | Mechanotransduction at the cell-matrix interface[END_REF] . Thus, we investigated the effects of substrate rigidity on FAs in astrocytes plated on polyacrylamide hydrogels of different rigidities: 1.26 kPa, 2 kPa, 9 kPa and 48 kPa (Fig. 2d), by immunostaining of the FA-associated protein, Paxillin [START_REF] Ladoux | Front-Rear Polarization by Mechanical Cues: From Single Cells to Tissues[END_REF][START_REF] Prager-Khoutorsky | Fibroblast polarization is a matrix-rigidity-dependent process controlled by focal adhesion mechanosensing[END_REF] . Quantification of the density of FAs (Extended Data Fig. 3d) indicates that astrocytes plated on soft substrates display FAs throughout the cell surface, whereas, FAs are predominantly at the periphery of cells adhering to stiffer substrates (Fig. 2d). Thus, changes in substrate rigidity alter FA distribution in astrocytes, allowing us to assess the impact of microtubule acetylation in this phenomenon. Following αTAT1 depletion, FAs were distributed throughout the cell surface independently of the rigidity of the substrate, i.e., αTAT1-depleted cells on stiff substrates showed a distribution of FAs similar as that of control/siαTAT1 cells plated on soft substrates (Fig. 2e, Extended Data Fig. 3e-3h). To confirm that the changes in FA distribution act through tubulin acetylation, we treated cells plated on different rigidities with Tubacin. Tubacin had no effect on FA distribution in cells plated on 48 kPa substrates (Fig. 2f, Extended Data Fig. 3e and3f). In contrast, Tubacin treatment of cells plated on 1.26 kPa substrates mimicked the phenotype (FAs at the cell periphery) observed in control/Tubacin-treated cells plated on stiff matrices (Fig. 2f, Extended Data Fig. 3e and3f). Since cell spreading on different substrate rigidities can affect the distribution of FAs, we plated αTAT1-depleted cells on micropatterned hydrogels to observe FAs in cells of similar spread area. In line with our prior results, cells on soft substrates or αTAT1-depleted cells on stiffer substrates displayed FAs throughout the cell spreading area (Extended Data Fig. 3i). Together with previous findings that microtubule acetylation controls membrane vesicle delivery at FAs and FA dynamics [START_REF] Bance | Microtubule acetylation but not detyrosination promotes focal adhesion dynamics and astrocyte migration[END_REF][START_REF] Tran | HDAC6 deacetylation of tubulin modulates dynamics of cellular adhesions[END_REF] , these results show that the rigidity-dependent microtubule acetylation controls the mechanosensitive distribution of FAs (Fig. 2g).

Microtubule acetylation controls cytoskeletal organization at focal adhesions

Next, we investigated the impact of αTAT1 depletion on the FA-associated cytoskeleton. In migrating astrocytes, the actin network comprises of longitudinal stress fibres connected to FAs at the front of the leader cells as well as inter-junctional transverse arcs connecting neighbouring cells at adherens junctions (Fig. 3a and Extended Data Fig. 4a) [START_REF] Peglion | Adherens junction treadmilling during collective migration[END_REF][START_REF] De Pascalis | Intermediate filaments control collective migration by restricting traction forces and sustaining cell-cell contacts[END_REF] . In αTAT1-depleted cells, the transverse arcs of actin were dramatically reduced and the longitudinal fibres did not extend to the front of migrating cells (Fig. 3a, Extended Data Fig. 4a). Associated with these longitudinal fibres, FAs were located further back within the protrusion rather than at the front of leader cells (Fig. 3a and Extended Data Fig. 4a) [START_REF] Bance | Microtubule acetylation but not detyrosination promotes focal adhesion dynamics and astrocyte migration[END_REF] .

Myosin light chain phosphorylation (pMLC) is predominantly seen at the leading edge of migrating control cells; however, in αTAT1-depleted cells, pMLC was barely visible at the leading edge of cells and was only associated with the remaining actin fibres at the cell centre, similar to Myosin IIa distribution (Fig. 3a, 3b and Extended Data Fig. 4a). Moreover, intermediate filaments (visualised using Vimentin), which play a major role in regulating FAs and collective migration of astrocytes [START_REF] De Pascalis | Intermediate filaments control collective migration by restricting traction forces and sustaining cell-cell contacts[END_REF] and normally extend from the perinuclear region to the cell periphery close to FAs [START_REF] De Pascalis | Intermediate filaments control collective migration by restricting traction forces and sustaining cell-cell contacts[END_REF][START_REF] Sakamoto | APC binds intermediate filaments and is required for their reorganization during cell migration[END_REF] , were absent from the front of αTAT1depleted cells and frequently appeared fragmented (Fig. 3c). We then looked closely at the effect of αTAT1 on the cytoskeletal organisation at FAs by using platinum-replica transmission electron microscopy (EM) on unroofed migrating astrocytes located at the wound edge. As by light microscopy, EM images showed that FAs connected to actin bundles were distributed further within the protrusion in case of siαTAT1 cells as compared to a highly organised and parallel set of FAs at the leading edge of control cells (Fig. 3d). From the high magnification views of FAs in the control cells, microtubules were often seen along actin cables reaching FAs (Fig. 3d, marked with white arrows in i.1). Intermediate filaments were also clearly visible, intertwined with the actin filaments at FAs (Fig. 3d, marked with yellow arrows in i. 2 and i. 3). In siαTAT1 cells, the actin bundles near FAs were strikingly thinner than those in controls (Fig. 3d). We consistently observed a lack of microtubules and intermediate filaments associated with FAs in αTAT1-depleted cells (Fig. 3d). All these results strongly support a role for αTAT1 in the cytoskeletal organisation at FAs in astrocytes. We then tested the effects of microtubule acetylation on the cytoskeletal organisation by treating HUVECs and astrocytes plated on soft substrates with Tubacin to increase microtubule acetylation. In Tubacin-treated cells, we observed prominently larger actin cables, increased Myosin-IIa filaments, together with larger and more peripheral FAs (Fig. 3e and Extended Data Fig. 3b), reminiscent of the phenotype observed in astrocytes plates on stiff substrates (Fig. 3f), confirming that microtubule acetylation regulates actomyosin organization.

Microtubule acetylation increases actomyosin contractility via GEF-H1 and Rho

One major impact of mechanosensing is the adaptation of traction forces to the rigidity of the substrate [START_REF] Schiller | beta1-and alphav-class integrins cooperate to regulate myosin II during rigidity sensing of fibronectin-based microenvironments[END_REF] . We thus investigated whether microtubule acetylation might affect traction forces by plating control or αTAT1-depleted astrocytes on crossbow-shaped micropatterned hydrogels. By traction force microscopy, we observed that αTAT1 depletion resulted in lower traction force production on 40 kPa substrates (Fig. 4a and4b). In contrast, overexpression of GFP-αTAT1 increased traction energies and forces and also rescued the effect observed on αTAT1 knockdown (siαTAT1 + GFP-αTAT1;

Fig. 4a and4b) when astrocytes were plated on 40 kPa. In addition, Tubacin-treated astrocytes on soft 2 kPa hydrogels showed increased traction energies and forces, comparable to control astrocytes plated on 40 kPa (Fig. 4c). We also studied the effects of microtubule acetylation on traction forces in Tubacintreated HUVECs plated on soft 2 kPa substrates. Similar to astrocytes, increased microtubule acetylation using Tubacin resulted in higher traction energies and forces in HUVECs (Fig. 4d). Thus, the level of microtubule acetylation dictates traction forces exerted on the substrate through FAs. This further illustrates the essential role of the mechanosensitive regulation of microtubules in force transmission.

