Safieddine Bouali
email: safieddine.bouali@isg.rnu.tn

Strange Attractor Morphogenesis by Sensitive Dependence on Initial Conditions

A feedback loop to a 3D chaotic system with only six-terms on the right-hand of the equations and only two nonlinearities is applied to intentionally build a minimalist novel 4D chaotic system. Simulations depict the coexistence of strange attractors not by the modification of an unique or several parameters but surprisingly by a slight modification of the intial conditions. It is acknowledged that a strange attractor is locally unstable but globally stable. Our experiementation displays that strange attractors could be unstable at all scales.

I.

Introduction

Morphology of the strange attractors as mathematical "objects" could have relevance to report the whole envelop of amplitudes and frequencies reached by the chaotic oscillations in the phase space. One could well argue that their shapes are just the condensed summary of long listings of time series. However, the morphological plasticity have a significance introduced by the sensitive dependence on parameters (SDP) of the given attractor and helps the modeler to configurate adaquately its system. Such pertinence arises more intensely when the chaotic behaviors become captured by the sensitive dependance on initial conditions (SDIC) as displayed in a previous 3D chaotic system [START_REF] Bouali | A 3D Strange Attractor with a Distinctive Silhouette. The Butterfly Effect Revisited[END_REF]. Its properties are in full contradiction to the findings of the Butterfly Effect [START_REF] Lorenz | The essence of chaos[END_REF], indicating that small gaps between the initial conditions lead the trajectories to different dynamics, but converge asymptotically toward an unique attractor. Locally unstable but globally stable, a strange attractor.. The primary objective of this paper is to experiment the findings of an interactive perturbation of this system of 2013 violating the SDIC scopes. To this end, a feedback loop altering the structural stability of the model and extending the model to the fourth dimension is applied. We expect the emergence of a morphogenesis at the sense of Thom (2018 for the last edition).

The experimentation represents a preliminary step to the topological study of strange attractors [START_REF] Gilmore | The Topology of Chaos: Alice in Stretch and Squeezeland[END_REF][START_REF] Letellier | Topology and Chaos Dynamics: Robert Gilmore's 70th Anniversary Celebration[END_REF]. The secondary objective is represented by the self-selected constraint to develop a 4D version with the minimum number of terms in the right-hand of the equations. In light of the coexisting attractors computed from such unique 4D dynamical system only with SDIC, we indicate its suitability as a chaotic generator for secure communication schemes. In section III, the morphological variety of the phase portraits are described as well as the feature of the (planar) rotated copy for any phase portrait found. Section IV presents remarks on the origin of the attractor plasticity and the high suitability of the 4D model as a chaotic generator for cryptology.

II. A Minimalist 4D chaotic system

To achieve our goal to intentionally construct a 4D chaotic model with the minimum extent possible set of terms on the right-hand side, we propose the following system:

{ 𝒙̇= 𝜶 𝒙 ( 𝟏 -𝒚 ) -𝜷𝒛 𝒚̇= -( 𝟏 -𝒙 𝟐 ) 𝒚 𝒛̇= -𝝁 𝒗 𝒗̇= -𝝆 𝒛
where the overdot denotes a time derivative, x, y, z, and v, the four state variables of the model, and α, β, µ, ρ assumed to be positive parameters. Such seven-term system embeding only two nonlinearity, a quadratic term, and one cubic item, respectively, xy, and x 2 y, expresses the minimalist extension of a 3D system having an exclusive silhouette (Bouali, idem, 2013) with a fourth state variable.

Requirement to investigate the patterns of its phase portrait lead us to retain the set of parameters P (α, β, µ, ρ) = (3, 0.1, 1, 0.2, 0.01).

II.1. Basic mathematical properties

The equilibrium coordinates could be found by setting: 𝒙̇= 𝒚̇= 𝒛̇= 𝒗̇= 𝟎. Thus, the solutions of the following system :

{ 𝟎 = 𝜶 𝒙 ( 𝟏 -𝒚 )-𝜷𝒛 𝟎 = -( 𝟏 -𝒙 𝟐 ) 𝒚 𝟎 = -𝝁 𝒗 𝟎 = -𝝆 𝒛 give the coordinates of equilibria.
In the first equation, and substituting z = 0 from the last relation, we obtain x = 0 or y = 1. Firstly, the second equation gives for x= 0, y= 0, and secondly for y =1, x = ±1. The elementary attributes of these three equilibria given by the corresponding eigenvalues λi are found by solving the characteristic equation |J -λI | = 0 where J, the Jacobian of the model, and I, the unit matrix. This kind of calculation is easy to achieve since the non-zero terms of the Jacobian are the least numerous (7 out of 16). In Table 1, coordinates, eigenvalues and features of the stability related to the equilibria are reported.

(λ 2 + 0.002)(1+λ)(3-λ)= 0 λ1 = -𝑖 10√5 λ2 = 𝑖 10√5 λ3 = -1 λ4 = 3 Index-1 Unstable: Spiral saddle point E1 (x1, y1, z1, v1) = (1, 1, 0, 0) & E2 (x2, y2, z2, v2) = (-1, 1, 0, 0) (λ 2 + 0.002)(6+λ 2 )= 0 λ1 = -𝑖 10√5 λ2 = 𝑖 10√5 λ3 = -𝑖√6 λ4 = 𝑖√6
Index-0 Neutrally stable: centers

(1) Index reports the number of eigenvalues with real parts Re(λ) > 0. From 1 to 4, it indicates the degree of instability. Index-0: null or negative real parts of all eigenvalues of the equilibrium characterize its stability.

II.2. A nonuniformly dissipativity

To qualify the dissipative nature of the 4D dynamical system, the divergence of the whole vector field could be derived from the following formulation:

𝒅𝒊𝒗. (𝑽𝒐𝒍𝒖𝒎𝒆) = 𝝏𝑽𝒐𝒍𝒖𝒎𝒆 𝝏𝒕 ⁄ 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝑻𝒓( 𝑱 ) = 𝝏𝒙∂ 𝒙 + 𝝏𝒚∂ 𝒚 + 𝝏𝒛∂ 𝒛 + 𝝏𝒗∂ 𝒗
To this end, Tr (J), the sum of the diagonal terms of the Jacobian should be negative to attest the dissipativity of the flow:

𝑻𝒓( 𝑱 ) = 𝛂 (𝟏 -𝐲) -(𝟏 -𝐱 𝟐 ) < 0
where α, positive value. Dissipativity and volume contraction of the flow are accurately identified when these state variables of the flow, x, and y (and not including either z or v) fulfil the required condition:

𝛂 (𝟏 -𝐲) < (𝟏 -𝐱 𝟐 )

The system arises nonuniformly dissipative. However, in the precise case of the above condition is met, orbits are ultimately limited in a specific fractal-dimensional subspace of zero volume.

The set of (x, y) that verifies this inequality marks a domain bounded by an separatrix where the flow of the dynamics is dissipative. Besides, the trajectories are not short-term transient dynamics and report the zones of chaotic behavior -and not that regular-within the phase space [START_REF] Joglekar | Scaling of Chaos Versus Periodicity: How Certain Is It that an Attractor Is Chaotic?[END_REF][START_REF] Lai | Transient Chaos, Complex Dynamics on Finite-Time Scales[END_REF][START_REF] Motter | Effective Dynamics in Hamiltonian Systems with Mixed Phase Space[END_REF][START_REF] Kantz | Repellers, Semi-Attractors, and Long-Lived Chaotic Transients[END_REF].

On the other hand, focusing the 4D model with its P parameters, it is pertinent to compute the Lyapunov exponents spectrum, the most useful measure to estimate the chaos. The system displays a chaotic nature since for β, varying from 0.01 to 0.21, the dominant Lyapunov exponent reaches a positive value (Fig. 1).

III. A Collection of Attractors with Distinct Morphologies

We simulate the 4D system keeping unchanged the P parameters and all the graphical representations of the trajectories will be projected within the (x, y, z) space. This option will assist the analysis of morphological forms and their basins of attraction.

III.1. A non-standard SDIC

Surprisingly, the representation of these normalized 3D projections of the space phases depicts a noteworthy morphological variety of the portraits even if the system and the values of its parameters are kept unchanged. A non-standard butterfly effect is highlighted. The first two projections (fig. 2a, and2b) display asymmetrical portraits and identifiable saddle-focus homoclinic bifurcations leading to a very complex dynamics. Furthermore, the third projection (fig. 2c) depicts embedded rolls without, here also, any axis of symmetry.

Figure 2d shows a lemon-like shape of attractor whose initial conditions seem to belong to a zone of regular motion within the basin of attraction.

