Philippe Grandclément

Light rings and light points of boson stars

This work is devoted to the study of photon trajectories around rotating boson stars. The basic properties of boson star models are given with a particular emphasis on the high compactness those objects can have. Using an effective potential method, circular orbits of photons around rotating boson stars are then obtained, at least for relativistic enough configurations. A particular class of light rings, in which the photons are at rest on a stable orbit, is exhibited. By this one means that the associated worldline is collinear to the Killing vector corresponding to the asymptotic time translation symmetry. It is proposed to call those orbits light points. Their existence is very specific to boson stars and the link between light points and ergoregions is investigated.

I. INTRODUCTION

Boson stars were first introduced in the late 1960s in [1][2][3]. They consists of a complex scalar field coupled to gravity. Various families of boson stars have been constructed, with various potentials for the scalar field, with or without rotation, in various dimensions (see Refs. [4][5][6][7] and references therein)... One of the main motivations for the study of boson stars is the fact that they can act as black hole mimickers [8]. What this means is that they can have a large mass with a small size without developing any horizon or singularity. Moreover, boson stars have no hard surface, the scalar field interacting with ordinary matter only through gravitation. Possible scenarios for the formation of boson stars have been studied. There has also been a number of studies that proposed observational means of discriminating boson stars from black holes. This could, for instance, be done by direct imaging of the Galactic center [9]. Observations of gravitational waves could also be a fruitful way to investigate the nature of the compact objects observed in the Universe [10,11].

Apart from being an alternative to black holes, boson stars are very interesting test beds for a lot of situations in which intense gravitational fields play a role. For instance, a class of peculiar orbits for massive particles is investigated in Ref. [12]. Other specific effects concerning the imaging of accretion disks around boson stars are studied in Refs. [9,13].

The main goal of the current paper is to investigate the existence of light rings around boson stars. Those are closed (circular) orbits of photons. They can only appear for very compact objects (dubbed ultracompact in Ref. [14]). It is known that black holes admit light rings and that they lie outside the horizon. The link between light rings and various instabilities was explored in a recent study by V. Cardoso et al. [14]. Orbits of photons around boson stars or black holes with scalar hair have already been discussed in Refs. [15,16], especially in the context of imaging those objects. In this work results are expanded to boson stars with higher angular momentum.

In a recent paper [17], the connection between light rings and the gravitational waves observed by the LIGO observatory [18] is studied. The first oscillations of the ringdown signal are shown to be the signature of the presence of a light ring more than of an apparent horizon. Two objects with exactly the same structure in terms of light rings would generate similar early ringdown signal, regardless of the presence of a horizon. This paper is organized as follows. In Sec. II the reader is reminded about models of boson stars. In particular, the high compactness of such objects is exhibited. Section III describes the study of light rings by means of an effective potential method. In Sec. IV, a particular class of light rings, in which the photons are at rest and on a stable orbit, is investigated. Those light points are a new class of orbits, typical of boson stars. Indeed, their appearance requires an intense gravitational field but also no horizon. In the case of black holes, stationary orbits are located exactly on the horizon and are unstable. Conclusions are given in Sec. V.

II. BOSON STAR MODELS

In this section we recall the setting used in Ref. [12] to compute boson star spacetimes. Boson stars consist of a complex scalar field Φ coupled to gravity. This is achieved by considering the action

S = (L g + L Φ ) √ -gdx 4 , (1)
where L g is the Hilbert-Einstein Lagrangian and L Φ is the Lagrangian of the complex scalar field. They are given by the standard expressions

L g = 1 16π R (2)
L Φ = - 1 2 ∇ µ Φ∇ µ Φ + V |Φ| 2 . (3)
∇ denotes the covariant derivative associated to g, and R is the Ricci scalar. V |Φ| 2 is a potential that can be chosen to construct various types of boson stars. In this paper one considers the simplest possible choice in which the scalar field is a free field, which implies that

V |Φ| 2 = m 2 h2 |Φ| 2 . ( 4
)
m is the mass of the individual boson and the factor m/h then appears as a scale factor for the various quantities (distances, masses, etc). Throughout this paper geometric units are used, such that G = 1 and c = 1.

Rotating boson stars are computed by demanding that the scalar field takes the form

Φ = φ (r, θ) exp [i (ωt -kϕ)] , (5)
where φ is the amplitude of the field (and so real), ω is a real constant (smaller than m/h) and k an integer known as the rotational quantum number. Because of the U (1) symmetry of the action (1), it does not depend on t and ϕ, leading to axisymmetric solutions. This is why the amplitude φ depends on (r, θ) only. The numbers ω and k appear as parameters of the various boson stars. Let us point out that the case k = 0 corresponds to spherically symmetric boson stars.

For the metric, quasi-isotropic coordinates are well adapted to the symmetry of the problem. The metric reads

g µν dx µ dx ν = -N 2 dt 2 + A 2 dr 2 + r 2 dθ 2 + B 2 r 2 sin 2 θ (dϕ + β ϕ dt) 2 . (6)
In the 3+1 language, N is the lapse, the shift vector is β i = (0, 0, β ϕ ) and the spatial metric is γ ij = diag A 2 , B 2 r 2 , B 2 r 2 sin 2 θ . The unknowns are the functions N , A, B and β ϕ , all of them depending only on (r, θ). The unknown fields obey the Einstein-Klein-Gordon system which results from the variation of the action (1) with respect to both the metric g and the scalar field Φ.

Those equations are solved by means of highly accurate spectral methods, implemented by the KADATH library [19]. This enables us to compute sequences of rotating boson stars for k ranging from 0 to 4. Numerical methods and tests are given in detail in Ref. [12]. We present here the sequences in a different way: by plotting the compactness of the various configurations. To do so, one needs to define the radius of the boson stars. This cannot be done unambiguously because the scalar field extends up to infinity and those objects have no true surface. In the following, the radius R of the boson star is defined as the radius for which the field is 0.1 times its maximal value, in the equatorial plane. The precise value of the thus-defined radius obviously depends on the value of the threshold (here 0.1). However, because of the fast decay of the field far from the origin, it is a small effect. Figure 1 shows the radii of the various boson star models. One can clearly see the known fact that the size of a boson star increases with ω and k.