The crucial role of microtubule acetylation in the controlling cytoskeletal organisation and traction forces led us to further investigate the molecular mechanisms involved in this process. We focused on RhoA, a small G protein of the Rho family, well-known for promoting stress fibre formation and actomyosin contractility, via its effector ROCK and MLC phosphorylation. Pulldown of GTP-bound active RhoA using GST-Rhotekin beads showed that αTAT1 depletion reduced RhoA activity (Fig. 4e), suggesting that microtubule acetylation may promote actomyosin contractility by activating RhoA. RhoA activation is mediated by Guanine nucleotide exchange factors (GEFs) [START_REF] Lawson | Rho GTPase signaling complexes in cell migration and invasion[END_REF] . Amongst these GEFs, GEF-H1

(also known as ARHGEF2) is a microtubule-bound RhoGEF which, when released from microtubules triggers the Rho-ROCK signalling cascade and cell contractility [START_REF] Krendel | Nucleotide exchange factor GEF-H1 mediates crosstalk between microtubules and the actin cytoskeleton[END_REF][START_REF] Ren | Cloning and characterization of GEF-H1, a microtubuleassociated guanine nucleotide exchange factor for Rac and Rho GTPases[END_REF] . Previously, substrate stiffness was suggested to correlate with GEF-H1 activity and actomyosin contractility [START_REF] Heck | Microtubules regulate GEF-H1 in response to extracellular matrix stiffness[END_REF] . In addition, GEF-H1 was recently shown to be controlled by the interaction of microtubules with integrin-mediated adhesions [START_REF] Rafiq | A mechano-signalling network linking microtubules, myosin IIA filaments and integrin-based adhesions[END_REF] , leading us to investigate whether integrin-mediated microtubule acetylation could affect the association of GEF-H1 with microtubules. In control astrocytes plated on rigid glass coverslips, GEF-H1 localised partially (approximately 42%) on microtubules but also in the cytosol, a localisation that corresponds to the active/released GEF-H1 (Fig. 5a and Extended Data Fig. 5a) [START_REF] Jiu | Vimentin intermediate filaments control actin stress fiber assembly through GEF-H1 and RhoA[END_REF] . In contrast, in αTAT1-depleted cells, GEF-H1 was predominantly localised on microtubules (approximately 60%; Fig. 5a and Extended Data Fig. 5a). Tubacin treatment of αTAT1-depleted cells led to the release of GEF-H1 into the cytosol, rescuing the effect of αTAT1 depletion and confirming the role of microtubule acetylation in the GEF-H1 localisation (Fig. 5a and Extended Data Fig. 5a). In control HUVECs (which exhibit relatively low levels of acetylated tubulin) GEF-H1 was mostly bound to microtubules (approximately 50%; Fig. 5b). Like in astrocytes, the treatment of HUVECs with Tubacin resulted in the release of GEF-H1 into the cytoplasm, with only 38% of GEF-H1 colocalising with microtubules (approximately 38%; Fig. 5b). GEF-H1 association with microtubules was confirmed using GFP-immunoprecipitation experiments using HEK cells transfected with GFP-Ctl or GFP-GEF-H1 (Fig. 5c). Like Nocodazole treatment which leads to the depolymerisation of microtubules, Tubacin treatment (which increases microtubule acetylation)

abolished the binding of GEF-H1 to microtubules (Fig. 5c). Altogether, these results show that microtubule acetylation promotes the release of GEF-H1 from microtubules into the cytoplasm.

We then tested whether substrate rigidity and αTAT1 play a role in the association of GEF-H1 with microtubules. Astrocytes plated soft 2 kPa substrates exhibit a higher microtubule-bound portion of GEF-H1 as compared to cells on stiff 48 kPa gels (Fig. 5d and Extended Data Fig. 5b). In αTAT1-depleted cells plated on stiff substrates, GEF-H1 was found predominantly bound to microtubules, mimicking the phenotype observed in cells plated on soft substrates (Fig. 5d and Extended Data Fig. 5b). These results strongly suggest that rigidity-dependent microtubule acetylation contributes to mechanotransduction by enabling RhoA activation and actomyosin contractility.

Microtubule acetylation modulates mechanosensitive cell migration by tuning mechanotransduction at focal adhesions

The involvement of microtubule acetylation in mechanotransduction and in the mechanosensitive regulation of FAs and actomyosin contractility led us to investigate its influence on cell migration, which has often been described as a mechanosensitive cellular response [START_REF] Sun | Integrin-mediated mechanotransduction[END_REF][START_REF] Trepat | Physical forces during collective cell migration[END_REF] . To this end, we developed a collective migration assay on hydrogels, where microdropping a small amount of a chemical (sodium hydroxide) induced a circular wound in the cell monolayer (Fig. 6A, Supplementary Video 3). Sodium hydroxide treatment did not noticeably alter the hydrogel properties (see Methods). Wild-type astrocytes migrated significantly slower on 2 kPa gels than on 48 kPa gels (Fig. 6B, Supplementary Video 4), implying that astrocyte migration speed is affected by substrate rigidity. Most importantly, αTAT1 depletion abolished the increase of cell speed observed on stiff 48 kPa gels (Fig. 6C, Supplementary Video 5), so that the migration speed of αTAT1-depleted cells plated on 48 kPa was similar to that of control and αTAT1-depleted cells on 2 kPa substrates. In addition, the effects of αTAT1 depletion in cells plated on 48 kPa gels is in line with our previous results described on the role of αTAT1 during collective migration on glass coverslips [START_REF] Bance | Microtubule acetylation but not detyrosination promotes focal adhesion dynamics and astrocyte migration[END_REF] . Thus, we demonstrate that microtubule acetylation is required for the mechanosensitive regulation of astrocyte collective migration.

Discussion

Our results show that microtubules are regulated in response to substrate rigidity sensing and in turn, play a key role in mechanotransduction by participating in mechanosensitive cellular responses, including translocation of the transcriptional coactivator YAP, focal adhesion distribution, actomyosin contractility, generation of traction forces and cell migration (Fig. 6d). Our results point to a mechanism by which Talin-mediated mechanosensing downstream of integrin-mediated signalling (Fig. 6d-i) controls the recruitment of αTAT1 to FAs (Fig. 6d-ii) and induces microtubule acetylation (Fig. 6d-iii). We

show that αTAT1 interacts with Talin and that, on rigid substrates, Talin is required for the recruitment of αTAT1 to FAs, and for the acetylation of microtubules (Fig. 6d-ii). Although we do not have any clear evidence of a direct interaction of Talin with αTAT1, the β 1 -integrin-and actomyosin-dependent association of αTAT1 with Talin suggest that tension-induced changes in Talin conformation may be involved in inducing αTAT1 recruitment to FAs and microtubule acetylation.

How αTAT1 enters the lumen of microtubules still remains unclear although one can speculate that αTAT1 accesses the lumen through microtubule lattice defects or through the open ends [START_REF] Janke | Causes and Consequences of Microtubule Acetylation[END_REF] . Growing microtubule ends are often seen in close proximity to FAs at the leading edge of migrating cells [START_REF] Seetharaman | Microtubules at focal adhesions -a double-edged sword[END_REF] (Extended Data Fig. 5C). It is very likely that, like other FA-associated proteins, a large proportion of talin-associate αTAT1 rapidly exchange with a cytosolic pool [START_REF] Legerstee | Dynamics and distribution of paxillin, vinculin, zyxin and VASP depend on focal adhesion location and orientation[END_REF] . This lead us to propose a model in which the increased interaction of αTAT1 with talin increases the local concentration of αTAT1 and facilitates its entry in the microtubule lumen through microtubule open ends in the vicinity of adhesions (Fig. 6Diii; Extended Data Fig. 5C).

We show that microtubule acetylation reorganises the actomyosin network and promotes traction forces. Therefore, we propose a feedback mechanism involving a crosstalk between microtubules and actin wherein, actomyosin-dependent mechanosensing promotes microtubule acetylation which, in turn, facilitates the release of GEF-H1 from microtubules into the cytosol (Fig. 6D-iv) to increase RhoA activity (Fig. 6D-v), cell contractility (Fig. 6D-vi) and traction forces (Fig. 6D-vii). It was recently shown that uncoupling microtubules from FAs results in a similar release of microtubule-bound GEF-H1 into the cytosol [START_REF] Rafiq | A mechano-signalling network linking microtubules, myosin IIA filaments and integrin-based adhesions[END_REF] , which then triggers myosin IIA assembly and increased cell contractility through RhoA.

Suppression of RhoA activity in the absence of αTAT1 might be due to the sequestering of GEF-H1 by non-acetylated microtubules. Whether microtubule acetylation directly or indirectly induces the release of GEF-H1 remains unclear. One can speculate that acetylation of Lysine 40 changes the conformation of α-tubulin and modifies the binding site for GEF-H1, thereby, inducing its release from microtubules.

Alternatively, acetylation might change the mechanical properties of the microtubule lattice and affect its curvature [START_REF] Trepat | Physical forces during collective cell migration[END_REF][START_REF] Legerstee | Dynamics and distribution of paxillin, vinculin, zyxin and VASP depend on focal adhesion location and orientation[END_REF] , which would in turn facilitate the release of GEF-H1 from microtubules. It is also possible that any of the previous mechanisms affect a binding of a partner of GEF-H1, which alters GEF-H1 conformation (via phosphorylation for instance) and promote its release into the cytosol.

Interestingly, microtubule acetylation also affects the association of intermediate filaments with actin bundles at FAs. Since intermediate filaments have also been involved in the control of FA dynamics, actomyosin contractility as well as GEF-H1 activity [START_REF] De Pascalis | Intermediate filaments control collective migration by restricting traction forces and sustaining cell-cell contacts[END_REF][START_REF] Jiu | Vimentin intermediate filaments control actin stress fiber assembly through GEF-H1 and RhoA[END_REF] , the microtubule-actin interplay described here may also involve intermediate filaments, whose role in mechanotransduction is still elusive.