III.2. A rotated copy by π radians for any attractor

On the other hands, simulations have showed that another type of SDIC operates, albeit with a null impact on morphology. Indeed, for a small variation of the initial conditions, an identical portrait of the given strange attractor could be created, however with a planar half-turn. This is in the same line of the Chua model [START_REF] Chua | The double scroll family[END_REF][START_REF] Chua | The genesis of Chua's circuit. AEÜ[END_REF], or the Bouali model [START_REF] Bouali | Feedback loop in extended van der Pol's equation applied to an economic model of cycles[END_REF] since under a peculiar specification of parameters, each system creates two strange attractors obviously with a weak difference of initial conditions. Although located in two sub-basin of attraction, the attractors are in reversed position. Nevertheless, this duplication is only the effect of the least spectacular SDIC The present 4D system creates a pair of portraits for any particular morphology detected in the entire basin of attraction (Fig. 3). To note that figure 3a is similar of that of figure 2a. Another example of a pair of portraits is given in Figure 4. Here, the figure 4b is similar of that of figure 2b. In fact, the powerful SDIC able to drive trajectories toward distinct strange attractors stemms from the original 3D system (Bouali, idem, 2013). Embedded attractors phenomenon had been also reported. However, no identifiable morphogenesis could be noticed despite the light dissimilarity of rolls and wings of the attractors in the three dimensional space.

Comparatively, therefore, it is the feedback of the fourth state variable by means of a supplementary dimension that triggered a complex mutation of the attractor morphology.

IV. Concluding Remarks

The Butterfly Effect as the most explicit feature of a strange attractor constitutes a blazing legacy of the wide scientific contribution Edward Norton [START_REF] Lorenz | Deterministic Nonperiodic Flow[END_REF]. The SDIC reports that a small gap between two initial conditions in its idealized meteorological system deflects their paths in a way that no prediction of their drift could be done. Indeed, their gap grows at exponentially distance. Such singular phenomenon states that a strange attractor is locally unstable. However, it is globally stable since the trajectories converge to the same attractor.

Our 4D system moves beyond this result.

The main outcome of our paper is precisely that a strange attractor is not only locally unstable but also it could be globally unstable. Such novel SDIC allows a marked volatility of the chaotic dynamics at two levels. On the one hand, the bifurcation to a different strange attractor by a small variation of the initial conditions. On the other hand, the jump toward a rotated copy of any strange attractor by a slight change of the previous initial conditions even if this result had been observed elsewhese. Such model could be used to preserve cryptography against hacker attacks of the communication systems based on the synchronization of chaotic oscillators [START_REF] Frasca | Synchronization in Networks of Nonlinear Circuits: Essential Topics with MATLAB® Code[END_REF][START_REF] Huang | Synchronization-based estimation of all parameters of chaotic systems from time series[END_REF][START_REF] Kocarev | General approach for chaotic synchronization with applications to communication[END_REF][START_REF] Pecora | Synchronization in chaotic systems[END_REF]. Even if the oscillator parameters usually used as secret keys are recovered, the chaotic system remains unknow due to its drastic SDIC associating both volatility of morphology or rotation. Indeed, capture of both parameters and initial conditions appears costly in lenght time of computation, and highly uncertain in case of jump from a chaotic generator to an other. In this case, the predictability horizon stands far beyond the computation time and the physical latences of the (hacking) circuit receiver.

Fig. 1 .

 1 Fig. 1. Lyapunov exponent spectrum for the 4D chaotic system keeping unchanged the specification of parameters P, except β which varies in the range [0.01, 0.21]

Fig. 2 .

 2 Fig. 2. Simulation of the 4D chaotic system with the (unchanged) set of parameters P. The 4D trajectories projected on the space (x, y, z) do not converge to an unique attractor whereas only the initial conditions have been modified. The Initial Conditions (a) ICa=( xa, ya, za, va)= (2, 2, 2, 2), (b) ICb=( xb, yb, zb, vb)= (0.5, 0.5, 0.5, 0.5), (c) ICc=( xc, yc, zc, vc)= (0.05, 0.05, 0.05, 0.05), and (d) ICd=( xd, yd, zd, vd)= (1, 1, 1, 1).

Fig. 3 .Fig. 4 .

 34 Fig. 3. A pair of 4D trajectories projected on the space (x, y, z) simulated with the (unchanged) set of parameters P and (a) ICa=( xa, ya, za, va)= (2, 2, 2, 2), and (b) ICb=( xb, yb, zb, vb)= (2, 2, -2, -2).

Table 1 . Index and Stability of the Equilibria Coordinates of the equilibia

 1

	Corresponding characteristic	Index and
	equation,	stability
	|J -λI | = 0, and eigenvalues
	E0 (x0, y0, z0, v0) = (0, 0, 0, 0)


Préparée à MINES ParisTech

Globalization and the labor market: A study of tradable and non-tradable jobs in France

Soutenue par

Philippe FROCRAIN

Le 28 novembre 2018

École doctorale n° 396

Économie, Organisations, Société

Spécialité

Économie et Finance

ABSTRACT

The distinction between manufacturing and service jobs has lost some of its relevance due to the growing interpenetration between industrial and service activities. On the other hand, globalization has made it increasingly necessary to distinguish between jobs exposed to international competition (tradable jobs) and those not exposed to it (non-tradable jobs), found in primary, secondary and tertiary sectors. This thesis begins by analyzing the developments and characteristics of tradable and non-tradable employment in France. Which jobs are tradable and which are non-tradable? Are they different in terms of skills, wages, and productivity? How are they distributed around the country? It then studies the interdependencies between these two categories of employment, making an empirical evaluation of the local multiplier effect of tradable jobs on non-tradable jobs in French local labor markets. Lastly, the thesis focuses on workers laid off due to firm closure. It sets out to determine whether the risk of job displacement, and the associated costs, are higher in the tradable sector than in the non-tradable sector.

MOTS CLÉS

Exposé ; Abrité ; Mondialisation ; Multiplicateur ; Licenciement ; Structure de l'emploi en France

On oppose traditionnellement les emplois industriels et les emplois de services. L'interpénétration croissante de l'industrie et des services rend cependant cette distinction de plus en plus inopérante. À l'heure de la globalisation, il apparaît en revanche pertinent de distinguer les emplois exposés à la concurrence internationale de ceux qui en sont abrités, que l'on trouve à la fois dans l'industrie et dans les services. Cette thèse analyse dans un premier temps les évolutions et les caractéristiques des emplois exposés et abrités en France. Quels sont les emplois exposés et abrités ? Se distinguent-ils en termes de qualifications, salaires, évolutions de la productivité ? Comment sont-ils répartis sur le territoire ? Elle étudie ensuite les interdépendances entre ces deux catégories d'emplois, en évaluant empiriquement l'effet d'entraînement de l'emploi exposé sur l'emploi abrité au niveau des zones d'emploi de France métropolitaine. L'étude s'intéresse enfin à un échantillon de salariés licenciés suite à la fermeture d'un site de production. Elle tente de déterminer si le risque de licenciement, et les coûts qui lui sont associés, sont plus élevés dans le secteur exposé que dans le secteur abrité.Je montre dans cette thèse que les emplois exposés représentent en France une part toujours plus faible de l'emploi total, que désormais une très large part de ces emplois sont dans les secteurs de services exposés et qu'ils sont mieux rémunérés et plus productifs que les emplois abrités, sans que ceci soit associé à des différences de qualification.KEYWORDSTradable ; Non-tradable ; Globalization ; Local multiplier ; Displacement ; French employment structure


Sarah Guimbal

Anne Morel

David Guérit

Manon Chardon

Anne Blangy
email: anne.blangy@crbm.cnrs.fr

Virginie Vives

Dock5 is a new regulator of microtubule dynamic instability in osteoclasts (Dock5 regulates microtubule dynamics in osteoclasts)

Keywords: APC Adenomatous polyposis coli Dock5, osteoclast, microtubule, knockout mice, Rac

Background information: Osteoclast resorption is dependent on a podosome-rich structure called sealing zone. It tightly attaches the osteoclast to the bone creating a favorable acidic microenvironment for bone degradation. This adhesion structure needs to be stabilized by microtubules whose acetylation is maintained by down regulation of deacetylase HDAC6 and / or of microtubule destabilizing kinase GSK3β activities. We already established that Dock5 is a guanine nucleotide exchange factor for Rac1. As a consequence, Dock5 inhibition results in a decrease of the GTPase activity associated with impaired podosome assembly into sealing zones and resorbing activity in osteoclasts. More, administration of C21, a chemical compound that directly inhibits the exchange activity of Dock5, disrupts osteoclast podosome organization and protects mice against bone degradation in models recapitulating major osteolytic diseases.

Results:

In this report, we show that Dock5 knockout osteoclasts also present a reduced acetylated tubulin level leading to a decreased length and duration of microtubule growth phases whereas their growth speed remains unaffected. Dock5 does not act by direct interaction with the polymerized tubulin. Using specific Rac inhibitors, we showed that Dock5 regulates microtubule dynamic instability through Rac-dependent and -independent pathways. The latter involves GSK3β inhibitory serine 9 phosphorylation downstream of Akt activation but not HDAC6 activity.

Conclusion:

We showed that Dock5 is a new regulator of microtubule dynamic instability in osteoclast.

Significance: Dock5 dual role in the regulation of the actin cytoskeleton and microtubule, which both need to be intact for bone resorption, reinforces the fact that it is an interesting therapeutic target for osteolytic pathologies.