With this definition of the radius, it is possible to compute the compactness which is the adimensional ratio M/R, where M is the standard Arnowitt-Deser-Misner mass. Compactness is shown in Fig. 2. First one can notice that the result depends very moderately on the value of k. The variations of both the mass and radius with k are such that the compactness depends almost only on ω. One can also notice that it can reach very high values of the order unity (remember that typical neutron stars have a compactness of about 0.2), illustrating the fact that boson stars are indeed good black hole mimickers. Let us also mention that spherically symmetric boson stars (i.e. for which k = 0) can only reach compactness of about 0.25. This is mainly due to the fact that they cannot attain small enough values of ω, due to the turning point in the sequences (see Fig. 1 for instance). The results obtained in the case k = 1 are consistent with those shown in Fig. 2 of Ref. [20].

III. LIGHT RINGS

Given the high compactness of boson stars, they can be expected to have light rings which are closed orbits of photons. In Ref. [15] light rings are mentioned (see Fig. 1 in the context of imaging them. We extend the study of light rings to boson stars with higher k. Mathematically speaking one needs to find closed null geodesics of the spacetime. Given the fact that boson stars are axisymmetric objects, one looks for circular orbits in the equatorial plane, as in the case of a Kerr black hole. In this work such orbits are localized using an effective potential method. Let us mention that, in Ref. [16] similar techniques are employed to study photon orbits, in a somewhat more general setting (i.e. for orbits not confined in the orbital plane). Their results appear to be consistent with this work, in the case k = 1.

The effective potential method is a standard technique to study orbits around axisymmetric and stationary objects and it proceeds as follows. Let us call U µ the tangent vector of a photon trajectory. As the orbits are in the equatorial plane, it reduces to (U t , U r , 0, U ϕ ). Boson star spacetimes admit two independent Killing vectors : (∂ t )

µ and (∂ ϕ ) µ . Their existence leads to two conserved quantities being the scalar products of U µ with the two Killing vectors:

U µ (∂ t ) µ = -N 2 + B 2 r 2 β ϕ 2 U t + B 2 r 2 β ϕ U ϕ = -E (7) U µ (∂ ϕ ) µ = B 2 r 2 β ϕ U t + B 2 r 2 U ϕ = L, (8)
where E and L are two constants along the geodesics. From those equations, one can express the components U t and U ϕ , as functions of E and L :

U t = β ϕ L + E N 2 (9)
U ϕ = L B 2 r 2 - β ϕ N 2 (β ϕ L + E) . (10)
Injecting those expressions into the fact that the geodesic is null (i.e. U µ U µ = 0) leads to an equation on U r . It can be put into the form (U r ) 2 + V eff = 0 where V eff is an effective potential, given by

V eff = 1 A 2 - (β ϕ L + E) 2 N 2 + L 2 B 2 r 2 . ( 11
)
Circular orbits are such that V eff = 0 and ∂ r V eff = 0. The second condition is necessary to prevent being only at the periastron or the apoastron of an elliptic orbit. The first condition leads to

E L = -β ϕ + N Br , (12)
where = ±1. Contrary to the massive particle case, E and L are not constrained independently but only via their ratio. This is basically due to the fact that there is no scale of mass in the case of photons.

The condition ∂ r V eff = 0, along with V eff = 0 gives an equation involving the radius of the orbit and the various metric fields :

I (r) = ∂ r β ϕ N B r 2 + ∂ r B B 3 - ∂ r N N B 2 r + 1 B 2 = 0. ( 13
)
This is a purely radial equation (remember one works in the equatorial plane), but it is not a second-order equation on r, the various fields depending on the radius. Figure 3 shows the functions I (r) for two values of ω, in the case k = 1. Let us first mention that choosing = +1 leads to functions I (r) that always remain strictly positive. This corresponds to the dashed curves of Fig. 3. When = -1 the functions I (r) can sometimes vanish, depending on the boson star considered. I (r) vanishes for the most relativistic boson stars being the ones with small values of ω. It follows that those boson stars admit light rings. In Fig. 3, the boson star k = 1; ω = 0.8 has no light ring (the solid blue curve is always positive), whereas the boson star k = 1; ω = 0.7 has two light rings, corresponding to the two values of r at which I (r) vanishes (see the solid black curve).

Figure 4 shows the value of the minimum of I (r), in the case = -1. When this minimum is below zero the boson star admits light rings. This corresponds to the configurations with low values of ω and it can happen for all values of k > 0. The case k = 0 is not shown in Fig. 4 because the minimum is far above zero. The spherically symmetric boson stars considered in this paper have no light ring. This may be linked to the fact that they cannot reach a high enough compactness. On the other hand, the authors of Ref. [15] mentioned the existence of such orbits, even for k = 0 boson stars. However, this is only possible for objects that are farther along the sequence than the configurations explored here (see Fig. 1 of Ref. [15]).

The shape of the function I (r) depicted in Fig. 3 is very generic : when the minimum of I (r) is negative there are two light rings, corresponding to two radii R minus and R plus . Moreover, it is easy to see that ∂ r V eff and I (r) have opposite signs. It follows that R minus corresponds to a minimum of V eff and is stable whereas R plus corresponds to a maximum of V eff and so to an unstable orbit. The radii of the light rings are shown in Fig. 5. By inserting Eq. ( 12) into Eqs. ( 9) and (10) The right-hand side of Eq. ( 14) must be evaluated at R minus or R plus depending on the light ring considered. The tangent vector of the trajectories is then given by U µ = 1, 0, 0, dϕ dt where the parameter along the trajectory has been chosen to be the coordinate t. Orbital frequencies of the inner (respectively outer) light rings are shown in Fig. 6. They are discussed further in Sec. IV.

IV. LIGHT POINTS

The most striking result from Fig. 6 is the fact that the orbital frequency of the inner light ring can change sign. This implies that it can even vanish, for certain boson stars. The associated trajectories are simply given by U µ = (1, 0, 0, 0). In term of spatial coordinates (r, θ, ϕ), those photons are not moving. They correspond more to light points than light rings. This effect is more than just a coordinate effect. Indeed those photons can be said to be at rest in the sense that their worldline is collinear to the Killing vector that corresponds to the asymptotic time translation symmetry. This notion of "at rest" is not new and is indeed identical to the one used when defining what an ergoregion is.

Even if those light points seem to be peculiar, a detailed study can confirm their existence and location. One first condition is that the vector U µ = (1, 0, 0, 0) must be a null vector. Given the fact that the quasi-isotropic metric is diagonal this condition is g tt = 0. It is the same as the one defining the boundary of an ergoregion (see Sec. IV-E of Ref. [12] for a detailed study of ergoregions of boson stars). So light points must be on the boundary of an ergoregion.