In response to substrate rigidity sensing, cells adapt essential functions such as migration [START_REF] Barriga | Tissue stiffening coordinates morphogenesis by triggering collective cell migration in vivo[END_REF] . In the absence of αTAT1, cells plated on stiff substrates produce less traction forces and migrate slower (Fig. 6d-viii). The relationship between forces and migration can be considered counter-intuitive for single cell migration where traction forces appear to correlate with migration speed; and contractile forces are required for migration but exaggerated traction forces tend to slow or even stop migration [START_REF] Leal-Egaña | The size-speed-force relationship governs migratory cell response to tumorigenic factors[END_REF] . In fact, measurements of traction forces exerted by astrocytes plated on substrates of various rigidities shows that there is a rigidity optimum (around 50 kPa), at which the forces are maximal (Extended Data Fig. 5d). When looking at microtubule acetylation, actomyosin contractility, or cell migration, it is clear that all of them increase up to about 48kPa but then reach a plateau or even decrease when cells are plated on glass, indicative of a non-monotonous response. In collective cell migration, the distribution of forces within the migrating monolayer, more than the intensity itself, is crucial for the efficient control of collective cell migration. Also, collective cell migration relies on the transmission of forces between the leaders and followers. Cell-cell junctions not only transmit forces between cells but help maintain the integrity of the monolayer [START_REF] Peglion | Adherens junction treadmilling during collective migration[END_REF][START_REF] Sunyer | Collective cell durotaxis emerges from long-range intercellular force transmission[END_REF] , which improves collective and directed cell migration. Alteration of microtubule acetylation did not induce any detachment of leader cells from followers and did not affect directionality or persistence of migration [START_REF] Bance | Microtubule acetylation but not detyrosination promotes focal adhesion dynamics and astrocyte migration[END_REF] . Therefore, we propose a model in which, on stiff substrates, increased microtubule acetylation would trigger higher traction forces in leader cells, which would transmit these pulling forces to followers and increase collective migration speed. Microtubule acetylation may also affect cell migration by controlling FA dynamics. We have previously demonstrated that microtubule acetylation promotes the fusion of Rab6-dependent post-Golgi carriers at FAs and increases FA turnover [START_REF] Bance | Microtubule acetylation but not detyrosination promotes focal adhesion dynamics and astrocyte migration[END_REF] , such that they are continuously renewed at the front of leader cells. Depletion of αTAT1 (as that of Rab6) alters the distribution of FAs in migrating cells [START_REF] Bance | Microtubule acetylation but not detyrosination promotes focal adhesion dynamics and astrocyte migration[END_REF] . The change in FA distribution between the cell center and the cell periphery suggests that rigidity-induced acetylation of microtubules may promote FA turnover during cell spreading, thereby, excluding them from the cell center. We speculate that the increase in FA turnover together with the increase in cell contractility participate in strengthening the traction forces exerted on FAs concentrated at the leader cell wound edge and improve astrocyte collective migration (Fig. 6d-viii).

In conclusion, our results depict a crosstalk between the actin and microtubule cytoskeletal networks (Fig. 6d), whereby microtubule acetylation, downstream of rigidity-dependent integrinmediated signalling, alters actomyosin contractility as well as focal adhesion distribution and dynamics to promote mechanosensitive migration of astrocytes, thus closing a crucial feedback loop governing mechanotransduction at FAs (Fig. 6d).

Methods

Cell culture

Primary astrocytes were obtained from E17 rat embryos [START_REF] Etienne-Manneville | In vitro assay of primary astrocyte migration as a tool to study Rho GTPase function in cell polarization[END_REF] . Use of these animals is in compliance with ethical regulations and has been approved from the Prefecture de Police and Direction départementale des services vétérinaires de Paris. Astrocytes were grown in 1g/L glucose DMEM supplemented with 10% FBS (Invitrogen, Carlsbad, CA), 1% penicillin-streptomycin (Gibco) and 1% Amphotericin B (Gibco) at 5% CO 2 and 37°C. Human umbilical vein endothelial cells (HUVECs) were cultured in EGM-2 BulletKit media (Lonza, catalogue no. CC-3162).

Cell nucleofection

Astrocytes were transfected with Lonza glial transfection solution and electroporated with a Nucleofector machine (Lonza). Cells were then plated on appropriate supports previously coated with poly-L-Ornithine (Sigma). Experiments are carried out 3 or 4 days post-transfection and comparable protein silencing was observed. siRNAs were used at 1 nmol and DNA was used at 5 µg. siRNA sequences used were:

Luciferase (control) UAAGGCUAUGAAGAGAUAC; αTAT1 rat (siαTAT1-1): 5'-ACCGACACGUUAUUUAUGU-3' and 5'-UUCGAAACCUGCAGGAACG-3'; αTAT1 rat (siαTAT1-2): 5'-UAAUGGAUGUACUCAUUCA-3' and 5'-UCAUGACUAUUGUAGAUGA-3'; Talin-1 rat (siTalin1): 5'-GGGACACCAUGGAGUACAGAA-3', 5'-GACCCCCACUUCCAGGAUAUU-3', 5'-GCUGGAAGCUGUGGAUAACCU-3', and 5'-GACCUCCACCCCUGAAGAUUU-3'; β 1 integrin (AUUGCCAGAUGGAGUAACA).

Constructs used were: GFP-αTAT1 and GST-αTAT1 (gift from Philippe Chavrier, Institut Curie and Guillaume Montagnac, Institut Gustave Roussy) and mCherry-vinculin. siαTAT1-1 and siαTAT1-2 are pools of two siRNAs each. For all experiments, siαTAT1-2 has been used due to better knockdown efficiency as seen in Extended Fig. 1a and previously characterized in [START_REF] Bance | Microtubule acetylation but not detyrosination promotes focal adhesion dynamics and astrocyte migration[END_REF] . A few major experiments have been carried out with siαTAT1-1 and the results are shown in the respective extended data figures.

Cell treatment

Cells

Preparation of homogenously coated polyacrylamide hydrogels

Protocol to prepare polyacrylamide hydrogels was adapted from [START_REF] Trepat | Physical forces during collective cell migration[END_REF]46,[START_REF] Tambe | Collective cell guidance by cooperative intercellular forces[END_REF] . Coverslips were plasma cleaned for 3 min and silanised for 10 min using a solution of 1% (v/v) 3-(trimethoxysilyl)propyl methacrylate and 1% (v/v) acetic acid in ethanol. Coverslips were then washed twice with absolute ethanol and dried. A solution of polyacrylamide (proportions of acrylamide and bisacrylamide in the solution define the rigidity of the hydrogel) was prepared. 2.5 µl 10% APS and 0.25 µl TEMED was added and mixed well with the solution. A 50 µl drop of the solution was added over each coverslip (20x20 mm) and immediately, an 18x18 mm coverslip was placed gently over the solution. The solution was allowed to polymerise for 1 h at room temperature. HEPES was added over the coverslips to detach the top glass.

The polymerised gel was then activated under UV for 5 min using Sulpho-SANPAH and washed with HEPES twice. The hydrogels were then coated with 100 µg/ml of rat tail collagen I overnight at 4 o C. The excess collagen was washed once with PBS and approximately 5 x 10 4 cells/ml were plated on hydrogels.

Micropatterning

Coverslips were plasma-cleaned for 45 s and incubated with 0.1 mg/ml poly-l-lysine/polyethylene glycol (PLL-PEG) diluted in 10 mM HEPES for 30 min at room temperature. Excess of PLL-PEG was allowed to slide down by gravity, and the coverslips were dried and stored at 4 o C overnight before printing.

Micropatterns were printed on previously prepared PLL-PEG coverslips for 3 min with specifically designed chrome masks and coated with 50:50 v/v fibronectin/fibrinogen mixture (20 μg/ml each) diluted in fresh NaHCO 3 , pH 8.3, 100 mM, for 30 min at room temperature. Micropatterned coverslips were washed thrice in NaHCO 3 and used immediately for transfer on hydrogels. Plated cells (approximately 6 x 10 4 cells/ml) were allowed to adhere for 16 h before imaging/fixation. Crossbow patterns have an area of ∼2500 µm 2 .