Introduction

Bone homeostasis is regulated through life by the balanced actions of highly specialized bone-forming osteoblasts and bone-resorbing osteoclasts. An important step of bone resorption is mineral phase dissolution by massive secretion of H + and Cl -ions. It exposes the collagen-rich underlying organic phase that becomes accessible to degradation by lysosomal proteases such as cathepsin K [START_REF] Cappariello | The Great Beauty of the osteoclast[END_REF]. To maintain this acidic microenvironment which is the key for an efficient resorption, osteoclasts tightly attach to the bone matrix by an actin-rich sealing zone composed of densely packed podosomes. Because of its nature, it was thought for a long time that its formation depended only on actin regulators whose roles have been extensively studied [START_REF] Lowe | Osteopetrosis in Src-deficient mice is due to an autonomous defect of osteoclasts[END_REF] [START_REF] Chellaiah | Rho-A Is Critical for Osteoclast Podosome Organization, Motility, and Bone Resorption[END_REF] [START_REF] Calle | WASp deficiency in mice results in failure to form osteoclast sealing zones and defects in bone resorption[END_REF] [START_REF] Destaing | The Tyrosine Kinase Activity of c-Src Regulates Actin Dynamics and Organization of Podosomes in Osteoclasts[END_REF] [START_REF] Croke | Rac deletion in osteoclasts causes severe osteopetrosis[END_REF] [START_REF] Touaitahuata | Tensin 3 is a new partner of Dock5 that controls osteoclast podosome organization and activity[END_REF]. A decade and a half ago, Jurdic's lab discovery that an intact microtubule network was necessary for this adhesion structure stability [START_REF] Destaing | Podosomes display actin turnover and dynamic self-organization in osteoclasts expressing actingreen fluorescent protein[END_REF] emphasized the importance to understand the signaling pathways linking both cytoskeletons. On one hand, it was shown that belt stabilization is correlated to microtubule acetylation which is maintained by downregulation of two important players. First, RhoA-mDia2 dependent tubulin deacetylase HDAC6 activity [START_REF] Destaing | A novel Rho-mDia2-HDAC6 pathway controls podosome patterning through microtubule acetylation in osteoclasts[END_REF] is negatively controlled by Pyk2 [START_REF] Gil-Henn | Defective microtubuledependent podosome organization in osteoclasts leads to increased bone density in Pyk2(-/-) mice[END_REF]. Independently, HDAC6 interaction with tubulin is antagonized by competitive binding of Cbl adaptor proteins [START_REF] Purev | c-Cbl and Cbl-b act redundantly to protect osteoclasts from apoptosis and to displace HDAC6 from betatubulin, stabilizing microtubules and podosomes[END_REF]. Second, the kinase GSK3β activity is inhibited by serine 9 phosphorylation by Akt. It allows stabilizing MAPs binding to growing microtubules [START_REF] Matsumoto | Regulation of bone resorption and sealing zone formation in osteoclasts occurs through protein kinase B-mediated microtubule stabilization[END_REF] and interaction between +TIP EB1 and actin belt associated cortactin [START_REF] Biosse Duplan | Microtubule dynamic instability controls podosome patterning in osteoclasts through EB1, cortactin, and Src[END_REF]. On the other hand, it was shown that microtubules stabilize sealing zones through direct linkage mediated by actin regulators. Indeed, the unconventional myosin X encircles nascent sealing zones and provides force to expand them into mature ones through interaction with microtubules [START_REF] Mcmichael | Myosin X regulates sealing zone patterning in osteoclasts through linkage of podosomes and microtubules[END_REF]. More, the Dynamin 2 GTPase acts as an adaptor molecule between microtubules and actin podosomes, preventing them to scatter from the densely packed adhesion structure [START_REF] Batsir | Dynamics of the sealing zone in cultured osteoclasts[END_REF]. We showed that the Dock family member Dock5 is an exchange factor for the actin regulator Rac1 [START_REF] Vives | The Rac1 exchange factor Dock5 is essential for bone resorption by osteoclasts[END_REF]) [START_REF] Ferrandez | Allosteric inhibition of the guanine nucleotide exchange factor DOCK5 by a small molecule[END_REF]. During osteoclast resorption cycle, αvβ3 integrins signaling induces Dock5 recruitment to the Src/Pyk2/p130Cas complex which ultimately leads to Rac1 activation and localization to the sealing zone [START_REF] Vives | The Rac1 exchange factor Dock5 is essential for bone resorption by osteoclasts[END_REF]. Dock5 plays an important role in this pathway because its deletion in knockout mice results in increased trabecular bone mass due to an impaired osteoclasts ability to assemble podosomes into sealing zones [START_REF] Vives | The Rac1 exchange factor Dock5 is essential for bone resorption by osteoclasts[END_REF], a phenotype also observed in knockout mice for Rac1 [START_REF] Croke | Rac deletion in osteoclasts causes severe osteopetrosis[END_REF] [START_REF] Magalhaes | Deleting Rac1 improves vertebral bone quality and resistance to fracture in a murine ovariectomy model[END_REF] and some of its guanine exchange factors [START_REF] Faccio | Vav3 regulates osteoclast function and bone mass[END_REF] [START_REF] Takegahara | Integral roles of a guanine nucleotide exchange factor, FARP2, in osteoclast podosome rearrangements[END_REF].

Interestingly, a group working on allergy showed that Dock5 is essential for the dynamic rearrangement of microtubules during mast cell degranulation in anaphylactic responses [START_REF] Ogawa | DOCK5 functions as a key signaling adaptor that links FcεRI signals to microtubule dynamics during mast cell degranulation[END_REF].

In this study, we tested the hypothesis that, in addition to its Rac1-dependent actin regulation [START_REF] Vives | The Rac1 exchange factor Dock5 is essential for bone resorption by osteoclasts[END_REF], Dock5 could also be involved in microtubule control in osteoclast. We showed that Dock5 regulates microtubules dynamic instability via Rac1-dependent and -independent pathways, the latter involving the Akt-GSK3β axis.

Results

Dock5 -/-osteoclasts present an abnormal microtubule acetylation pattern associated with a perturbed microtubule dynamic instability

Western blot analysis of proteins from wild-type and Dock5 knockout osteoclasts showed a significant decrease of tubulin acetylation in the absence of Dock5 (figure 1A). Confocal immunofluorescence microscopy allowed us to take a closer look at the microtubules distribution and acetylation pattern (figure 1B). As expected [START_REF] Akisaka | Differential distribution of posttranslationally modified microtubules in osteoclasts[END_REF], microtubules of wild-type osteoclasts were typically organized (i) radially from the perinuclear region to the periphery where the podosome belt was located and (ii) circularly at the periphery. Both were modified with large evenly distributed acetylation patches. Although Dock5 -/-osteoclasts microtubules kept the radial orientation, the circular ones were missing at the periphery, which was consistent with the absence of podosome assembly into a belt.

More, tubulin acetylation, that was present in small patches, was decreased in density compared to their wild-type counterpart (figure 1C).

Since tubulin acetylation is associated with microtubule stabilization [START_REF] Matsumoto | Regulation of bone resorption and sealing zone formation in osteoclasts occurs through protein kinase B-mediated microtubule stabilization[END_REF], we reasoned that this acetylation defect could affect their dynamic instability. To answer this question, we expressed the EB3 +TIP fused to GFP in wild-type and Dock5 knockout osteoclasts to visualize their growing microtubule plus ends using time-lapse video microscopy (supplementary figure 1).

Tracking of EB3-GFP at the +TIP of microtubules (figure 2A) showed a significant decrease of microtubule polymerization duration (figure 2B) and length (figure 2C) in the absence of Dock5. Polymerization speed however remained unchanged (figure 2D). More, the mean total number of actively growing microtubules was similar for both genotypes per image of the movies (figure 2E) but the total comet tracks number was significantly increased in Dock5 -/-osteoclasts over a 1 min period (figure 2F), suggesting an increased number of catastrophes.

These results show that the belt disruption and the resulting decreased resorption activity observed in Dock5 knockout osteoclasts [START_REF] Vives | The Rac1 exchange factor Dock5 is essential for bone resorption by osteoclasts[END_REF] are associated with a decrease in acetylated tubulin and a perturbed microtubule dynamic instability.