However, not all the null curves are geodesics. The geodesic equation is

dU µ dt + Γ µ αβ U α U β = C (t) U µ
where C (t) is a function along the curve. Note that, a priori, one cannot make C (t) = 0 because there is no guarantee that t is an affine parameter. Γ denotes the four-dimensional Christoffel symbols. Inserting U µ = (1, 0, 0, 0) in the geodesic equation leads to the condition

Γ µ tt = 1 2 g µα [-∂ α g tt ] = C (t) U µ , (15)
where all the time derivatives have been set to zero. Moreover, ∂ t g tt and ∂ ϕ g tt vanish due to the stationarity and axisymmetry of the problem. ∂ θ g tt is also zero because the considered orbits are in the equatorial plane which is a surface of symmetry. Equation ( 15) then reduces to g µr [-∂ r g tt ] = 2C (t) U µ . Given the form of the metric (i.e. the use of quasi-isotropic coordinates), one can show that g tr , g θr and g ϕr all vanish. It first implies that one must have C (t) = 0 which means that t is indeed an affine parameter. Only the radial component of the geodesic equation is then not trivially satisfied and it reduces to

∂ r g tt = 0. (16)
This condition on the derivative, along with the fact that g tt must vanish at the light points implies that they can only be situated exactly where an ergoregion starts to form. This is illustrated in Fig. 7, in which the quantity -g tt is plotted, in the plane z = 0, for three different boson stars. The curve ω = 0.8 never goes to zero ; hence the associated boson star has no ergoregion. The curve ω = 0.6 gets below zero and vanishes at two different radii. They are the inner and outer radii of the ergoregion (for boson stars ergoregions have the shape of a torus ; see Sec. IV-E of Ref. [12]). The curve ω ≈ 0.638 corresponds to the case in which an ergoregion appears. g tt vanishes at just one point, which is also the minimum of the curve. This point is such that g tt = 0 and ∂ r g tt = 0 and so corresponds to a light point. Figure 8 gives another illustration of the location of the light points (for the sequence k = 2). The solid black curve denotes the inner radius of the ergoregion and the dashed black one denotes the outer one. Those two curves join at ω ≈ 0.638 which is the frequency at which an ergoregion starts to develop. The blue curve shows the radius of the inner light rings for which dϕ/dt < 0 whereas the red curve denotes the same radius but for configurations with dϕ/dt > 0. It is clear from Fig. 8 that the light point lies at the intersection of the various curves, being the point where the ergoregion starts to develop. This is not so surprising. Ergoregions are defined as being locations where nothing can remain at rest, due to the frame-dragging effect generated by rotation. When it just appears only massless particles moving at the speed of light can overcome this effect and remain at rest. In other words, at the onset of the ergoregion, the frame-dragging effect is just compensated by the velocity of light. Each sequence of boson stars with k > 0 admits a single light point. The values of ω and R minus are summarized in Table I.

It is worth discussing the existence of light points in the case of black holes. First, they cannot exist around rotating Kerr black holes. Indeed, as previously seen, light points exist only when the ergoregion is limited to a single ring. In the case of a Kerr black hole, it has a nonzero volume as soon as a = 0, so as soon as one deviates from the Schwarzschild case. When a = 0, photons at rest do indeed exist but they are on orbits exactly on the horizon (they are the null generators of the event horizon). However, and this is the main difference from the boson star case, they correspond to unstable orbits. Stable light points are therefore very specific to boson star spacetimes.

The results can be checked by integrating numerically the geodesic equation. This is done using the tool Gyoto [21]. One can compute the radius of a light ring, put a photon there, with the right angular frequency [i.e. given by Eq. (14)] ; and check whether the computed orbit is indeed circular. Figure 9 shows such computations, in the case k = 2. The first panel corresponds to the inner light ring (R minus = 3.11139 and dϕ/dt = -0.0404863) and the second panel corresponds the outer light ring (R plus = 7.07489 and dϕ/dt = -0.0637682), both for ω = 0.7. The photon on the inner light ring remains nicely on a circular orbit. The one on the outer light ring starts by describing a circular orbit but is eventually kicked out. This is consistent with the fact that only the inner light ring is stable, corresponding to a minimum of the effective potential. The last panel shows the light point of the k = 2 sequence (i.e. ω = 0.6387 and R = 1.99). As expected, the photon stays at the same spatial location and so appears as a single point in Fig. 9.

V. CONCLUSION

In this paper, models of rotating boson stars are studied. They were obtained numerically in a previous work [12]. It is confirmed that rotating boson stars are objects that can reach very high compactness. This implies that boson stars can exhibit effects in which strong gravity is crucial such as the existence of closed orbits of photons. The existence of those orbits, called light rings, is confirmed for a large class of boson stars. Light rings could have strong implications on the stability of the boson stars themselves [14].

A new class of light rings, which corresponds to stable trajectories in which the photons are at rest, with respect to an observer at infinity, is exhibited. It is proposed to call those orbits light points. Their existence is closely linked to the appearance of an ergoregion. They are very specific of boson stars in the sense that they cannot exist around black holes. Indeed in that case they are located exactly on the horizon and correspond to unstable trajectories. It is believed that this is the first time that those stable light points are discussed. Their existence has been confirmed by a direct integration of the geodesic equation.

FIG. 3 :FIG. 4 :

 34 FIG.3: Function I (r) for k = 1. The blue curves correspond to ω = 0.8 and the black ones correspond to ω = 0.7. The dashed lines denote the cases in which = +1 and the solid lines denote the cases in which = -1.

FIG. 5 :

 5 FIG.5: Radii of the inner light rings (first panel) and the outer ones (second panel).

FIG. 6 :

 6 FIG. 6: Orbital frequency dϕ dt of the inner light rings (first panel) and the outer ones (second panel).

FIG. 7 :

 7 FIG.7:-gtt for three boson stars with k = 2. The upper curve has no ergoregion. The lower one has a standard ergoregion in the shape of a torus. The middle curve corresponds to the critic case in which an ergoregion just starts to appear.

FIG. 8 :

 8 FIG.8: For boson stars with k = 2, inner and outer radii of the ergoregion (black curves). The red and blue curves denote the radius of the inner light ring, with different signs in the orbital frequency. All the curves intersect at the location of the circle which is the light point.

FIG. 9 :

 9 FIG. 9: Orbits integrated numerically. The first panel corresponds to a stable inner light ring whereas the second one depicts the orbit of an unstable outer one (for k = 2 and ω = 0.7). The last panel shows the light point of the k = 2 sequence.

  in that reference) for both k = 0 and k = 1 boson stars, Radii of the various boson stars, defined as the radius for which the field is 0.1 times its maximal value, in the equatorial plane.