Traction force experiments on micropatterned substrates

After printing and coating micropatterned coverslips, gel mixtures were prepared as follows: 40.4 kPa -100 µl 40% acrylamide, 120 µl 2% bisacrylamide, 280 µl water (final concentration of 8% acrylamide and 0.48% bisacrylamide in water); 2.61 kPa -50 µl 40% acrylamide, 25 µl 2% bisacrylamide, 425 µl water (final concentration of 8% acrylamide and 0.048% bisacrylamide in water). 5 µl of fluorescent microbeads (FluoSpheres; Molecular Probes) were incubated with 0.1 mg/ml PLL-PEG on a rotator for 1 h at 4°C prior to mixing with the gel mixture. The PLL-PEG coated beads were washed and centrifuged thrice at 1000 rpm for 10 min with 10 mM HEPES. These beads were then mixed with 165 µl of the gel mixture. After adding 1 µl APS and 1 µl TEMED, the solution was added as a 25 µl drop on a silanised 20x20 mm coverslip. The protein-coated micropatterned coverslip was then gently placed on the gel mixture. The solution was allowed to polymerize for 25 min at room temperature. After detaching the micropatterned glass from the polymerized gel, 6 x 10 4 cells/ml were plated on the gels and allowed to adhere and spread for 16 h before TFM experiments. Stacks of single micropatterned cells were acquired before and after trypsin treatment. Acquisitions were performed with a Nikon Eclipse Ti-E epifluorescence inverted microscope, with a 40× 1.49 NA air objective equipped with a pco.edge sCMOS camera and Metamorph software. Cells were maintained at 5% CO 2 and 37°C in normal astrocyte medium during acquisition.

Migration assays

In vitro wound healing assays: cells were plated on appropriate supports (dishes, plates, coverslips or glass-bottom MatTek) and grown to confluence. The cell monolayer was scratched with a p200 pipette tip to induce migration and imaged during 8h.

Migration on soft substrates: cells were plated on hydrogels of different rigidities in 6-well glass bottom plates (MatTek) and allowed to grow for 72 h after transfection before creating a chemical wound. 0.05 M NaOH was gently dropped on the cells using a microinjector. NaOH is used to instantly kill and detach cells localized in a region of a few hundred microns. The NaOH is added with a microinjector which delivers a fraction of a nanoliter of NaOH (0.05M) on a cell monolayer, which immediately mixes with 2 ml of culture medium that is present in the culture dish. The dead cells and debris were gently washed away using PBS. Fresh medium was added to the cells. Cells were placed on the microscope for live imaging. In these conditions, the hydrogel is only very locally and briefly exposed to the chemical. As cells migrate into this space (wounded area of the monolayer), the fibronectin coating and the surface of the hydrogel remains intact with negligible and unnoticeable changes. recorded on a CCD camera with a Leica software, or imaged at the basal plane using Yokogawa CSU-W1, Nikon TiE Spinning disk confocal microscope, at 100x magnification.

TIRF microscopy

Acquisitions were performed with a Nikon Eclipse Ti-E epifluorescence inverted microscope, with a 60×

1.49 NA oil objective equipped with a pco.edge sCMOS camera with Metamorph software. Cells were maintained at 5% CO 2 and 37°C in normal astrocyte medium. After acquiring a 15 min movie of GFP-αTAT1 and mCherry-vinculin, Nocodazole or Y27 were added and images were acquired every 2 min for 1 h.

Transmission electron microscopy

Adherent migrating astrocytes plated on glass coverslips were disrupted by sonication as described previously [START_REF] Heuser | The production of 'cell cortices' for light and electron microscopy[END_REF] . Coverslips were unroofed by scanning the coverslip with rapid sonicator pulses in KHMgE buffer (70 mM KCl, 30 mM HEPES, 5 mM MgCl 2 , 3 mM EGTA, pH 7.2). Paraformaldehyde 2%/glutaraldehyde 2%-fixed cells were further sequentially treated with 0.5% OsO 4 , 1% tannic acid and 1% uranyl acetate prior to graded ethanol dehydration and Hexamethyldisilazane substitution (HMDS, Sigma-Aldrich). Dried samples were then rotary-shadowed with 2 nm of platinum and 5-8 nm of carbon using an ACE600 high vacuum metal coater (Leica Microsystems). Platinum replicas were floated off the glass by 5% hydrofluoric acid, washed several times by floatation on distilled water, and picked up on 200 mesh formvar/carbon-coated EM grids. The grids were mounted in a eucentric side-entry goniometer stage of a transmission electron microscope operated at 80 kV (Philips, model CM120) and images were recorded with a Morada digital camera (Olympus). Images were processed in Adobe Photoshop to adjust brightness and contrast and presented in inverted contrast.

Image analysis

Normalised mean intensity levels of acetylated and detyrosinated tubulin for immunofluorescence images were calculated as shown in equation 1:

1 = /
YAP nuclear/cytoplasmic ratio was calculated as shown in equation 2:

2 = ℎ ℎ
FA density was calculated as shown in equation 3:

3 = ℎ ℎ
Different regions of cells (entire cell, cell periphery-8 μm, 8 μm-16 μm, 16 μm-cell center) were analysed as depicted in Extended Data Fig. 2d.

For migration assays, nuclei of cells were manually tracked to determine the speed directionality and persistence of migration using the following formulae: Mean velocity ( ), persistance ( ) and directionality ( ) of cell migration are calculated shown in equations 4-6: for a given ( , ) coordinate of leading cell nucleus,

4 = ∑ 5 = ( -) + ( -) ∑ ( - ) + ( - ) 6 = |( - )| ∑ ϑ ( - ) + ( - )
Proteins on beads were washed twice with 100 μL of 25 mM NH 4 HCO 3 and we performed on-beads digestion with 0.2 μg of trypsine/LysC (Promega) for 1 h in 100 µL of 25 For identification, the data was searched against the Homo sapiens (UP000005640) SwissProt database using Sequest HF through proteome discoverer (version 2.2). Enzyme specificity was set to trypsin and a maximum of two missed cleavage site were allowed. Oxidized methionine, N-terminal acetylation, and carbamidomethyl cysteine were set as variable modifications. Maximum allowed mass deviation was set to 10 ppm for monoisotopic precursor ions and 0.6 Da for MS/MS peaks. The resulting files were further processed using myProMS 50 v3.6 (work in progress). FDR calculation used Percolator and was set to 1% at the peptide level for the whole study. The label free quantification was performed by peptide Extracted Ion Chromatograms (XICs) computed with MassChroQ version 2.2 [START_REF] Valot | MassChroQ: a versatile tool for mass spectrometry quantification[END_REF] . For protein quantification, XICs from proteotypic peptides shared between compared conditions (TopN matching)

with no missed cleavages were used. Median and scale normalization was applied on the total signal to correct the XICs for each biological replicate. To estimate the significance of the change in protein abundance, a linear model (adjusted on peptides and biological replicates) was performed and p-values were adjusted with a Benjamini-Hochberg FDR procedure with a control threshold set to 0.05. Upregulated proteins with at least 3 proteotypic peptides (fold change > 1.5 and p-value < 0.05) and the unique proteins identified only in the GFP-αTAT1 were used for gene ontology (GO) enrichment analysis by using GO::TermFinder tools (10.1093/bioinformatics/bth456) through myProMS.

The mass spectrometry proteomics data have been deposited to the ProteomeXchange Consortium via the PRIDE [START_REF] Vizcaino | update of the PRIDE database and its related tools[END_REF][START_REF] Perez-Riverol | Extended Data Figure 2: a, b. Astrocytes were treated with (a) RGD control or RGD peptides, (b) solvant (Ctl) or MnCl 2 , and (a. i, b. i) stained for acetylated tubulin, α-tubulin, paxillin or DAPI; (a. ii, b. ii) ratio of the intensities of Acetylated tubulin over total Tubulin; n = 146 for RGD control, 99 for RGD peptide, 116 for Ctl and 105 for MnCl 2 . c, d. Astrocytes were transfected with siCtl or siβ 1 integrin; (c. i) Western blots showing levels of β 1 integrin, Acetylated tubulin and GAPDH; (c. ii, iii) ratio of intensities of β 1 integrin or Acetylated tubulin over GAPDH, normalised to the levels in siCtl respectively; (d. i) stained for Acetylated tubulin, α-tubulin and DAPI; (d. ii) ratio of the intensities of Acetylated tubulin over total Tubulin; n = 122 for siCtl and 117 for siβ 1 integrin. e. Astrocytes were treated with Ctl, Src kin 1 or PF-562271; (e. i) stained for Acetylated tubulin, α-tubulin and Paxillin; (e. ii) ratio of the intensities of Acetylated tubulin over total Tubulin intensity; n = 78 for Ctl, 70 for Src kin[END_REF] partner repository with the dataset identifier PXD015871 (username: reviewer89842@ebi.ac.uk; password: EC4DSdRf; project name: Potential interactors of alpha-tubulin acetyltransferase 1 (αTAT1/MEC17).