Dock5 regulates microtubules dynamics through Rac-dependent andindependent pathways

It has been shown that Dock5 is a guanine exchange factor for Rac1 whose activity is decreased in Dock5 knockout osteoclasts [START_REF] Vives | The Rac1 exchange factor Dock5 is essential for bone resorption by osteoclasts[END_REF]. That made us wonder if the microtubule phenotype we described in Dock5 -/-osteoclasts was Rac-dependent. We reasoned that, if it is the case, we should reproduce it when we specifically inhibit Dock5-dependent Rac activation in wild-type osteoclasts using C21, an allosteric inhibitor of Dock5 exchange activity [START_REF] Vives | The Rac1 exchange factor Dock5 is essential for bone resorption by osteoclasts[END_REF] [START_REF] Ferrandez | Allosteric inhibition of the guanine nucleotide exchange factor DOCK5 by a small molecule[END_REF] (Xu et al., 2017a) (figure 3A). Western blot analysis revealed that, opposite to what was observed in Dock5 knockouts, C21 treatment significantly increased tubulin acetylation level (figure 3B). It was confirmed by confocal immunofluorescence microscopy that showed that, in addition to the podosome belt disruption, C21 treatment induced a more intense microtubules acetylation compared to wild-type osteoclasts (figure 3C). We observed the same phenotype using EHT 1864, a compound that inhibits Rac activation (supplementary figure 2A) and downstream signaling [START_REF] Shutes | Specificity and mechanism of action of EHT 1864, a novel small molecule inhibitor of Rac family small GTPases[END_REF]. Tubulin acetylation augments with increasing concentrations of the drug (supplementary figure 2B) in osteoclasts that show in addition to the expected actin belt perturbation an overall disrupted microtubule network (supplementary figure 2C). In order to check if the acetylation alteration affected microtubules dynamics, we treated or not EB3-GFP expressing wild-type osteoclasts with C21 and recorded the resulting comets using time-lapse video microscopy. Their tracking showed a significant decrease of microtubule polymerization duration, length and speed after C21 treatment (figure 3D). More, C21 impacted neither the total number of actively growing microtubules on each image of the movies nor the comet tracks number over a 1 min period (figure 3E-F).

Overall, these results show that Rac activity inhibition does not reproduce the microtubule phenotype observed in Dock5 knockouts. However, diminution of Dock5-dependent Rac activation by C21, which is associated with a decreased resorption function [START_REF] Vives | The Rac1 exchange factor Dock5 is essential for bone resorption by osteoclasts[END_REF] [START_REF] Vives | Pharmacological inhibition of Dock5 prevents osteolysis by affecting osteoclast podosome organization while preserving bone formation[END_REF], induces an increase in tubulin acetylation and also affects microtubule dynamics. It suggests that Dock5 regulates microtubules both dependently and independently of its Rac guanine exchange factor activity, the combination of both pathways resulting in the global decrease of tubulin acetylation quantified in Dock5 knockout osteoclasts.

Dock5 regulates microtubules through a distant signaling pathway

Consistently with its Rac-dependent actin regulation role, we previously showed that Dock5 colocalizes with vinculin, an actin cloud-associated protein, at the podosome belt of osteoclasts [START_REF] Touaitahuata | Tensin 3 is a new partner of Dock5 that controls osteoclast podosome organization and activity[END_REF]. Since we uncovered a new microtubule-regulating function, we wondered whether a Dock5 protein fraction could also be associated to microtubules. To answer this question, we separated soluble and polymerized microtubules. Western blot analysis showed that Dock5 was not associated to the polymerized fraction, which contained acetylated tubulin and could be depolymerized upon nocodazole treatment (figure 4A). These results suggest that Dock5 microtubules regulation is not mediated by direct association but through a distant signaling pathway.

Dock5 -/-osteoclasts show an increased GSK3β activity

In order to identify the Rac-independent mechanism involved in Dock5 knockout microtubule destabilization, we searched for deregulations of pathways involved in microtubules acetylation in osteoclasts. HDAC6 is a deacetylase whose activity is kept to a minimum [START_REF] Gil-Henn | Defective microtubuledependent podosome organization in osteoclasts leads to increased bone density in Pyk2(-/-) mice[END_REF] [START_REF] Purev | c-Cbl and Cbl-b act redundantly to protect osteoclasts from apoptosis and to displace HDAC6 from betatubulin, stabilizing microtubules and podosomes[END_REF] to allow a stable microtubule network. We therefore checked whether HDAC6 activity was higher Dock5 -/-osteoclasts but we could not find any differences between genotypes (figure 4B). We next focused on the Akt-GSK3β axis. Akt inhibits GSK3β; a serine threonine kinase GSK3β which destabilizes microtubules by regulating microtubules associated proteins interactions [START_REF] Matsumoto | Regulation of bone resorption and sealing zone formation in osteoclasts occurs through protein kinase B-mediated microtubule stabilization[END_REF]. Western blot analysis revealed a GSK3β decreased inhibitory serine 9 phosphorylation suggesting an increased activity in Dock5 -/-osteoclasts (figure 4C). Interestingly, Akt phosphorylation on serine 473, readout of the kinase activity, was not modified (figure 4D), leading us to believe that Dock5 acts downstream in this pathway. Mature Dock5 -/-osteoclasts mainly present isolated or clustered podosomes as if they were not able to assemble or maintain them into a belt [START_REF] Vives | The Rac1 exchange factor Dock5 is essential for bone resorption by osteoclasts[END_REF].

Since acetylated microtubules were shown to be necessary for podosome belt assembly/stabilization [START_REF] Destaing | Podosomes display actin turnover and dynamic self-organization in osteoclasts expressing actingreen fluorescent protein[END_REF] [START_REF] Destaing | A novel Rho-mDia2-HDAC6 pathway controls podosome patterning through microtubule acetylation in osteoclasts[END_REF], we wondered whether restoration of a normal acetylation level in Dock5 -/-osteoclasts would rescue the actin belt. Therefore, wild-type and Dock5 knockout osteoclasts were cultured in the presence of GSK3β inhibitor lithium chloride. Although the treatment increased tubulin acetylation at a similar level between the two genotypes, it was not enough to allow the actin belt formation/stabilization in the absence of Dock5 (figure 4E). The same result was obtained using CHIR99021, a more selective GSK3β inhibitor (data not shown).

There is a close relationship between microtubules and the actin cytoskeleton in osteoclasts. Microtubules grow toward isolated/clustered podosomes during osteoclast differentiation and then form a ring at the cell periphery where the podosome belt is located [START_REF] Akisaka | Differential distribution of posttranslationally modified microtubules in osteoclasts[END_REF]. During this process, they are necessary for podosomes transition from rings to belt [START_REF] Destaing | Podosomes display actin turnover and dynamic self-organization in osteoclasts expressing actingreen fluorescent protein[END_REF]. To our knowledge, no data have been made available about the microtubule network in mice knockout for actin regulators, but we would expect an effect on the microtubules dynamics. Since actin regulator Rac activity is also decreased in Dock5 -/-osteoclasts [START_REF] Vives | The Rac1 exchange factor Dock5 is essential for bone resorption by osteoclasts[END_REF], we decided to reestablish not only GSK3β inhibition but also Rac activation hypothesizing that both might be needed for the podosome belt rescue. Contrary to our expectations, lithium chloride treatment of Rac L61 infected knockout osteoclasts did not rescue the adhesion structure (data not shown). However, we cannot exclude the fact that a better controlled regulation of Rac activation level and/or localization is important for this process.

Discussion

Osteoclast resorption function depends on podosome assembly into sealing zones. More and more evidence show now that the interplay between the actin cytoskeleton and microtubules is crucial for this adhesion structure formation/stabilization but the overall mechanism is still poorly understood.

We already established that Dock5 is essential for osteoclast sealing zone formation through Rac-dependent actin regulation. In this report, we show that Dock5 is also involved in the control of microtubule dynamic instability through both Racdependent and -independent pathways. Indeed, Dock5 deletion results in decreased microtubules growth length and duration associated to reduced acetylated tubulin level. This post translational modification is a hallmark of stability because it increases mechanical resilience to ensure the persistence of long-lived microtubules [START_REF] Xu | Microtubules acquire resistance from mechanical breakage through intralumenal acetylation[END_REF] [START_REF] Janke | Causes and Consequences of Microtubule Acetylation[END_REF]. Its downregulation, resulting from either HDAC6 inhibition or GSK3β activation, has been correlated to microtubule dynamics alteration in different cell types [START_REF] Garza | Disruption of the psychiatric risk gene Ankyrin 3 enhances microtubule dynamics through GSK3/CRMP2 signaling[END_REF] [START_REF] Bacon | Histone deacetylase 3 indirectly modulates tubulin acetylation[END_REF]) [START_REF] Qu | Stabilization of dynamic microtubules by mDia1 drives Tau-dependent Aβ1-42 synaptotoxicity[END_REF]. Dock5 belongs to the Dock family of guanine nucleotide exchange factors which are atypical activators of small G proteins through evolutionarily conserved DHR2 domain [START_REF] Laurin | Insights into the biological functions of Dock family guanine nucleotide exchange factors[END_REF]. Accordingly, Dock5 published functions are mainly related to Rac activation [START_REF] Omi | Mutation of Dock5, a member of the guanine exchange factor Dock180 superfamily, in the rupture of lens cataract mouse[END_REF] [START_REF] Vives | Pharmacological inhibition of Dock5 prevents osteolysis by affecting osteoclast podosome organization while preserving bone formation[END_REF] [START_REF] Vives | The Rac1 exchange factor Dock5 is essential for bone resorption by osteoclasts[END_REF] [START_REF] Ferrandez | Allosteric inhibition of the guanine nucleotide exchange factor DOCK5 by a small molecule[END_REF] [START_REF] Biswas | MBD3/NuRD loss participates with KDM6A program to promote DOCK5/8 expression and Rac GTPase activation in human acute myeloid leukemia[END_REF] in various cellular processes such as neutrophil chemotaxis and superoxide production [START_REF] Watanabe | DOCK2 and DOCK5 act additively in neutrophils to regulate chemotaxis, superoxide production, and extracellular trap formation[END_REF], epithelial cell invasion and metastasis [START_REF] Frank | The focal adhesion-associated proteins DOCK5 and GIT2 comprise a rheostat in control of epithelial invasion[END_REF] or lens protection (Xu et al., 2017a) [START_REF] Omi | Mutation of Dock5, a member of the guanine exchange factor Dock180 superfamily, in the rupture of lens cataract mouse[END_REF]. Ogawa et al. were the first ones to describe a Racindependent role in microtubule network remodeling during mast cell degranulation.