		40
							k=4
			k=0
			k=1
		30	k=2 k=3
			k=4
	_ R[h /m]	20					k=0
		10
		0	0.6	0.7	0.8 ω [m/h _ ]	0.9	1
	0 0.2 1 1.2 FIG. 1: 0.6 0.4 0.6 0.8 Compactness	0.7	ω[m/h _ ]	0.8	0.9 k=0 k=1 k=2 k=3 k=4	1

FIG. 2: Compactness of the various boson stars.

TABLE I :

 I Parameters of the light points for k = 1 to k = 4.

	k	ω	Rminus
	1 0.6582 0.779
	2 0.6387 1.99
	3 0.6383 3.36
	4 0.6401 4.83

Acknowledgments

The author thanks V. Cardoso and E. Gourgoulhon for convincing him to study light rings around boson stars.


Olivier Cases

Antoine Obry

Sirine Ben-Yacoub

Sébastien Augustin

Antoine Joseph

Géraldine Toutirais

Manuel Simonutti

Annabel Christ

Pascal Cosette

Renata Kozyraki
email: renata.kozyraki@inserm.fr

Impaired vitreous composition and retinal pigment epithelium function in the FoxG1::LRP2 myopic mice

Keywords: High myopia, Endocytosis, Vitreous, low density lipoprotein receptor-related protein 2

High myopia (HM) is one of the main causes of visual impairment and blindness all over the world and an unsolved medical problem. Persons with HM are predisposed to other eye pathologies such as retinal detachment, myopic retinopathy or glaucomatous optic neuropathy, complications that may at least partly result from the extensive liquefaction of the myopic vitreous gel. To identify the involvement of the liquid vitreous in the pathogenesis of HM we here analyzed the vitreous of the recently described highly myopic low density lipoprotein receptor-related protein 2 (Lrp2)-deficient eyes. Whereas the gel-like fraction was not apparently modified, the volume of the liquid vitreous fraction (LVF) was much higher in the myopic eyes. Biochemical and proteome analysis of the LVF revealed several modifications including a marked decrease of potassium, sodium and chloride, of proteins involved in ocular tissue homeostasis and repair as well as of ADP-ribosylation factor 4 (ARF4), a protein possibly involved in LRP2 trafficking. A small number of proteins, mainly comprising known LRP2 ligands or proteins of the inflammatory response, was over expressed in the mutants. Moreover the morphology of the LRP2-deficient retinal pigment epithelium (RPE) cells was affected and the expression of ARF4 as well as of proteins involved in degradative endocytosis was strongly reduced. Our results support the idea that impairment of the RPE structure and most likely endocytic function may contribute to the vitreal modifications and pathogenesis of HM.

A C C E P T E D

M A N U S C R I P T

Introduction

The human LRP2 gene (2q23.3-3.1) encodes the multiligand endocytic receptor LRP2/Megalin, essential for the renal reabsorption of plasma proteins and the homeostasis of various vitamins and nutrients. Mutations in LRP2 have been associated with the Donnai-Barrow syndrome, a rare genetic syndrome characterized by facial, skeletal and auditory findings, high myopia (axially elongated eyes) and low-molecular weight proteinuria [START_REF] Kantarci | Mutations in LRP2, which encodes the multiligand receptor megalin, cause Donnai-Barrow and facio-oculoacoustico-renal syndromes[END_REF][START_REF] Pober | A review of Donnai-Barrow and facio-oculoacoustico-renal (DB/FOAR) syndrome: clinical features and differential diagnosis, Birt[END_REF].

Mutations in LRP2 and high myopia were also described in patients with the Stickler syndrome suggesting that LRP2 might directly be associated with the highly myopic phenotype. We indeed recently reported that Foxg1-Cre mediated ablation of the Lrp2 gene in the ocular tissues resulted in excessively long eyes essentially due to increased vitreous chamber depth [START_REF] Cases | Foxg1-Cre Mediated Lrp2 Inactivation in the Developing Mouse Neural Retina, Ciliary and Retinal Pigment Epithelia Models Congenital High Myopia[END_REF]. We moreover showed that the mutant eyes displayed several features of myopic retinopathy including chorioretinal atrophy, loss of retinal photoreceptors and posterior staphyloma as well as a decrease in retinal ganglion cell axons and progressive optic nerve atrophy, known causes of visual deterioration and blindness [START_REF] Cases | Foxg1-Cre Mediated Lrp2 Inactivation in the Developing Mouse Neural Retina, Ciliary and Retinal Pigment Epithelia Models Congenital High Myopia[END_REF][START_REF] Ouyang | Diagnosis and treatment of myopic traction maculopathy[END_REF][5][START_REF] Jonas | Histological changes of high axial myopia[END_REF]. These data indicate that LRP2 expressed in the ocular tissues, mainly the retinal pigment (RPE) and ciliary body (CE) epithelia is necessary for normal eye growth and vision but do not identify the underlying mechanism.

A potential regulator of eye growth is the vitreous humor, the colorless, highly aqueous gel-like matrix that fills the posterior segment of the eye [START_REF] Le Goff | Adult vitreous structure and postnatal changes[END_REF]. In addition to the structural components collagen and hyaluronic acid, the vitreous contains various low molecular weight solutes such as ions, glucose, lactate or proteins. Soluble vitreous proteins may be essential for local nutrient supply and coordination of the eye growth and modifications in their concentration may reflect or anticipate ocular states of health and disease [START_REF] Le Goff | Adult vitreous structure and postnatal changes[END_REF][START_REF] Monteiro | Vitreous humor in the pathologic scope: insights from proteomic approaches[END_REF].