Rho activation assay

Rho activation assay was performed using a RhoA Pull-down Activation Assay Biochem Kit from Cytoskeleton Inc (BK036-S). In short, cells were lysed in ice-cold Cell Lysis Buffer plus 1x protease inhibitor cocktail. Cells were centrifuged at 10,000 g at 4°C for 10 min to pellet cell membranes and insoluble material. Part of the supernatant was stored as input for western blot. The remaining supernatant was divided equally into two parts (300-800 μg protein/tube). 1/15 th the volume of Loading Buffer was added to each tube (final conc. 15 mM). Then, 1/100 th the volume of GTPγS was added to one of the tubes and used as positive control (final conc. 0.2 mM). All tubes were incubated at room temperature for 15 min. The reaction was stopped by adding 1/10 th the volume of STOP Buffer to all tubes (final conc. 60 mM).

Rhotekin-RBD (50 μg) beads were resuspended and added to the tubes. Tubes were rotated at 4 o C for 1 h and centrifuged at 5000 g at 4 o C for 3 min. Beads were washed with 500 μl each of Wash Buffer. 10-20 ul of Laemmli sample buffer was added to each tube.

Western blot

Cells lysates were obtained with Laemmli buffer composed of 60 mM Tris-HCl pH6.8, 10% glycerol, 2%

SDS and 50 mM DTT with the addition of anti-protease (cOmplete cocktail, Roche 11 873 588 001).

Samples were boiled 5 min at 95°C before loading on polyacrylamide gels. Transfer occurred at 100V for 1 h on nitrocellulose membranes. Membranes were blotted with TBST (0.2% Tween) and 5% milk and incubated 1 h with the primary antibody and 1 h with HRP-conjugated secondary antibody. Bands were revealed with ECL chemoluminescent substrate (Biorad).

Primary antibodies used: Antibody anti-Acetylated Tubulin (1:10000; clone 6-11B-1, mouse monoclonal, T-6793 Sigma Aldrich), anti-αTAT1 (1:400; HPA046816, rabbit, Atlas Antibodies), anti-Poly-Glu Tubulin

Graphs and statistics

All graphs and statistics were done using Graphpad Prism.

Data availability

The data supporting the findings of this study are available within the manuscript. The mass spectrometry proteomics data have been deposited to the ProteomeXchange Consortium via the PRIDE

Main figure legends

( 1 :

 1 1000; AbC0101, ValBiotech), anti-α-Tubulin (1:1000; MCA77G, rat, Biorad), anti-β 1 integrin (ab52971, Abcam), anti-Talin (1:1000; T3287, mouse monoclonal, clone 8D4, lot 035M4805V; Sigma-Aldrich), anti-GEF-H1 (1:100; ab155785, Abcam), anti-GAPDH (MAB374, lot 2689153 Millipore). Secondary HRP antibodies were all purchased from Jackson ImmunoResearch. Unmodified western blots of Figures1, 4

and 5 ,

 5 as well as Extended Data Figures 1, 2 and 3 can be found in the corresponding source data files.

Figure 1 :

 1 Figure 1: Integrin-mediated signalling and substrate rigidity regulate microtubule acetylation. a, b, c.

Figure 2 :

 2 Figure 2: Microtubule acetylation tunes cell mechanosensitivity. a, b. Astrocytes transfected with siCtl

Figure 3 : 2 )

 32 Figure 3: Microtubule acetylation reorganises the actomyosin and intermediate filament networks. a,

Figure 4 :

 4 Figure 4: Microtubule acetylation promotes traction force generation and Rho activation. a, b, c, d. (a-

Figure 5 :

 5 Figure 5: Microtubule acetylation promotes the release of GEF-H1 into the cytosol. a. Migrating

Figure 6 : 3 (

 63 Figure 6: Microtubule acetylation is required for mechanosensitive migration. a, b, c. Phase contrast

  NaOH treatment did not affect the weight of the gel -indicative of gel swelling-; it did not induce any detectable bead movementsindicative of gel contraction or expansion of the gel-, nor did it affect the traction forces of cells, indicating that cells did not sense any difference in substrate rigidity. Acquisition was started 30 min after wounding. Supplementary Videos were acquired with a Zeiss Axiovert 200M or a Nikon Eclipse Ti-E

	epifluorescence inverted microscope with cells maintained at 5% CO 2 and 37°C. All images were
	acquired with dry objective 10X 0.45 NA and an EMCCD camera/pco.edge sCMOS camera and

Metamorph software. Images were acquired every 15 min for 24 h. Nuclei of leader cells were manually tracked with Fiji software (Manual Tracking plugin).

  mM NH 4 HCO 3 . Sample were then loaded onto a homemade C18 StageTips for desalting. Peptides were eluted using 40/60

	MeCN/H 2 O + 0.1% formic acid and vacuum concentrated to dryness. Online chromatography was
	performed with an RSLCnano system (Ultimate 3000, Thermo Scientific) coupled to an Orbitrap Fusion
	Tribrid mass spectrometer (Thermo Scientific). Peptides were trapped on a C18 column (75 μm inner
	diameter × 2 cm; nanoViper Acclaim PepMapTM 100, Thermo Scientific) with buffer A (2/98 MeCN/H 2 O
	in 0.1% formic acid) at a flow rate of 4.0 µL/min over 4 min. Separation was performed on a 50 cm x 75
	μm C18 column (nanoViper Acclaim PepMapTM RSLC, 2 μm, 100Å, Thermo Scientific) regulated to a
	temperature of 55°C with a linear gradient of 5% to 25% buffer B (100% MeCN in 0.1% formic acid) at a

flow rate of 300 nL/min over 100 min. Full-scan MS was acquired in the Orbitrap analyzer with a resolution set to 120,000 and ions from each full scan were HCD fragmented and analyzed in the linear ion trap.

  Data Figures 1, 2 and 3 can be found in the corresponding source data files. Other raw data generated during this study are available on reasonable request.

	52,53	partner	repository	with	the	dataset	identifier	PXD015871	(username:

reviewer89842@ebi.ac.uk; password: EC4DSdRf; project name: Potential interactors of alpha-tubulin acetyltransferase 1 (αTAT1/MEC17). Unmodified western blots of Figures

1, 4 and 5

, as well as Extended

Acknowledgements

This work was supported by the La Ligue contre le cancer (S-CR17017), Centre National de la Recherche Scientifique and Institut Pasteur. SS is funded by the ITN PolarNet Marie Curie grant and Fondation pour la Recherche Médicale (FDT201904007930) and was enrolled at the Ecole Doctorale FIRE Frontières de l'Innovation en Recherche et Éducation (Frontières du Vivant FdV -Programme Bettencourt), CRI and the Université de Paris. We would like to thank Anna Akhmanova for providing GFP-Talin construct, Cécile Leduc for FA distribution macro, Bertille Bance and Franck Coumailleau for discussions, Nisha Mohd Rafiq for fixation protocols and discussions, Emma van Bodegraven for critical reading of the manuscript and discussions.

Code availability

Codes for focal adhesion distribution and traction forces are available from the authors upon request.

where is the number of time points acquired and is cell velocity.

Analysis of TFM on micropatterns was performed with a custom-designed macro in Fiji based on work by 49 . The topmost planes of beads before and after trypsinisation were selected and aligned using a normalized cross-correlation algorithm (Align Slices in Stack plugin). The displacement field was computed from bead movements using particle image velocimetry (PIV). Parameters for the PIV were three interrogation windows of 128, 64, and 32 pixels with a correlation of 0.60. Traction forces were calculated from the displacement field using Fourier transform traction cytometry and a Young's modulus of 40 kPa or 2 kPa, a regularization factor of 10 -9 , and a Poisson ratio of 0.5.

For the localisation of GEF-H1 on microtubules, the percentage GEF-H1 on microtubules was calculated as shown in equation 7:

× 100

Immunoprecipitations and pulldown assays

Cell lysates were prepared by scraping cells in 1X lysis buffer (10X buffer recipe: 50 mM Tris pH 7.5, triton 20%, NP40 10%, 2 M NaCl with Complete protease inhibitor tablet (Roche, Indianapolis, IN)) and centrifuged for 30 min at 13,000 rpm 4°C to pellet cell debris. Soluble detergent extracts were incubated with GFP-Trap agarose beads (Chromotek, GTP-20) for immunoprecipitation and with GST or GST-αTAT1 beads for 2 hr at 4°c prior to washing three times with wash buffer (50 mM ris pH 7.5, 150 mM NaCl, 0.1% triton , 1 mM EDTA, 2.5 mM MgCl 2 ). The resin was then mixed with Laemmli buffer and used for western blot and mass spectrometry analysis. HEK293 cells were used for mass spectrometry analysis in order to have enough amount of protein from cells expressing αTAT1.