In these cells, Dock5 acts as a key signaling adaptor that brings Akt into the vicinity of GSK3β, allowing it to phosphorylate GSK3β on serine 9 and inhibit its activity [START_REF] Ogawa | DOCK5 functions as a key signaling adaptor that links FcεRI signals to microtubule dynamics during mast cell degranulation[END_REF]. The fact that Akt activation remains unchanged whereas that of GSK3β is increased in Dock5 knockout osteoclasts suggests that the same mechanism takes place in both cell types. Ubiquitously-expressed serine threonine kinase GSK3β is a critical negative regulator of microtubule dynamic instability [START_REF] Xu | Glycogen synthase kinase 3β orchestrates microtubule remodeling in compensatory glomerular adaptation to podocyte depletion[END_REF] [START_REF] Kumar | GSK3beta phosphorylation modulates CLASP-microtubule association and lamella microtubule attachment[END_REF] [START_REF] Barnat | The GSK3-MAP1B pathway controls neurite branching and microtubule dynamics[END_REF]. It phosphorylates microtubulebinding proteins such as MAP1B [START_REF] Barnat | The GSK3-MAP1B pathway controls neurite branching and microtubule dynamics[END_REF], CRMP-2 [START_REF] Liz | Neuronal deletion of GSK3β increases microtubule speed in the growth cone and enhances axon regeneration via CRMP-2 and independently of MAP1B and CLASP2[END_REF]) [START_REF] Garza | Disruption of the psychiatric risk gene Ankyrin 3 enhances microtubule dynamics through GSK3/CRMP2 signaling[END_REF], CLASP-2 [START_REF] Kumar | GSK3beta phosphorylation modulates CLASP-microtubule association and lamella microtubule attachment[END_REF] [START_REF] Schmidt | Agrin regulates CLASP2-mediated capture of microtubules at the neuromuscular junction synaptic membrane[END_REF], Tau [START_REF] Xu | Glycogen synthase kinase 3β orchestrates microtubule remodeling in compensatory glomerular adaptation to podocyte depletion[END_REF] [START_REF] Gąssowska | Extracellular α-synuclein leads to microtubule destabilization via GSK-3β-dependent Tau phosphorylation in PC12 cells[END_REF] or APC [START_REF] Asada | LKB1-mediated spatial control of GSK3beta and adenomatous polyposis coli contributes to centrosomal forward movement and neuronal migration in the developing neocortex[END_REF] [START_REF] Zhou | NGF-induced axon growth is mediated by localized inactivation of GSK-3beta and functions of the microtubule plus end binding protein APC[END_REF], preventing them from stabilizing microtubules. Some of them are recruited by EB1 at growing microtubule ends that can be assimilated to a molecular platform establishing the link between microtubules and other structures [START_REF] Akhmanova | Control of microtubule organization and dynamics: two ends in the limelight[END_REF]. In osteoclasts, EB1 is enriched in microtubules that are directed toward podosomes [START_REF] Akisaka | Differential distribution of posttranslationally modified microtubules in osteoclasts[END_REF]. Depletion of Akt results in EB1/APC loss of interaction leading to reduced sealing zone and bone resorption, which are recovered upon GSK3β inhibition [START_REF] Matsumoto | Regulation of bone resorption and sealing zone formation in osteoclasts occurs through protein kinase B-mediated microtubule stabilization[END_REF]. More, EB1 which is indispensable for podosome patterning, interacts with podosome-containing acetylated cortactin in the belt [START_REF] Biosse Duplan | Microtubule dynamic instability controls podosome patterning in osteoclasts through EB1, cortactin, and Src[END_REF]. It would be interesting to test if either one of these proteins phosphorylation status or localization is impacted in the absence of Dock5.

Together, our previous and present results show that Dock5 is necessary for osteoclast resorption function by regulating cytoskeleton elements that are important for podosome belt establishment. In the absence of Dock5, actin and microtubules dynamics are impacted by Rac activation decrease and GSK3β inhibition. We were surprised that a simultaneous recovery of their physiological levels could not rescue podosome belts in Dock5 -/-osteoclasts. This process might need a finely tuned amount of Rac activation and/or localization that can hardly be achieved though primary cells infection. Similarly, although lithium chloride treatment increases in fine microtubule acetylation level, the latter is much higher than in control osteoclasts (figure 4E). That might affect the expected rescue because both positive and negative regulation of acetylation impact microtubules growth characteristics [START_REF] Qu | Stabilization of dynamic microtubules by mDia1 drives Tau-dependent Aβ1-42 synaptotoxicity[END_REF] [START_REF] Zilberman | Regulation of microtubule dynamics by inhibition of the tubulin deacetylase HDAC6[END_REF]. Nevertheless, further data are consistent with our model. Among them, the fact both C21 and EHT 1864-dependent inhibition of Rac1 activity did not reproduce Dock5 -/-osteoclasts phenotype suggesting that at least part of it is independent from this GTPase. In the same line of idea, since Rac1 signaling pathway has been shown to downregulate Op18 catastrophe promoting activity through its phosphorylation [START_REF] Watabe-Uchida | The Rac activator DOCK7 regulates neuronal polarity through local phosphorylation of stathmin/Op18[END_REF] [START_REF] Wittmann | Regulation of microtubule destabilizing activity of Op18/stathmin downstream of Rac1[END_REF], we wondered whether Op18 phosphorylation levels were decreased in Dock5 -/- osteoclasts which show a reduced Rac activity [START_REF] Vives | The Rac1 exchange factor Dock5 is essential for bone resorption by osteoclasts[END_REF]. Interestingly, they were not modified by the absence of Dock5 expression in osteoclasts (unpublished data). Dock5 is not the only member of the Dock family to regulate microtubules independently from its guanine exchange factor activity. Dock7 controls the genesis of neurons from radial glial progenitor cells during cortical development through antagonistic interaction with the microtubule growth-promoting function of TACC3 [START_REF] Yang | DOCK7 interacts with TACC3 to regulate interkinetic nuclear migration and cortical neurogenesis[END_REF]. And finally, Dock3 participates in BDNF-dependent primary hippocampal neurons axonal outgrowth by also regulating (i) Rac1-dependent actin reorganization and (ii) microtubule assembly through GSK3 binding, recruitment to the plasma membrane and inactivation. The resulting decrease in CRMP-2 and APC phosphorylation allows their association to tubulin dimers and promotes microtubules polymerization [START_REF] Namekata | Dock3 Stimulates Axonal Outgrowth via GSK-3β-Mediated Microtubule Assembly[END_REF].

Altogether, our data involve Dock5 as a new regulator of microtubule dynamic instability in osteoclast through Rac-dependent and -independent pathways.

The fact that part of it is mediated by Rac supports the therapeutic use of C21, a chemical inhibitor of Dock5-dependent activation of Rac1, to treat osteolytic pathologies.

Materials and Methods

Ethics statement

Mice sacrifice and bone marrow harvest were performed in compliance with local animal welfare laws, guidelines and policies, according to the rules of the regional ethical committee.

Reagents.

C21 (N-(3,5-dichlorophenyl)-benzenesulfonamide) was synthetized by Roowin, lithium chloride was purchased from Fluka (#62476), EHT1864 from Santa Cruz (#sc-361175) and nocodazole from Sigma (#M1404).

Bone Marrow Macrophages (BMMs) isolation and differentiation in osteoclasts

Bone marrow cells were purified from long bones of 6-to 8-week-old C57BL/6 Dock5 wild-type and knockout mice sacrificed by cervical dislocation. 3.10 6 bone marrow cells were cultured for 3 days in αMEM containing 10% heat-inactivated fetal calf serum (Biowest), 2 mM glutamine and 100U/ml penicillin-streptomycin supplemented with 100 ng/ml M-CSF (Miltenyi, #130-101-703) in 100mm-diameter petri dishes.

Medium was changed and the cells cultured for another 3 days to obtain BMMs. For osteoclast differentiation, 3.10 4 BMMs/well were seeded in 24-well tissue culture plates and cultured with 30 ng/ml M-CSF and 50 ng/ml RANKL (Miltenyi, #130-094-076) for 4-6 days. Media was changed every 2 days.