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

Vitreous liquefaction, i.e. progressive increase in liquid vitreous occurring simultaneously with a decrease in gel volume, begins at the age of four years and is associated with normal vitreous ageing [START_REF] Le Goff | Adult vitreous structure and postnatal changes[END_REF][START_REF] Sebag | Ageing of the vitreous[END_REF]. In highly myopic eyes vitreous liquefaction occurs earlier, it is more extensive and has been associated with the progression of myopia-associated pathologies including nuclear cataract, glaucoma and rhegmatogenous retinal detachment [START_REF] Ouyang | Diagnosis and treatment of myopic traction maculopathy[END_REF]5,[START_REF] Shui | The gel state of the vitreous and ascorbate-dependent oxygen consumption: relationship to the etiology of nuclear cataracts[END_REF][START_REF] Watanabe | Stickler syndrome with rhegmatogenous retinal detachment[END_REF][START_REF] Holekamp | Myopia and axial length contribute to vitreous liquefaction and nuclear cataract[END_REF][START_REF] Satofuka | Rhegmatogenous retinal detachment associated with primary congenital glaucoma[END_REF][START_REF] Lai | Peripheral and posterior pole retinal lesions in association with high myopia: a cross-sectional community-based study in Hong Kong[END_REF]. A fundamental alteration of the myopic vitreous biochemistry was proposed more than 50 years ago [15] and liquid vitreous modifications were identified in experimentally induced myopia in the chick [START_REF] Seko | Disturbance of electrolyte balance in vitreous of chicks with form-deprivation myopia[END_REF][START_REF] Seltner | A role for the vitreous humor in experimentally-induced myopia[END_REF]. However the electrolytic balance, soluble protein content and the pathogenic potential of the liquid vitreous in congenital HM are still unknown. Furthermore the origin of the liquid vitreous remains unclear although disruption of the blood-ocular barriers and the ensuing leakage of serum components were proposed to contribute to vitreous liquefaction in both myopic patients and animal models [START_REF] Hosaka | Permeability of the blood-retinal barrier in myopia. An analysis employing vitreous fluorophotometry and computer simulation[END_REF][START_REF] Yoshida | Inward and outward permeability of the blood-retinal barrier in experimental myopia[END_REF][START_REF] Kitaya | Changes in blood-retinal barrier permeability in form deprivation myopia in tree shrews[END_REF].

To answer these questions and gain some insight into the pathogenesis of the disease and its complications we here analyzed the liquid vitreous chamber content of the highly myopic Lrp2-deficient eyes, examined the morphology and endocytic function of the outer blood-retinal barrier forming RPE cells and identified ARF4 as a novel factor potentially involved in LRP2-mediated endocytosis.

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Materials and Methods

2.1.Ethics statement

Animal procedures were conducted in strict compliance with approved institutional protocols (INSERM and comité d'éthique en experimentation animale Charles Darwin N°5, permit number 01519.01) and in accordance with the provisions for animal care and use described in the European Communities council directive of 22 September 2010 (2010/63/EU). Deep anesthesia for terminal procedures (perfusion) was provided with a ketamine/xylazine cocktail (80mg/10mg/kg).

Animals

Foxg1.Cre (129.Cg-Foxg1 tm1(cre)Skm /J) mice were purchased from the Jackson Laboratory (Bar Harbor, Maine). Lrp2 Lox/Lox mice have been described elsewhere [START_REF] Cases | Foxg1-Cre Mediated Lrp2 Inactivation in the Developing Mouse Neural Retina, Ciliary and Retinal Pigment Epithelia Models Congenital High Myopia[END_REF]. The average optical parameters of the Lrp2 FoxG1.cre-KO at three-months were measured by In vivo Magnetic Resonance Imaging and were as previously described [START_REF] Cases | Foxg1-Cre Mediated Lrp2 Inactivation in the Developing Mouse Neural Retina, Ciliary and Retinal Pigment Epithelia Models Congenital High Myopia[END_REF] (control mice: axial length = 3.50 mm; Equatorial diameter = 3.15 mm; Vitreous chamber depth = 1.00 mm; Lrp2 FoxG1.cre-KO : axial length = 5.00 mm; Equatorial diameter = 4.00 mm; Vitreous chamber depth = 2.50 mm).

Liquid vitreous humor collection

Three-month-old mice were euthanized by CO 2 inhalation followed by decapitation. Eyes were weighed, oriented posterior-side up in a Petri dish. Optic nerve was sliced to allow collection of the liquid vitreous. Using a micromanipulator (UltraMicroPump UMP3, World Precision Instrument, Sarasota, FL), a glass pipette with a tip broken to 4μm diameter was introduced into the posterior chamber through the head of the optic nerve of the eye in situ.

The pipette was attached to a micro syringe pump controller (SYS-Micro4, World Precision Instrument, Sarasota, FL) permitting 5 millisecond pulses of suction to the back of the pipette.

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humor could be retrieved from each eye in control conditions. In mutant conditions up to 20 l could be retrieved from one eye. Concerning the biochemical analysis, normal or mutant liquid humors were pooled to obtain a necessary 100 l volume.

2.4.Vitreous gel collection

Three-month-old mice were euthanized by CO 2 inhalation followed by decapitation. Eyes were weighed. The scleral tissue posterior to the limbus was grasped and a microsurgical blade was used to make an incision in the cornea from limbus to limbus. The lens and the vitreous were eviscerated using a fine needle holder and the vitreous was carefully dissected and analyzed.

In vivo injection of tracers

Two-month-old mice (n = 3 for each genotype) were used to retro-orbital-injections with fluorochrome labeled-albumin (100μg/ml albumin from bovine serum diluted in PBS, Alexa

Fluor ® 488 conjugate, Thermofisher, USA) or -transferrin (50μg/ml transferring from human serum, Alexa Fluor ® 488 conjugate, Thermofisher). Fifteen minutes after injection, contralateral eye was removed and frozen in isopentane. Serial frozen sections were directly processed for confocal microscopy.

Immunohistochemistry -Immunofluorescence

Adults (n = 5 for each genotype) were anesthetized and perfused transcardially with 4% PFA in 0.12 M phosphate buffer, pH 7.4. After perfusion, eyes were removed from the skull and postfixed overnight in fresh fixative. Serial frozen sections were processed for immunocytochemistry using sheep anti-Lrp2 (1/2000) and mouse anti-alpha smooth muscle actin (1/250; ab7817, Abcam), anti-occludin (1/100; OC-3F10, ThermoFisher). Alexa 488-or 594-conjugated antibodies (1:200, Invitrogen) were used for secondary detection. Nuclear

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staining was achieved in Hoechst 33342. Fluorescent images were obtained using an Olympus confocal microscope (FV-1200-IX83).

Protein analysis

Samples

Vitreous samples were stored at -80°C until analyzed. Protein concentration of these samples was determined using the Bradford method. Twenty-five µg of samples were loaded in Laemmli buffer on 7% polyacrylamide gels and allowed for a short period of migration (1 h). After Coomassie staining, the protein band was excised and the proteins within the bands were reduced in 5mM dithiothreitol and cysteins were irreversibly alkylated in 25 mM iodoacetamide. After washing steps with water, gel bands were submitted to protein digestion (trypsin from Promega, 0.5 µg per band). Several steps of peptide extraction were then performed in H 2 0/CH 3 CN solutions (1:1) acidified with TFA (1%) and the peptide fractions were combined and evaporated.