Mass spectrometry

Author contributions

Competing interests

Authors declare no competing interests. Statistical tests: One-way ANOVA followed by Tukey's multiple comparison's test (for a).

Extended Data


L Lavoine

A Hammond

A Recati

D S Petrov

T Bourdel
email: thomas.bourdel@institutoptique.fr

Beyond-mean-field effects in Rabi-coupled two-component Bose-Einstein condensate

We theoretically calculate and experimentally measure the beyond-mean-field (BMF) equation of state in a coherently coupled two-component Bose-Einstein condensate (BEC) in the regime where averaging of the interspecies and intraspecies coupling constants over the hyperfine composition of the single-particle dressed state predicts the exact cancellation of the two-body interaction. We show that with increasing the Rabi-coupling frequency Ω, the BMF energy density crosses over from the nonanalytic Lee-Huang-Yang (LHY) scaling ∝ n 5/2 to an expansion in integer powers of density, where, in addition to a two-body BMF term ∝ n 2 √ Ω, there emerges a repulsive three-body contribution ∝ n 3 / √ Ω. We experimentally evidence these two contributions, thanks to their different scaling with Ω, in the expansion of a Rabi-coupled two-component 39 K condensate in a waveguide. By studying the expansion with and without Rabi coupling, we reveal an important feature relevant for observing BMF effects and associated phenomena in mixtures with spin-asymmetric losses: Rabi coupling helps preserve the spin composition and thus prevents the system from drifting away from the point of the vanishing mean field.

Vacuum effects are among the most striking features of quantum field theories. The high degree of control of cold gases has made these systems ideal candidates to identify and measure the role of quantum fluctuations in matter fields. For instance, using Bose gases, quantum phonon fluctuations [START_REF] Armijo | Direct Observation of Quantum Phonon Fluctuations in a One-Dimensional Bose Gas[END_REF], phononic Lamb shift [START_REF] Rentrop | Observation of the Phononic Lamb Shift with a Synthetic Vacuum[END_REF], dynamical phononic Casimir effect [START_REF] Jaskula | Acoustic Analog to the Dynamical Casimir Effect in a Bose-Einstein Condensate[END_REF], as well as evidence of Hawking-like radiation from analog gravity configurations [START_REF] Steinhauer | Observation of quantum Hawking radiation and its entanglement in an analogue black hole[END_REF] have been reported.

For a weakly interacting Bose gas quantum fluctuations cause the BEC depletion [START_REF] Chang | Momentum-Resolved Observation of Thermal and Quantum Depletion in a Bose Gas[END_REF][START_REF] Lopes | Quantum Depletion of a Homogeneous Bose-Einstein Condensate[END_REF] and lead to the socalled LHY correction to the mean-field (MF) equation of state [START_REF] Lee | Eigenvalues and Eigenfunctions of a Bose System of Hard Spheres and Its Low-Temperature Properties[END_REF]. This BMF correction has found experimental verification a few years ago [START_REF] Papp | Bragg Spectroscopy of a Strongly Interacting 85 Rb Bose-Einstein Condensate[END_REF][START_REF] Navon | Dynamics and Thermodynamics of the Low-Temperature Strongly Interacting Bose Gas[END_REF]. More recently, selftrapped quantum droplets were stabilized against collapse by BMF effects in single component dipolar BECs [START_REF] Ferrier-Barbut | Observation of Quantum Droplets in a Strongly Dipolar Bose Gas[END_REF][START_REF] Chomaz | Quantum-Fluctuation-Driven Crossover from a Dilute Bose-Einstein Condensate to a Macrodroplet in a Dipolar Quantum Fluid[END_REF] and in two-component BEC mixtures [START_REF] Petrov | Quantum mechanical stabilization of a collapsing Bose-Bose mixture[END_REF][START_REF] Cabrera | Quantum liquid droplets in a mixture of Bose-Einstein condensates[END_REF][START_REF] Semeghini | Self-Bound Quantum Droplets of Atomic Mixtures in Free Space[END_REF][START_REF] Guo | Yang effects in the ultracold mixture of 23 Na and 87 Rb with attractive interspecies interactions[END_REF]. In the latter case, the MF interaction is greatly reduced by the compensation between intraspecies repulsion and interspecies attraction while the BMF energy originating from quantum fluctuations remains finite.

In this Letter we consider a Bose gas formed by atoms with two internal levels, which are coherently coupled [START_REF] Goldstein | Quasiparticle instabilities in multicomponent atomic condensates[END_REF]. The BEC order parameter is then a two-component vector and the relative phase excitations are gapped due to the coherent drive. Such systems have been thoroughly analyzed on the MF level and used in variety of experiments on coherent Rabi oscillations [START_REF] Matthews | Watching a Superfluid Untwist Itself: Recurrence of Rabi Oscillations in a Bose-Einstein Condensate[END_REF][START_REF] Williams | Excitation of a dipole topological state in a strongly coupled two-component Bose-Einstein condensate[END_REF], internal selftrapping effects [START_REF] Zibold | Classical Bifurcation at the Transition from Rabi to Josephson Dynamics[END_REF], ferromagnetic classical bifurcation [START_REF] Nicklas | Rabi Flopping Induces Spatial Demixing Dynamics[END_REF], Kibble-Zurek mechanism [START_REF] Nicklas | Observation of Scaling in the Dynamics of a Strongly Quenched Quantum Gas[END_REF], control of the twobody interaction [START_REF] Hanna | Creation and manipulation of Feshbach resonances with radiofrequency radiation[END_REF][23][START_REF] Sanz | Interaction control and bright solitons in coherently-coupled Bose-Einstein condensates[END_REF], and magnetic domain wall dynamics [START_REF] Farolfi | Quantum-torque-induced breaking of magnetic domain walls in ultracold gases[END_REF]. For sufficiently attractive interspecies interaction such Rabi-coupled BECs are predicted to sustain droplets in the symmetric case, where the Rabi coupling is resonant and the intraspecies coupling constants are equal to each other. The stabilization mechanism is interpreted from the few-body perspective as an emergent three-body repulsion [23] or as a many-body BMF effect due to a structural change in the excitation spectrum [START_REF] Cappellaro | Equation of state and self-bound droplet in Rabi-coupled Bose mixtures[END_REF].

Here, we analytically calculate the BMF energy density in the experimentally relevant case of an asymmetric Rabi-coupled Bose mixture in the whole range of Rabi frequencies Ω. We show that the BMF energy crosses over from the LHY law ∝ n 5/2 for small Ω to the regular expansion in integer powers of n for large Ω. The quadratic term in the latter limit can be understood as a renormalization of the two-body interaction, the cubic term as an emergent three-body interaction, each term having a different scaling with Ω. In the large Ω limit, we experimentally evidence these two contributions by preparing a condensed 39 K spin mixture at the point of vanishing mean field [START_REF] Skov | Observation of a Lee-Huang-Yang Fluid[END_REF] and by quantitatively measuring its expansion in a waveguide as a function of Ω. In the absence of Rabi coupling, we observe that asymmetric losses between the two spin states drive the spin composition away from the point of zero MF interaction and thus strongly affect the expansion.

We consider a Bose gas of N atoms possessing two internal states, σ =↑, ↓, coherently coupled with the Rabi frequency Ω and detuning δ. In the rotating wave approx-imation the Hamiltonian of the mixture reads [START_REF] Goldstein | Quasiparticle instabilities in multicomponent atomic condensates[END_REF][START_REF] Abad | A study of coherently coupled two-component Bose-Einstein condensates[END_REF]

Ĥ = σ -Ψ † σr ∇ 2 r 2 Ψσr + σσ Ψ † σr ξ σσ Ψσ r + σ,σ g σσ 2 Ψ † σr Ψ † σ r Ψσr Ψσ r   dr, (1)
where Ψσr is the annihilation Bose field operator, ξ = -(δσ z + Ωσ x )/2 is the single particle spin Hamiltonian written in terms of the Pauli matrices acting in the |↑ -|↓ space, and g σσ = 4πa σσ are the coupling constants for the σ-σ interaction with the scattering lengths a σσ . In Eq. ( 1) and in the rest of the paper we adopt the units = m = 1, where m is the mass of the particles. Assuming zero-temperature and weak interactions, we follow the usual Bogoliubov procedure by separating the dominant condensate contribution and writing the field operator as Ψσr =

√ n σ + φσr , where n σ are the condensate densities and φσr annihilate particles with nonzero momenta. Neglecting φ and substituting Ψσr = √ n σ into Eq. ( 1) we obtain the MF energy density