Active Rac quantification

Differentiated osteoclasts were treated with 100 µM C21, 10 µM EHT1866 or vehicle for 1 or 2 h and lysed in 25mM HEPES (pH 7.5), 1% Igepal, 5% glycerol, 100mM NaCl, 10mM MgCl2, 5mM NaF, 1mM Na3VO4 and Protease Inhibitor Cocktail (Sigma, #P8340). Cleared lysate from 3 wells of a 6-well plate was incubated with 2 µg of GST-Pak CRIB bound to glutathione-coupled Sepharose beads (GE Healthcare, #17-0756-01) for 45 min at 4°C. Beads were washed three times in lysis buffer before addition of Laemmli sample buffer. Rac was revealed by western blot and the amount of active Rac was determined relative to the total amount of the protein in whole-cell lysates.

Retroviral transduction of BMMs pMXS-EB3-GFP or pMXS-Rac L61-GFP was co-transfected with pC57GP-Gag-pol and pCSIG-VSV-G constructs into Hek 293T cells using Jet-PEI (Polyplus transfection, #101-10 N). Six hours later, the medium was replaced with fresh one and cells were grown for an additional 72 h. The conditioned medium containing recombinant retroviruses was collected and filtered through 0.45 µm-pore-size filters.

BMMs were cultured with fresh retroviral supernatant diluted to half in αMEM containing 10% fetal calf serum, 2 mM glutamine, 8 µg/ml polybrene (Sigma) and 100 ng/ml M-CSF for 24 h and then selected for expression of the construct with 3 µg/ml puromycin (Sigma) for 24 h. Puromycin-resistant BMMs were used for osteoclast differentiation.

Time-lapse microscopy and microtubules dynamic instability analysis EB3-GFP expressing osteoclasts were placed on a 37°C heated stage in a 5% CO 2 chamber. Images were acquired using a Yokogawa CSU-X1 spinning-disk confocal unit (Andor Technology) and an EMCCD iXon Ultra camera coupled to a Nikon Ti Eclipse microscope (Nikon Instruments) controlled by the Andor iQ3 software (Andor Technology). They were collected every second after 488-nm laser illumination using a 60X plan apo lambda 1.4 NA objective over a 2 min period and processed using ImageJ PureDenoise plugin. Ten EB3-GFP comets appearing in at least 3 consecutive sections were manually tracked per osteoclast with ImageJ MtrackJ plugin. Comet count and comet tracks count were obtained using an ImageJ customwritten macro [START_REF] Sánchez-Huertas | Non-centrosomal nucleation mediated by augmin organizes microtubules in post-mitotic neurons and controls axonal microtubule polarity[END_REF]. HDAC6 activity measurement.

HDAC6 activity was assessed using the Fluorogenic HDAC6 assay kit (Bioscience, #50076) according to the manufacturer's instructions.

Soluble and polymerized tubulin fractionation

Soluble and polymerized tubulin was fractionated as described [START_REF] Larrieu | Chemical inhibition of NAT10 corrects defects of laminopathic cells[END_REF].

Briefly, osteoclasts pre-treated or not with 10 µM nocodazole for 2 h were lysed in Microtubules Stabilization Buffer (MSB: 85 mM PIPES pH 6.9, 1 mM EGTA, 1 mM MgCl 2 , 2M glycerol, 0.5% triton, 4 µg/ml taxol) supplemented with proteases inhibitors. Lysates were kept in ice for 2 min and then centrifuged for 10 min at 13000 rpm. Laemmli buffer was added to the supernatants (soluble fraction). Pellets (polymerized fraction of proteins) were washed once in MSB without triton and resuspended in Laemmli buffer.

Western Blot

Whole osteoclast extracts were made in Laemmli buffer, resolved on SDS-PAGE and electrotransferred on nitrocellulose membranes. Immunoblotting was performed using the following primary antibodies: mouse anti-lys40 acetylated alpha tubulin (1/1000, Sigma, #T6793), mouse anti-alpha tubulin (1/1000, Sigma, #T6074), rabbit anti-GAPDH (1/2000, Cell Signaling, #2118), mouse anti-Rac1 (1/5000, Millipore, #05-389), rabbit anti-ser9 phospho GSK3β (1/1000, Cell Signaling, #9322), rabbit anti-GSK3β (1/1000, Cell Signaling, #12456), rabbit anti-ser473 phospho Akt (1/1000, Cell Signaling, #4060), rabbit anti-Akt (1/1000, Cell Signaling, #9272), rabbit anti-Dock5 (1/2000, homemade) and rabbit anti-histone H3 (1/5000, Abcam, #1791).

Signals were visualized by the ECL Western Lightning Plus detection system (Perkin Elmer) with horseradish peroxidase-conjugated secondary antibodies (GE Healthcare) and quantified using ImageJ.

Immunofluorescence and microscopy

Osteoclasts differentiated on 13 mm-diameter glass coverslips were fixed for 20 min, in 10 µM Taxol (Sigma, #T7402) and 3,2% paraformaldehyde in PHEM (60 mM Pipes, 25 mM Hepes, 10 mM EGTA, 4 mM MgSO 4 , pH 6.9). After permeabilization with 0.1% Triton X100 for 1 min and blocking with 1% BSA in PBS for 15 min, osteoclasts were incubated for 1 h with Alexa Fluor 647-labelled phalloidin (1/1000, Life Technologies, #A22287), mouse anti-lys40 acetylated alpha tubulin (1/500, Sigma, #T6793), rat anti-alpha tubulin (1/200, Santa Cruz, #sc-53030) or rabbit anti-Dock5 (1/100, homemade) and revealed with the adapted Alexa Fluor 488 or 546conjugated secondary antibodies (1/1000, Life Technologies). Preparations were mounted in Citifluor mouting medium (Biovalley) and imaged with Leica SP5-SMD confocal microscope using 40X HCX Plan Apo CS oil 1.3NA or 63X HCX Plan Apo CS oil 1.4NA objectives. All imaging was performed at the Montpellier RIO Imaging facility (http://www.mri.cnrs.fr/en/). Immunofluorescent signals integrated density was quantified using ImageJ.

Statistical analysis

Graphs are represented as the mean ± SD. Statistical significance was assessed with the nonparametric statistical tests mentioned in figure legends using GraphPad Prism (GraphPad Software, Inc).

Figure legends

Figure 1 :

 1 Figure 1: Microtubule acetylation in Dock5 -/-osteoclasts (A) Representative western blot showing acetylated tubulin, total tubulin and GAPDH in Dock5 +/+ and Dock5 -/-osteoclasts (left). Acetylated tubulin levels from 7 independent experiments were normalized to total tubulin and represented on a bar graph (right). *, p<0.05 using Wilcoxon signed rank test. (B) Actin (blue), acetylated (green) and total (red) tubulin immunofluorescence staining of Dock5 +/+ and Dock5 -/- osteoclasts. For each genotype, bottom panels show an enlarged view of the top panels boxed areas. Scale bar= 20 µm. (C) Graph showing acetylated tubulin integrated density from immunofluorescent staining of 13-14 Dock5 +/+ and Dock5 -/- osteoclasts from different experiments. **, p<0.01 as determined by two-tailed Mann Whitney test.

Figure 2 :

 2 Figure 2: Microtubule dynamic instability in Dock5 -/-osteoclasts (A) GFP-positive comets were tracked in Dock5 +/+ and Dock5 -/-osteoclasts infected with EB3-GFP. Scale bar= 20 µm. Bar graphs showing microtubule growth (B) duration, (C) distance and (D) speed as well as (E) comets number / µm 2 and (F) comet tracks number / µm 3 / min of 18-19 osteoclasts from 4 independent experiments. (B-F) ns, not significant; ***p<0.0001; ****p<0.00001 as determined by two-tailed Mann Whitney test.

Figure 3 :

 3 Figure 3: C21 effect on microtubule acetylation and dynamic instability Representative western blots showing (A) active and total Rac1 as well as (B) acetylated tubulin, total tubulin and GAPDH from wild-type osteoclasts treated (C21) or not (vehicle) with 100 µM C21 for 1 hour (left). Acetylated tubulin mean levels from 3 independent experiments were normalized to total proteins and represented on a bar graph (right). (C) Actin (blue), acetylated (green) and total (red) tubulin immunofluorescence staining of Dock5 +/+ osteoclasts treated (+C21) or not (-C21) with 100 µM C21 for 1 hour. For each condition, bottom panels show an enlarged view of the top panels boxed areas. Scale bar= 20 µm. Bar graphs showing (D) microtubule growth duration, distance and speed, (E) comets number / µm 2 as well as (F) comet tracks number / µm 3 / min in Dock5 +/+ osteoclasts expressing EB3-GFP and treated (C21) or not (vehicle) in the same conditions as above in 14-20 osteoclasts from 3 independent experiments. (B) *, p<0.05 using Wilcoxon signed rank test. (D-F) ns, not significant; ***, p<0.0001; ****, p<0.00001 as determined by two-tailed Mann-Whitney test.