Liquid nanochromatography and mass spectrometry

For each sample, peptides were dissolved in 0.1% formic acid in water. All experiments were carried out with a linear quadrupole ion trap-Orbitrap mass spectrometer (LTQ Orbitrap Elite, Thermo Scientific) equipped with a nano-ESI source and coupled to a nanoliquid chromatrography (Easy-nLC II, Thermo Scientific). The sample was loaded onto an enrichment column (Cap Trap C 8 , 0.5 x 2 mm, Michrom Bioresources) and the separation was performed by using a reversed phase column (C 18 , L153, ID 5µm, 100 Å pore size, Nikkyo Technos, Japan). The gradient (mobile phase A: H 2 O/0.1% FA; mobile phase B: CH 3 CN/0.1% FA) was delivered at a flow rate of 300 nl/min for 120 min.

Tryptic peptides were eluted from the reverse-phase column into the mass spectrometer, using a linear gradient from 2% to 45% of B. The capillary voltage was set at 1.5 kV and

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the source temperature at 200 °C. The mass spectrometer was operated in the datadependent mode. Survey full scan mass spectra (from m/z 300 to 2000) were acquired in the Orbitrap with a resolution of 30,000. The mass spectrometer selected the 20 most intense ions for fragmentation.

Protein identification

Raw data files were processed using Proteome Discoverer carbamidomethylation of cystein, and oxidation of methionine. The parent ion and daughter ion tolerances were 10 ppm and 0.5 Da, respectively. The results were filtered by applying the identity threshold score of 20. The false positive rates (FDR) were obtained using a reverse database search approach ranged from 1.65 to 2.18 %. Within Proteome Discoverer, a further filtering step was applied: only peptides associated to "high confidence" (HC) were kept and at least two HC peptides were needed for protein identification.

Relative quantification by Progenesis LC-MS

Raw data were imported in Progenesis LC-MS software (v4.0.4441.29989). One sample was set as a reference, and the retention times of all other samples within the experiment were aligned. After alignment, data were normalized. Normalization results in a unique factor for each sample that corrects all peptides abundances in the sample. Statistical analysis was performed using normalized abundances for one-way analysis of variance (ANOVA) calculations. Features presenting p-value <0.05 and q-value <0.05 were

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selected. MS/MS spectra from selected peptides were exported from the Progenesis LC-MS software as a mgf file and used for peptide identification. Search parameters were identical to those described above when using Proteome Discoverer. Conflicting features were discarded for quantification. The total cumulative abundance of the protein was calculated by summing the abundances of all peptides.

SDS-PAGE and Western blot analysis

Three-month-old mice (n = 5 for each genotype) were used. Retinal, RPE and vitreous lysates were prepared as previously described [START_REF] Cases | Foxg1-Cre Mediated Lrp2 Inactivation in the Developing Mouse Neural Retina, Ciliary and Retinal Pigment Epithelia Models Congenital High Myopia[END_REF]. SDS-PAGE and immunoblotting were performed as previously described [START_REF] Cases | Foxg1-Cre Mediated Lrp2 Inactivation in the Developing Mouse Neural Retina, Ciliary and Retinal Pigment Epithelia Models Congenital High Myopia[END_REF]. Briefly

2.8.Transmission Electron Microscopy

Three-month-old (n = 3 for each genotype) mice were anesthetized and perfused transcardially with 2% paraformaldehyde 2.5%glutaraldehyde in 0.1M PBS (pH 7.4). Retinas were sliced in 200-μm-thick sections, postfixed 2 h in 1% osmium tetroxide, dehydrated in alcohol, cleared in acetone, and embedded in Epon. For light microscopy, transverse serial sections (1 μm) were cut, heat dried, and stained with toluidine blue. Ultrathin sections were cut and stained with lead citrate and examined with a Philips (Aachen, Germany) CM100 electron microscope. For each genotype, three different animals were analyzed.

Blood sample collection

After an overnight fast, mice were shortly anesthetized with isoflurane (Forene r , Abbot, France) and blood was collected at 9 a.m. by retro-orbital puncture. For biochemical analysis 300 l of blood was collected into a heparinized tube.

Biochemical analyses

About 12 biochemical parameters were measured, including electrolytes and ions (sodium, potassium, chloride, calcium, iron) and other metabolites (creatinine, total protein, albumin, transferrin) in the blood as well as the liquid vitreous fraction. These parameters were

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determined using an Olympus AU 400 analyzer (Beckman Coulter, Inc, Brea CA, USA) with kits and controls supplied by Olympus or other suppliers. The number of control and mutant eyes analyzed were 30 and 6 respectively. For all the samples three independent analyses were performed.

Data analyses and statistics

Data are expressed as the means ± s.e.m. Differences between the experimental groups were evaluated using ANOVA followed, when significant (P < 0.05), by the Tukey-Kramer test. When only two groups were compared, Mann-Whitney or Student's t test was used as appropriate. The statistical analyses were performed using Graph Prism Software and Sigmaplot.

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Results

The electrolytic composition of the liquid vitreous fraction is modified in the

Lrp2 FoxG1.cre-KO mutants.

The excessive axial elongation of the Lrp2-deficient eyes is essentially due to the increased depth of the vitreous chamber [START_REF] Cases | Foxg1-Cre Mediated Lrp2 Inactivation in the Developing Mouse Neural Retina, Ciliary and Retinal Pigment Epithelia Models Congenital High Myopia[END_REF], a feature of high myopia [5,[START_REF] Meng | Axial length of myopia: a review of current research[END_REF]. We previously showed that the major modifications of the vitreous chamber depth (VCD) occurred during the first three months of life (P90) and that at this age the mutant VCD was already 2.5-fold longer than the control one [START_REF] Cases | Foxg1-Cre Mediated Lrp2 Inactivation in the Developing Mouse Neural Retina, Ciliary and Retinal Pigment Epithelia Models Congenital High Myopia[END_REF]. As a consequence, the mean mass of the Lrp2-deficient eye differed greatly from those displayed by control eyes at P90 (Lrp2-deficient: m = 42.15 ± 2.3 mg, n = 24; control: m = 24.15 ± 0.81, n = 28; p-value = 1.849 10 -35 < 0.001). We therefore isolated the vitreous chamber content of P90 control and Lrp2-deficient eyes, as described in the material section and [START_REF] Harocopos | Importance of vitreous liquefaction in age-related cataract[END_REF]. A gel-like and a clear with no visible blood or pigment, acellular LVF were obtained from both the control and Lrp2-deficient eyes. In control mice the vitreous chamber was essentially filled by a gel-like material attached to the lens; it also contained low amounts of a liquid vitreous. The mean mass of the retro-lental gel-like fraction or the lens of the Lrp2-deficient eye did not differ from those displayed by control eyes (Lrp2deficient: lens, m = 6.02 ± 0.09 mg and vitreous gel, m = 9 ± 0.11 mg, n = 24; control: lens, m = 6.07 ± 0.8 mg and vitreous gel, m = 9.01 ± 0.12 mg, n = 28; lens, p-value = 0.289, vitreous gel, p-value = 0.992). Moreover, the biochemical composition of the gel was globally preserved as suggested by the normal expression of the main vitreous components collagen, fibrillin and fibronectin (Fig. 1A,B).