E MF = δ(1 -α 2 ) -2Ωα 2(1 + α 2 ) n + g ↑↑ α 4 + g ↓↓ + 2g ↑↓ α 2 (1 + α 2 ) 2 n 2 2 , (2)
where n = n ↑ + n ↓ and α = n ↑ /n ↓ . In the limiting case of vanishing density (or interactions), E MF gets minimized for α = α 0 = δ/Ω + 1 + δ 2 /Ω 2 , consistent with the condensation in the dressed state |-= (α 0 |↑ + |↓ )/ 1 + α 2 0 , which is the ground state of ξ. The coefficient in front of n 2 /2 in Eq. ( 2) is then the MF coupling constant corresponding to the scattering length a --= (a ↑↑ α 4 0 + a ↓↓ + 2a ↑↓ α 2 0 )/(1 + α 2 0 ) 2 [START_REF] Sanz | Interaction control and bright solitons in coherently-coupled Bose-Einstein condensates[END_REF][START_REF] Search | Manipulating the speed of sound in a two-component Bose-Einstein condensate[END_REF]. For a ↑↑ > 0, a ↓↓ > 0, and a ↑↓ < 0, a --exhibits a minimum as a function of δ (or α 0 ). We are interested in the particular configuration of the scattering lengths (controlled by the magnetic field) and the RF drive parameters where this minimum touches zero. This translates into two conditions:

δa = 0 and α 0 = β, (3)
where we have introduced the scattering length detuning δa = a ↑↓ + √ a ↑↑ a ↓↓ and the interaction asymmetry parameter β = (a ↓↓ /a ↑↑ ) 1/4 . Note that for finite n the energy E MF , upon minimization with respect to α, is not, in general, a quadratic function of n since the optimal polarization parameter α does depend on n. In particular, the system can feature a three-body attraction already on the MF level [30]. However, the minimum of a --is a special point where both terms on the right-hand side of Eq. ( 2) are minimized at α = α 0 independent of n. At this point E MF is thus quadratic in n. If, in addition, we tune δa to zero [configuration (3)], the minimum of a -- also vanishes and the condensate becomes noninteracting on the MF level.

We shall now discuss the influence of BMF effects on the equation of state. In the Bogoliubov approach the leading BMF term is obtained by expanding the Hamiltonian (1) up to quadratic terms in φσ and by summing the zero point energies of the corresponding Bogoliubov modes. In the symmetric case (a ↑↑ = a ↓↓ and δ = 0) the calculation has been performed in Ref. [START_REF] Cappellaro | Equation of state and self-bound droplet in Rabi-coupled Bose mixtures[END_REF]. The general asymmetric case is technically more difficult because of cumbersome expressions for the Bogoliubov modes [31]. However, under the conditions (3), these expressions simplify and read

E p,-= p 2 /2, E p,+ = (p 2 /2 + Ω)(p 2 /2 + Ω -2g ↑↓ n),
where p is the momentum and Ω = Ω(α 0 + 1/α 0 )/2. The BMF energy density can then be reduced to the form [31]

E BMF = 8(-g ↑↓ n) 5/2 15π 2 I Ω -2g ↑↓ n , (4)
where I(y) = (15/4)

1 0
x(1 -x)(x + y)dx. Equation (4) remains a good approximation for the BMF energy density as long as |δa/a ↑↓ | 1 and |α 0 -β| 1. The function I(y) is a monotonically growing function, which tends to 1 for y = Ω/(-2g ↑↓ n) → 0. This is the limit of two uncoupled condensates where Eq. ( 4) reduces to the LHY form, responsible for the BMF stabilization of quantum droplets in binary mixtures [START_REF] Petrov | Quantum mechanical stabilization of a collapsing Bose-Bose mixture[END_REF]. In the opposite limit I(y) can be expanded in powers of 1/y 1, the first two leading terms being I(y) ≈ (15π/128)(4

√ y + 1/ √ y).
The substitution of this expansion into Eq. ( 4) gives

E BMF ≈ Ω 2 √ 2π g 2 ↑↓ n 2 2 + 3 4 √ 2π Ω |g ↑↓ | 3 n 3 6 , (5)
which is qualitatively different from the LHY n 5/2 scaling. The two terms in Eq. ( 5) can be interpreted as a BMF-renormalized two-body interaction and an emergent three-body term, both with a specific scaling with Ω. Interestingly, both these terms agree with exact two-body [31] and three-body calculations [32] in the limit Ω|g ↑↓ | 1. Higher-order terms require taking into account interactions between excited (noncondensed) atoms and are thus not captured by the quadratic Bogoliubov Hamiltonian. For the same reason, Eq. ( 5) contains no LHY term ∝ n 5/2 , expected to be "induced" by the BMF-renormalized two-body interaction. This term, however, is of very high order ∝ ( Ωg 2 ↑↓ n) 5/2 and is negligible for our experimental parameters.

In our experiment we directly measure the BMF energy at the point of vanishing MF through the released energy in a one-dimensional expansion. We work with the second and third lowest Zeeman states of the lowest manifold of 39 K, namely |↑ = |F = 1, m F = -1 and |↓ = |F = 1, m F = 0 . At a magnetic field of 56.830(1) G, the three relevant scattering lengths are a ↑↑ = 33.4 a 0 , a ↓↓ = 83.4 a 0 and a ↑↓ = -53.2 a 0 , where a 0 is the atomic Bohr radius [START_REF] Tiemann | Beyond Born-Oppenheimer approximation in ultracold atomic collisions[END_REF]. The minimum of a -- is then -0.2 a 0 . We have checked that the corresponding residual MF energy is a small correction as compared to the BMF energy for our parameters. The experiment starts with a quasi pure BEC in state |↑ . The atoms are optically trapped in an elongated harmonic potential with frequencies (ω x , ω y , ω z ) = (137, 137, 25.4) Hz.

The coherent mixture in |-is prepared in an adiabatic passage, in which the radio-frequency (RF) detuning is swept from δ = 7.5 Ω to its final value δ ≈ 0.23 Ω, for which α 0 ≈ β. During the RF sweep, a --decreases from 33.3 a 0 to a value close to 0. Its shape and duration of 9 ms are chosen in order to be adiabatic not only with respect to the internal-state dynamics but also with respect to the radial evolution of the wave function, which progressively shrinks and approaches the ground state of the radial harmonic confinement. We have checked that we cannot detect any residual oscillations of the radial size after the RF sweep. In addition, the axial trap frequency is progressively turned off during the sweep and adjusted such that the gas is neither axially expanding nor shrinking in a three-dimensional expansion after the sweep. The radially confined condensate is then free to expand in the axial direction for a time of 75 ms. During the final stage of the expansion, we sweep back the RF frequency to its initial value to map the dressed states |and |+ back onto |↑ and |↓ , which we independently detect by fluorescent imaging after a short Stern-Gerlach separation.

Interestingly, we observe that the atoms remain in state |- [START_REF]We have found that at low Rabi frequencies (Ω/2π ≤ 7 kHz), magnetic field noise around these frequencies transfers a small fraction of the atoms to state |+ during the condensate expansion (3% at Ω/2π = 6.9 kHz, 4.5% at Ω/2π = 5.5 kHz, 10% at Ω/2π = 3.9 kHz[END_REF], which might be surprising as losses are known to take place mostly in state |↓ [START_REF] Semeghini | Self-Bound Quantum Droplets of Atomic Mixtures in Free Space[END_REF][START_REF] Cheiney | Bright Soliton to Quantum Droplet Transition in a Mixture of Bose-Einstein Condensates[END_REF] and thus could lead to the creation of |+ . In order to understand the spin dynamics, we can model the losses by adding -iΓ/2 to the second diagonal term in ξ. Diagonalizing the resulting matrix, we see that for Γ Ω the initial state |-remains essentially unchanged, except that it decays with the rate ≈ Γ| -| ↓ | 2 = Γ/(1 + α 2 0 ). In our experiment Γ is at most ∼ 20 s -1 , which is much lower than Ω. We thus conclude that the spin-dependent loss in our case reduces to an effective loss in the dressed state, with Γ weighted by the fraction of the lossy component.