Figure 4 :

 4 Figure 4: Defective signaling pathways in Dock5 -/-osteoclasts (A) Western blot showing Dock5 association to soluble (S) or polymerized (P) tubulin from wild-type osteoclasts treated (+) or not (-) with 10 µM nocodazole for 1 hour. It is representative of 3 independent experiments. (B) HDAC6 activity in Dock5 +/+ and Dock5 -/-osteoclasts. Representative western blots showing (C) GSK3β-and (D) Akt phosphorylation levels in Dock5 +/+ and Dock5 -/-osteoclasts (left). Signals were normalized to the total protein levels from 3-5 independent experiments (right). (E) Dock5 +/+ and Dock5 -/-osteoclasts were treated (LiCl) or not (-) with 20 mM lithium chloride for 5 hours and tubulin acetylation was observed by western blot (left). The

Acknowledgments

We acknowledge the Montpellier RIO Imaging (https://www.mri.cnrs.fr/en/) and the RAM (http://www.ram.cnrs.fr/) facilities. We thank J. Boudeau (CRBM, Montpellier, France) for EB3-GFP expressing vector, Carlos Sanchez-Huertas (CRBM, Montpellier, France) for helpful discussions and Anne Morel (CRBM, Montpellier, France) for the management of Dock5 transgenic mice colonies.

Funding

This study was supported by the French Centre National de la Recherche Scientifique (CNRS), Montpellier University and grants from the French Fondation pour la Recherche Médicale (Grant # DEQ20160334933) and from the GEFLUC Languedoc Roussillon (grant #A.P. 2015) to A.B. and a grant from the Fondation ARC 2015 (grant # PJA 20151203109) to V.V..

Conflict of interest statement

The authors have declared no conflict of interest.


Ryusei Oketani, Marine Hoquante, Clément Brandel, Pascal Cardinael, Gérard Coquerel* Normandie Université, Laboratoire SMS-EA3233, Université de Rouen, F-76821, Mont Saint Aignan, France Supporting Information Placeholder ABSTRACT: Herein, we report the chiral symmetry breaking of 2-methoxy-1-naphthamide atropisomers through temperature cycling without the use of any racemization reagent. The racemization rate (k1) controls the deracemization process when the cooling of the slurry is slow enough to keep the system close to equilibrium. The productivity appears proportional to the racemization rate (k1) multiplied by the solubility.

Chirality has attracted research interest for many years. Among many of its fascinating facets, chiral resolution, i.e., the process by which two mirror-image molecules are separated from their mixture, is of paramount importance in chemistry. [START_REF] Ribó | Spontaneous Mirror Symmetry Breaking and Origin of Biological Homochirality[END_REF][START_REF] Chubb | Symmetry Breaking in Development and Stochastic Gene Expression[END_REF][START_REF] Pillai | Symmetry Breaking in Space-Time Hierarchies Shapes Brain Dynamics and Behavior[END_REF] In the context of the pharmaceutical industry, this has led to the stringent requirement that any new chiral pharmaceutical ingredient be developed in its active enantiopure form. [START_REF] Francotte | Chirality in Drug Research[END_REF][START_REF] Nguyen | Chiral Drugs: An Overview[END_REF] Hence, resolution procedures are important; optimization facilitated by a fundamental understanding of the resolution process is key to the successful application of these techniques in industry.

Among the different chiral-resolution techniques available to the engineer, [START_REF] Guy | CRC Handbook of Optical Resolutions via Diastereomeric Salt Formation, Edited by David Kozma[END_REF][START_REF] Coquerel | Preferential Crystallization[END_REF][START_REF] Anderson | Developing Processes for Crystallization-Induced Asymmetric Transformation[END_REF][START_REF] Breuer | Industrial Methods for the Production of Optically Active Intermediates[END_REF] deracemization is an attractive process because its theoretical yield is 100%, whereas the yield from other methods cannot exceed 50%. Although many techniques, for example grinding, [START_REF] Noorduin | Complete Chiral Symmetry Breaking of an Amino Acid Derivative Directed by Circularly Polarized Light[END_REF][START_REF] Engwerda | Solid Phase Deracemization of an Atropisomer[END_REF] ultrasound, [START_REF] Rougeot | Ultrasound-Enhanced Deracemization: Toward the Existence of Agonist Effects in the Interpretation of Spontaneous Symmetry Breaking[END_REF] homogenization, [START_REF] Iggland | Complete Solid State Deracemization by High Pressure Homogenization[END_REF] temperature cycling, [START_REF] Suwannasang | A Novel Design Approach to Scale Up the Temperature Cycle Enhanced Deracemization Process: Coupled Mixed-Suspension Vessels[END_REF][START_REF] Breveglieri | Deracemization of NMPA via Temperature Cycles[END_REF] and microwave [START_REF] Kacker | Novel Design Integrating a Microwave Applicator into a Crystallizer for Rapid Temperature Cycling. A Direct Nucleation Control Study[END_REF] have been applied to deracemization processes, temperature cycling is considered to be the most favourable technique from an industrial perspective because of its simplicity and applicability to a wide range of compounds. Also, in the context of the universal origin of chirality, temperature cycling is found in the natural day/night cycle, which might induce (or promote) deracemization in a natural system. Hence, clarification of the key parameter(s) that control(s) deracemization processes through temperature cycling is urgently required, not only from an industrial viewpoint, but also from the perspective of gaining a universal understanding of symmetry breaking.

While many molecules have been deracemized, several researchers have attempted to experimentally identify the key parameters that control these processes. [START_REF] Fytopoulos | A Population Balance Model for Temperature Cycling-Enhanced Deracemization[END_REF][START_REF] Noorduin | Complete Deracemization by Attrition-Enhanced Ostwald Ripening Elucidated[END_REF][START_REF] Xiouras | Coupling Viedma Ripening with Racemic Crystal Transformations: Mechanism of Deracemization[END_REF][START_REF] Xiouras | Particle Breakage Kinetics and Mechanisms in Attrition-Enhanced Deracemization[END_REF][START_REF] Uwaha | A Model for Complete Chiral Crystallization[END_REF][START_REF] Katsuno | Effect of Nucleation on Chirality Conversion Induced by Random Fluctuation[END_REF][START_REF] Uwaha | Simple Models for Chirality Conversion of Crystals and Molecules by Grinding[END_REF][START_REF] Uwaha | Steady Chirality Conversion by Grinding Crystals -Supercritical and Subcritical Bifurcations[END_REF][START_REF] Uwaha | Mechanism of Chirality Conversion by Grinding Crystals: Ostwald Ripening vs Crystallization of Chiral Clusters[END_REF][START_REF] Iggland | A Population Balance Model for Chiral Resolution via Viedma Ripening[END_REF][START_REF] Iggland | On the Effect of Initial Conditions in Viedma Ripening[END_REF][START_REF] Hein | Pasteur's Tweezers Revisited: On the Mechanism of Attrition-Enhanced Deracemization and Resolution of Chiral Conglomerate Solids[END_REF][START_REF] Gherase | Experimental and Theoretical Study of the Emergence of Single Chirality in Attrition-Enhanced Deracemization[END_REF][START_REF] Uchin | Model of Temperature Cycle-Induced Deracemization via Differences in Crystal Growth Rate Dispersion[END_REF] The results of these studies suggest that racemization and crystal-growth/dissolution rates are essential factors that govern the efficiency of a deracemization procedure. This observation is in good agreement with that of Iggland and Mazzotti, who focused on computational simulations of the isothermal Viedma ripening process, and showed that the ratio of these rates is the key criterion that controls the deracemization rate. [START_REF] Iggland | A Population Balance Model for Chiral Resolution via Viedma Ripening[END_REF] However, in the case of deracemization by temperature cycling, dissolution/growth rates are mainly controlled by the temperature profiles.

Consequently, when dissolution and crystal growth are performed smoothly, racemization rates should be, in principle, the key parameter in deracemization kinetics. In addition, if the racemization rate alone is a good proxy for estimating deracemization, it will provide a high level of practical convenience.

Despite this background, the quantitative role of racemization rate on deracemization kinetics has not been considered experimentally yet because of the difficulties associated with experimental systems. Indeed, most of the published deracemization studies focus on chiral compounds containing stereogenic centres that follow complicated deracemization kinetics because an additional component (e.g. a base) is added to the system, thereby adding an additional level of complexity to the problem. [START_REF] Yuchun | Kinetics of Base Catalyzed Racemization of Ibuprofen Enantiomers[END_REF] The deracemization of atropisomers elegantly circumvents this issue. [START_REF] Engwerda | Solid Phase Deracemization of an Atropisomer[END_REF] Indeed, atropisomerism arises from axial chirality and is due to restricted rotation about a single bond; accordingly, its racemization is a unimolecular process that should follow first-order reaction kinetics characterized by a single parameter.

In this context, we prepared naphthamide 1 as a compound that racemizes in a simple manner and forms in conglomerate system. Indeed, compound 1 exhibits frozen atropisomerism in its solidstate, whereas it spontaneously racemizes in solution. Naphthamide 1 has previously been reported by Sakamoto et al., [START_REF] Sakamoto | Asymmetric Photocycloaddition in Solution of a Chiral Crystallized Naphthamide[END_REF] and our sample was synthesized using a similar route (Scheme 1), the details of which can be found in the Supporting Information. Large single crystals (5-30 mg) for racemization-rate experiments were also prepared by the slow evaporation of a saturated 1:1 (v/v) CHCl3:heptane solution. Scheme 1. Synthesis of the 2-methoxy-1-naphthamide 1.