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Remarkably, the volume of the mutant LVF was at least 8-fold higher at P90 (Lrp2deficient: V = 18.5 ± 0.91 l; control: V = 2.07 ± 0.15 l; p-value = 3.48 10 -54 <0.001). LVF was analyzed as in the schematic experimental work-flow described in Fig 1C . We first characterized the concentration of several soluble constituents including sodium, potassium, chloride, lactate, glucose as well as the total iron content (Table 1) and compared them to their respective serum levels. As expected the concentration of the circulating electrolytes was similar in control and mutant mice (Table 1) and the values were in accordance with previously published results [START_REF] Champy | Genetic background determines metabolic phenotypes in the mouse[END_REF]. The concentration of the same electrolytes in the LVF was different from that of serum: the sodium, potassium, chloride, lactate and glucose content was higher in both the control and mutant LVF, the calcium and phosphorus content was lower in both genotypes (Table 1). Comparative analysis of the control and mutant LVF showed however that whereas the levels of lactate and glucose were not modified, the potassium level was two-fold decreased in the mutants. A slight decrease was also observed in the sodium and chloride content of the mutant LVF. On the contrary, the calcium and phosphorus levels were at least two-fold increased in the mutant LVF. Of note the total iron content was dramatically increased in the mutant LVF and its total antioxidant capacity (i.e. protein thiols, vitamins A, E) [24,[START_REF] Kampa | A new automated method for the determination of the Total Antioxidant Capacity (TAC) of human plasma, based on the crocin bleaching assay[END_REF] was decreased (Table 1). Finally, the osmolality of the mutant LVF was lower than that of the controls (307.8 mOsm/L versus 346.1 mOsm/L) and the total protein concentration of the control and mutant LVF was of 0.3 g/L and 0.9 g/L respectively (Table 1).

The proteome of the liquid vitreous fraction is modified in the Lrp2

FoxG1.cre-KO mutants.

Mass spectrometry overwiew.

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To investigate the proteome of the control and mutant LVF we used label-free liquid nano chromatography-mass spectrometry analysis. Three samples of each condition (normal sample corresponding to the pool of 10 LVFs and individual mutant sample) were analyzed independently by nanoLC-MSMS (biological replicates). Each sample was analyzed in duplicate only for evaluation of analytical robustness (technical replicates).

In order to evaluate the reproducibility of the experiments, different linear regressions were performed by plotting the logarithm of the number of peptide spectrum matches (PSMs)

for different samples of the same group (Ci vs Cj for controls) and for samples from different groups (Mi vs Ci for mutant vs control) (Figure 2A-B). The regression coefficient measured to evaluate the robustness of the technical scenario between biological replicates was found around 0.970. Besides, the regression coefficient is highly lower when comparing samples from the 2 groups, already arguing for the different LVF protein content.

Using the so-called spectral counting semi-quantitative approach with highly stringent criteria (at least two unique matched peptides for each protein and an adjusted p-value < 0.05) nano-LC-ESI-MS/MS experiments identified 679 proteins in the control LVF and 366 in the mutant LVF including 312 proteins that were common in the two groups (Figure 2C, TableS1 for all peptide sequences, Table S2 for all protein identifications, Table S3 for all proteins found in normal LVF and Table S4 for all proteins found in mutant LVF). More than 80% of the proteins identified in the control LVF were previously reported in proteome studies of human and mouse vitreous [START_REF] Wu | Identification and localization of major soluble vitreous proteins in human ocular tissue[END_REF][START_REF] Skeie | Proteomic interactions in the mouse vitreous-retina complex[END_REF][START_REF] Koss | Proteomics of vitreous humor of patients with exudative age-related macular degeneration[END_REF][START_REF] Murthy | Proteomic analysis of human vitreous humor[END_REF][30][START_REF] Skeie | Proteomic insight into the molecular function of the vitreous[END_REF], validating our approach. Of the 679 proteins identified by mass spectrometry in controls, the majority concerned house-keeping metabolic pathways such as glycolysis or gluconeogenesis, but also more specific ones such as phagosome maturation. Interestingly, proteins involved in vision and recently inventoried in proteomic studies of the vitreous were also identified, including opticin, cystatin C, retinol-binding protein, protein dickkopf homolg 3 or S-arrestin [START_REF] Murthy | Proteomic analysis of human vitreous humor[END_REF].

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When discarding proteins common to control and mutants samples, it resulted that 367 proteins appeared unique to the control LVF. Among the latter, proteins involved in ubiquitination process and phagosome maturation were the more represented pathways. In addition, numerous components directly or indirectly involved in the cytoskeleton dynamics (actin, septins, coronins …) were specifically observed in control LVF. Interestingly, all the 8 components of the T-complex, a multimeric chaperonin dedicated to ATP-dependent protein folding and in particular to folding of cytoskeleton proteins (including actin and tubulin) [START_REF] Liang | Molecular chaperones and the cytoskeleton[END_REF] were also present in this list.

Differential Expression

Nowadays, MS1-based measurements are considered more accurate than spectral counting in terms of sensitivity and also dynamic range. Accordingly, protein abundance in the LVF of control and mutant samples was compared using a 2-way ANOVA from the MS1based measurement. This resulted in 387 proteins displaying decreased abundance and in 74 proteins displaying increased abundance when mutant samples were compared with the controls (Tables S5 andS6).

To obtain a global view of the biological processes, molecular function, and cellular components represented by these proteins a gene ontology analysis was performed and protein-protein interaction schemes were constructed. Most of the proteins under-expressed in the mutant LVF were involved in intracellular signaling, ocular function, ocular tissue repair and homeostasis, cell structure, growth and metabolism and included members of the cytochrome P450 and proteasome systems, the neurotrophin pathway, as well as proteins involved in photo-transduction, glycolysis, the function of the citric acid cycle (Figure 3, Table 2) and several crystallins (Figure 4A-B). It is of interest that ARF4, a protein thought to

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be important for the trafficking of the light receptor rhodopsin [START_REF] Deretic | Molecular assemblies that control rhodopsin transport to the cilia[END_REF] as well as of LRP2 [START_REF] Follit | Arf4 Is Required for Mammalian Development but Dispensable for Ciliary Assembly[END_REF] (Figure 4C) was strongly decreased in the mutant LVF.