For each value of the Rabi frequency Ω, we first measure the condensate density profile after expansion and extract the condensate sizes from fits with 1D Thomas-Fermi density profiles ∝ 1-(z/R TF ) 2 (see inset of Fig. 1). As a function of the detuning δ, a clear minimum in size appears at δ/Ω ≈ 0.23, i.e., a position where we expect a --, and thus also the MF energy to be minimized [36]. As explained above, under the conditions (3), the MF term vanishes at its minimum (a --≈ 0) and the expansion of the cloud is governed by the BMF term. In Fig. 1, we plot the measured minimal size as a function of the Rabi frequency Ω/2π. For Rabi frequencies Ω/2π between 6 kHz and 38 kHz, we observe a slow increase of the measured size as a function of Ω, which corresponds to an increase of the BMF energy as predicted previously. More precisely, the size after a long time of flight is expected to scale with E LHY ∝ Ω 1/4 for a dominant two-body term at large Ω, in qualitative agreement with our observed behavior. At low Rabi frequencies, below 6 kHz, we observe an increase of the size which we attribute to low frequency magnetic field noise. In a 50 µs Ramsey sequence, we have measured a standard deviation of the magnetic field ∆B ∼ 0.8(2) mG corresponding to ∆δ ∼ 560(140) Hz. These fluctuations lead to an increased average value of the scattering length ∆a --≈ 48(∆δ/Ω) 2 a 0 , which can explain our observed increasing sizes for low values of Ω [37].

In order to precisely model our experiment we solve the wave-function evolution through a single component one-dimensional nonlinear Schrödinger equation in which we properly account for the BMF energy E BMF [START_REF] Antoine | GPELab, a Matlab Toolbox to Solve Gross-Pitaevskii Equations II: Dynamics and Stochastic Simulations[END_REF]. The axial initial wave function is taken to be ∝ 1 -(z/R TF ) 2 , such that the axial density profile is the one of the initial 3D Thomas-Fermi condensate before the RF sweep. The spin healing length is smaller than the radial cloud size and the BMF energy is treated in a local density approx-imation in the radial direction [START_REF] Lavoine | Beyond-mean-field crossover from one dimension to three dimensions in quantum droplets of binary mixtures[END_REF]. Since the chemical potential can be of order of the radial confinement energy ω ⊥ (especially at large Ω), we take into account, as a function of the 1D density, the correction to the Gaussian radial profile due to the dominant two-body BMF contribution. The 1D energy density is then found by radial integration, and the chemical potential entering the Schödinger equation by partial derivation with respect to the 1D density. The magnetic field noise is accounted for by an increase of a --entering the MF term.

As can be seen from Fig. 1 this model quantitatively reproduces our experimental data. The black curves are generated by using Eq. ( 4) and assuming the measured finite ∆B (solid) and ∆B = 0 (dashed). For this calculation the initial atom number 1.05 × 10 5 has been adjusted to match the experimental data and corresponds within 5% to a calibration using the condensation temperature (determined with ∼20% accuracy). The initial peak density is n ∼ 5 × 10 14 cm -3 and the three-body loss coefficient K ---/3! = 4 × 10 -28 cm 6 s -1 has been adjusted to match our observed ∼ 30% atom loss during the expansions [40].

The other curves in Fig. 1 are obtained by running the same code with ∆B = 0 using Eq. ( 5) (red dash-dotted) or restricting it only to the two-body term (green dotted). They show that in our explored range Ω/2π > 3 kHz the BMF energy is very well approximated, at least theoretically, by the sum of the dominant two-body contribution E BMF ∝ √ Ω and a smaller three-body term ∝ 1/ √ Ω. To give concrete numbers, for Ω/2π = 10 kHz, Ω/(-2g ↑↓ n) ≈ 1 and the initial two-body and three-body energies per particle are calculated to be 20 and 3.2 Hz, respectively.

In order to better understand how much of this theory is experimentally tested, we first note that the scaling E BMF ∝ √ Ω is clearly evidenced in our data for Ω/2π > 6 kHz. Assuming that the next-order term scales as ∝ 1/ √ Ω we try to verify if the corresponding coefficients are consistent with Eq. [START_REF] Chang | Momentum-Resolved Observation of Thermal and Quantum Depletion in a Bose Gas[END_REF]. To this end we fit the data with the prediction of our dynamical model based on Eq. ( 5) with two free parameters: the total atom number and a prefactor η, inserted by hand in Eq. ( 5) in front of the three-body term. Repeating this fitting procedure for different values of ∆B we find η = 0.85 [START_REF] Cheiney | Bright Soliton to Quantum Droplet Transition in a Mixture of Bose-Einstein Condensates[END_REF], where the error bar is dominated by the uncertainty in our determination of the magnetic field noise. Our measurements as a function of Ω are thus in quantitative agreement with Eq. ( 5). The two terms, named two-body and three-body according to their predicted density scalings, are independently measured from their specific Ω dependence.

Finally, we can also repeat the expansion measurement while removing the RF coupling field at the end of the sweep such that we are left with two uncoupled condensates. The losses then dominantly take place in state |↓ and contrary to the coupled case, α quickly deviates from its initial value β. Moreover, we observe after a Stern-Gerlach separation that the two clouds behave differently in the expansion (see Fig. 2). The |↓ condensate does not expand much whereas the |↑ condensate exhibits a double structure with a low energy central part. This behavior is reminiscent of previous observations in droplet configuration where excess |↑ atoms are expelled from the droplet region [START_REF] Cabrera | Quantum liquid droplets in a mixture of Bose-Einstein condensates[END_REF][START_REF] Semeghini | Self-Bound Quantum Droplets of Atomic Mixtures in Free Space[END_REF][START_REF] Ferioli | Dynamical formation of quantum droplets in a 39 K mixture[END_REF]. In addition, we find that the condensate 1D Thomas-Fermi radius in state |↓ is 31 µm, a value that is significantly lower than the expected radius of 57 µm for our parameters in the single component simulation with Ω → 0, i.e. with the BMF energy density scaling with n 5/2 . This difference indicates a significant role of transient MF effects in the expansion dynamics of the central region. Here, |↑ atoms, which are more abundant than expected and which require some time to escape, create an excessive effective trapping for |↓ atoms forcing their slower expansion.

In conclusion, we have studied both theoretically and experimentally, the BMF equation of state of a coherently coupled two-component BEC in the asymmetric case. The BMF correction is most manifest when the spin polarization and interactions are tuned to the special point given by Eqs. [START_REF] Jaskula | Acoustic Analog to the Dynamical Casimir Effect in a Bose-Einstein Condensate[END_REF] where MF effects are minimized. The BMF energy density as a function of n then interpolates between the usual ∝ n 5/2 LHY form in the uncoupled limit to a qualitatively different behavior in the strong-coupling regime where one can introduce effective BMF two-body and three-body interactions with specific scalings as a function of the Rabi coupling strength Ω. The experimentally measured condensate expansions are governed by the BMF energy and are in good agreement with the theory at large Ω. We detect not only the dominant two-body BMF term ∝ √ Ω, but also evidence the smaller three-body term ∝ 1/ √ Ω, opening the path to the creation of coherently coupled quantum droplets in which the two-body interaction (MF+BMF) is compen-sated by BMF three-body effects [23,[START_REF] Bulgac | Dilute Quantum Droplets[END_REF]. Exploration of the small Ω regime will require a further reduction of the magnetic field fluctuations. Interestingly, a coherent Rabi coupling helps to preserve the spin composition and thus prevents the system from dynamically drifting away from the point of vanishing mean field and thus facilitates direct measurements of the BMF equation of state.

FIG. 1 :

 1 FIG.1: Minimal Thomas-Fermi radius of the condensate after 75 ms of expansion as a function of the Rabi coupling strength (for δ/Ω = 0.23). Each point is obtained by averaging 15 fluorescence images and the error bars correspond to the single shot standard deviation. The curves correspond to quasi-1D extended Gross-Pitaevskii simulations (see text) with or without magnetic field noise and assuming different forms of the BMF term. Black solid curve: full BMF term with noise; black dashed curve: full BMF without noise; green dotted curve: two-body BMF term without noise; red dash-dotted curve: two-body plus three-body BMF term without noise. Inset: Experimental Thomas-Fermi radius of the condensate after 75 ms of expansion as a function of δ/Ω, for a Rabi frequency Ω/2π = 12.29 kHz. The curve is a parabolic guide to the eye.

FIG. 2 :

 2 FIG. 2: Density profiles for |↑ (black) and |↓ (red) atoms after 75 ms of expansion at Ω = 0. The blue dashed curve is a 1D Thomas-Fermi fit of the density profile in state |↓ .

We thank L. Tarruell for useful discussions. This research has been supported by CNRS, Ministère de l'Enseignement Supérieur et de la Recherche, Labex PALM, Region Ile-de-France in the framework of DIM Sirteq, Paris-Saclay in the framework of IQUPS, ANR Droplets (19-CE30-0003), Simons Foundation (Grant No. 563916: localization of waves). AR acknowledge financial support from Provincia Autonoma di Trento, the FISh project of the Istituto Nazionale di Fisica Nucleare, and the Italian MIUR under the PRIN2017 project CEnTraL.