Sakamoto et al. reported the racemization rate constant (k1) for 1 in methanol (MeOH), tetrahydrofuran (THF), and a 1:1 mixture of MeOH and THF. [START_REF] Sakamoto | Asymmetric Photocycloaddition in Solution of a Chiral Crystallized Naphthamide[END_REF] Here, we measured k1 in various solvents including alcohols, ketones, nitriles, and ethers. A large single crystal (consisting of only one chiral form) was dissolved in a solvent and the time dependence of the optical rotation at 365 nm was recorded over a few hours using a polarimeter. Hence, the slope of ln[αt/α0] as a function of time was used to determine the first-order racemization rate constant k1. The racemization rate was evaluated in terms of racemization rate constant k1.

The solubilities and racemization rate constants (k1) of 1 in the solvents considered are summarized in Table S1 in the Supporting Information. Generally, k1 was higher in the less-polar solvents. For example, k1 increased linearly with increasing solvent chain length in the nitrile solvents. Although the racemization rate constants were highest in the ether solvents, solubility was relatively low. The racemization rate constants were much lower in the alcohol series than in other types of solvent. These observations are ascribable to hydrogen bonding between the carbonyl oxygen of 1 and the solvent, which reduces the rate of racemization. Although Sakamoto et al. reported that 1 retains its chirality in MeOH through hydrogen-bonding interactions, [START_REF] Sakamoto | Asymmetric Photocycloaddition in Solution of a Chiral Crystallized Naphthamide[END_REF] our systematic study reveals that 1 racemizes at a rate depending on the solvent polarity; this trend is particularly conspicuous within the same solvent series.

Deracemization experiments involving 1 were carried out in tBuOH, MTBE, MIBK, and iPrCN by temperature cycling of the saturated solution in equilibrium with the crystals, as described in the following procedure. First, the atropisomer was dissolved in the solvent and the solution was filtered through a membrane filter. The filtered solution was maintained at 20 °C until crystallization spontaneously occurred. The first crystals are usually slightly (< 3%) randomly enriched with one enantiomer. Temperature cycling was then performed between 20 and 25 °C for 5-7 d. The enantiomeric excess of the solid was monitored by optical rotation every 24 h. Since the cooling from 25 to 20 °C corresponds to the crystalgrowth step, it was conducted more slowly than the other steps in order to prevent an uncontrolled nucleation of the counter enantiomer. We showed in this study that the racemization of 1 occurs very slowly in cold methanol (MeOH); therefore, these conditions were selected to evaluate optical rotation and to mitigate the racemization of 1.

It should be noted that 20% of the crystal mass was deliberately dissolved during heating in order to directly compare the deracemization rates in different solvents (Figure 1). The solution equilibria after the heating and cooling steps were confirmed in the same manner used in gravimetric-solubility experiments; i.e., the times for dissolution and crystal growth were sufficient as not to be the rate-limiting steps in the experiment.

Compound 1 was efficiently deracemized within a few days with final enantiomeric excesses (ees) greater than 90% in tBuOH, MTBE, MIBK, and iPrCN, through temperature cycling, with or without an initial chiral bias (Figure 2). The crystal form obtained by deracemization was the same as that reported earlier. [START_REF] Sakamoto | Asymmetric Photocycloaddition in Solution of a Chiral Crystallized Naphthamide[END_REF] The evolution of enantiomeric excess was sigmoidal in nature. This behaviour is often observed in solid-state deracemizations. The dominant enantiomer was determined by the initial chiral bias, except in the iPrCN system. Indeed, in the latter case, when the process was slightly (-)-enriched, deracemization occurred with the formation of the (opposite) (+)-isomer. Although we observed this phenomenon on two occasions, we are currently unable to provide an explanation for this observation.

To compare the deracemization rates under different conditions, Iggland and Mazzotti used the time required for the racemic mixture (0% ee) to reach 90% ee in order to evaluate isothermal Viedma ripening in their computational work. [START_REF] Iggland | A Population Balance Model for Chiral Resolution via Viedma Ripening[END_REF] Here, we define the time required to reach 90% ee from a slightly enriched state (10% ee), to be t90-10, which excludes unreproducible behaviour often observed at the start of the deracemization process. For example, chirality in the solid state can deviate to one enantiomer and then heads in the opposite direction. The t90-10 values reported in Figure 2 are the averages of experiments conducted in triplicate.

According to Iggland and Mazzotti, deracemization kinetics is correlated theoretically to the ratio between the racemization rate and the crystal growth/dissolution rate. [START_REF] Iggland | A Population Balance Model for Chiral Resolution via Viedma Ripening[END_REF] However, as mentioned earlier, the cooling/heating steps were sufficiently slow, see also supporting information, to ensure that the suspension remains close to equilibrium; consequently these steps are not rate limiting and can be excluded when considering the overall deracemization rate in the experiment. Thus, herein, we considered the influence of the racemization rate on the global deracemization rate.

Figure 3a displays a plot of the natural logarithm of k1 as a function of t90-10 using the experimental data collected herein for 1 in the four different solvents. It reveals that the deracemization rate can be estimated by simply measuring the racemization rate in the case of temperature cycling. Contextualising to real phenomena, these parameters can be intuitively interpreted as follows. Slow racemization in solution retains chirality for long periods of time and crystallization occurs with the same chirality before dissolution; therefore, deracemization proceeds at a sluggish pace. By contrast, the solution remains racemic during fast racemization, even during crystallization, which contributes to deracemization.

This behaviour qualitatively rationalizes other results obtained experimentally for deracemization; for example, Breveglieri et al. recently reported the deracemization of N-(2-methylbenzylidene)phenylglycine amide (NMPA) with DBU through temperature cycling, and concluded that a high operating temperature leads to fast deracemization. [START_REF] Breveglieri | Deracemization of NMPA via Temperature Cycles[END_REF] Racemization of NMPA with DBU can be considered to be a catalytic reaction; consequently, the racemization rate is higher at higher operating temperatures. The faster deracemization kinetics at higher temperature for NMPA with DBU is, according to our interpretation, mostly due to the high racemization rate of NMPA. Moreover, we found that the deracemization productivity, which is defined by how much of the desired enantiomer is produced per unit time and volume of suspension, correlates linearly with the product of k1 and solubility, as shown in Figure 3b. In terms of the global outcome of the process, k1 controls the deracemization time, and solubility determines the scale of the symmetry breaking; hence, their product provides the productivity. For example, MTBE yields the fastest deracemization, but it is the worst medium in terms of productivity because of the poor solubility. This contrast illustrates that the order of the productivity differs from that of the deracemization rate. Thus, to achieve practically feasible applications of the process, it is very important to consider the balance between k1 and the solubility.

In summary, we reported the first deracemization study of a single-component atropisomer through temperature cycling without the use of any racemization reagent. We also measured the racemization rates of the compound in different solvents to understand its influence on the overall deracemization rate. Our study suggests that the racemization rate (k1) is the key parameter that determines the deracemization kinetics during temperature cycling. When considering once again the report of Breveglieri et al., [START_REF] Breveglieri | Deracemization of NMPA via Temperature Cycles[END_REF] the faster deracemization kinetics at higher temperature for NMPA is probably also due to the higher solubility of NMPA.

Nevertheless, when the productivity of the global process is of interest, the product of racemization rate by solubility becomes the key parameter to consider. In this context, further process understanding can also be gained by studying the influence of impurities and cooling rate on the process kinetics, in addition to the starting particle-size distribution.

The Supporting Information is available free of charge on the ACS Publications website.

Synthesis of 2-methoxy-1-naphthamide, and (2-methoxynaphthalen-1-yl)(pyrrolidin-1-yl)methanone, racemisation kinetics, crystal growth rate, deracemisation experiments (PDF), Movies monitoring crystal growth in MIBK (AVI).

Figure 2 .

 2 Figure 2. Enantiomeric excess (ee) values of the collected crystals of 1 as functions of time during deracemization experiments, and the corresponding t90-10 values in (a) MIBK, (b) tBuOH, (c) MTBE, and (d) iPrCN. Each point includes uncertainty (< ±1.5%) due to the polarimetery technique.

Figure 3 .

 3 Figure 3. (a) Time required to achieve an ee of 90% (t90-10) as a function of natural logarithm of the racemization rate (k1). (b) The product of k1 and solubility as a function of deracemization productivity.

E-mail: gerard.coquerel@univ-rouen.fr Ryusei Oketani: 0000-0001-7860-2456 Gérard Coquerel: 0000-0001-8977-8676 The authors declare no competing financial interests. This research received funding as part of a CORE project (October 2016-September 2020) from the Horizon 2020 Research and Innovation Programme of the European Union under Marie Sklodowska-Curie grant agreement No. 722456 CORE ITN.

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