The proteins over-expressed in the mutant LVF included transport proteins (afamin, albumin, apolipoprotein A-IV, apolipoprotein E, clusterin, ceruloplasmin, corticosteroidbinding globulin, retinol-binding protein, selenoprotein, transcobalamin, transferrin, transthyretin, vitamin-D binding protein); acute-phase response proteins (a-1-antitrypsin, a-2-HS-glycoprotein, angiotensinogen, chitinase 3-like 1); proteins involved in cell growth, maintenance and metabolism (carbonic anhydrase I, gelsolin, liver carboxypeptidase N); complement system proteins, immunoglobulins (illustrated in Figure 5), cell adhesion proteins (zinc-alpha-2-glycoprotein), fibrinolytic and clotting system proteins (anti-thrombin-III, alpha-2-antiplasmin, alpha-2 macroglobulin, plasminogen, prothrombin, kininogen, kallikrein). Among them at least 30, including alpha-amylase, albumin, transferrin, retinolbinding protein, selenoprotein, transcobalamin, clusterin, transthyrtein, angiotensinogen, carbonic anhydrase 1, alpha-2 macroglobulin, alpha-2 antiplasmin and anti-thrombin-III are known LRP2 ligands (Figure 6 and [START_REF] Kozyraki | Multiligand endocytosis and congenital defects: roles of cubilin, megalin and amnionless[END_REF]).

Remarkably most of the proteins overexpressed in the mutant LVF, including the LRP2 ligands may originate from the serum, suggesting impaired function of the blood-ocular barriers, tissues that normally express LRP2. We therefore first investigated the structure of the CE and RPE in control and mutant eyes using transmission electron (TEM) and confocal microscopy.

Morphological modifications of the LRP2-deficient cells

Two barriers have been described in the eye: the blood-retinal (BRB) and the bloodaqueous barrier (BAB). The BRB is particularly restrictive and regulates ion, protein and TJs between the RPE cells [START_REF] Cunha-Vaz | Blood-retinal barrier[END_REF][START_REF] Rizzolo | Barrier properties of cultured retinal pigment epithelium[END_REF][START_REF] Willermain | Origins and consequences of hyperosmolar stress in retinal pigmented epithelial cells[END_REF]. TJs between the non-pigmented layer of the CE as well as the endothelial cells of the iris vasculature provide the blood-aqueous barrier [START_REF] Freddo | A contemporary concept of the blood-aqueous barrier[END_REF][START_REF] Coca-Prados | The blood-aqueous barrier in health and disease[END_REF].

TEM of control and Lrp2-deficient ciliary processes did not reveal any difference in the formation and structure of the TJs and adherent junctions (AJ) at the level of the nonpigmented ciliary epithelial cells (Fig. 7A-B). Furthermore the expression of occludin, a membrane protein essential for both the establishment and maintenance of TJs, appeared preserved both in the mutant ciliary body vasculature and between the non-pigmented cells of the Lrp2-deficient eye (Fig. 7C-D).

TEM analysis of normal RPE cells was as previously described [START_REF] Lehmann | Plasma membrane protein polarity and trafficking in RPE cells: past, present and future[END_REF] (Fig. 7E-F). The RPE cells carried extremely long apical microvilli (20-30 μm) surrounding the outer rod segments (Fig. 7E). The basal plasma membrane of RPE cells displayed highly convoluted micro-infolds and interacted with the underlying choroidal capillaries through the Bruch's membrane (Fig. 7E). TJs followed by AJs were easily observed in the control RPE (Fig. 7F).

In the mutants the outer rod segments were absent and the morphology of the apical domain of RPE cells was dramatically modified; the apical microvilli were lost and replaced by short, convoluted, less ordered structures (Fig. 7G-H) suggestive of apical dedifferentiation. In addition the AJs between adjacent RPE cells appeared shorter compared with the controls and the TJs were smaller or even absent (Fig. 7I-J) suggesting that the barrier function of the mutant RPE cells might be compromised.

Confocal microscopy and western blot analyses showed ectopic, strong expression of alpha-smooth muscle actin (Fig. 7K-L) in the mutant RPE, increased expression of the mesenchymal proteins collagen type 1 and decreased expression of zonula occludens 1 (ZO-

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1), a protein associated with both TJs and AJs (Fig. 7M). These protein modifications are consistent with the TEM observations and apical dedifferentiation of the LRP2-deficient RPE cells [START_REF] Bastiaans | Thrombin induces epithelial-mesenchymal transition and collagen production by retinal pigment epithelial cells via autocrine PDGF-receptor signaling[END_REF], and this was further confirmed by the increased expression of phosphorylated Erk1/2 (Fig. 7M), a signaling marker activated in RPE cells upon initiation of transdifferentiation to a mesenchymal-like morphology [START_REF] Kimura | Attenuation of EMT in RPE cells and subretinal fibrosis by an RAR-γ agonist[END_REF].

Endocytosis as well as lysosomal degradation of material derived through the autophagic or phagocytotic pathways are essential RPE functions and their impairment triggers retinopathy [START_REF] Guha | Approaches for detecting lysosomal alkalinization and impaired degradation in fresh and cultured RPE cells: evidence for a role in retinal degenerations[END_REF]. In this context it is interesting to note that the loss of LRP2 function in the zebrafish pronephros has previously been associated with abrogation of endocytosis due to the impaired formation of the endocytic apparatus [START_REF] Anzenberger | Elucidation of megalin/LRP2-dependent endocytic transport processes in the larval zebrafish pronephros[END_REF]. Conversely inactivation of the lysosomal integral protein-2 (LIMP2) impairs the lysosomal pathway in the marginal cells of stria vascularis as well as membrane expression of Lrp2, a defect preceding overt hearing loss in the LIMP2 null mutants [START_REF] Knipper | Deafness in LIMP2-deficient mice due to early loss of the potassium channel KCNQ1/KCNE1 in marginal cells of the stria vascularis[END_REF]. We therefore analyzed the expression of the classical early endocytic and lysosomal markers EEA1, cathepsin D and LIMP2 in control and LRP2deficient RPE extracts. EEA1 levels were decreased in the mutants suggesting that the early steps of endocytosis may be perturbed (Fig. 7N). With respect to cathepsin D two bands were detected in the control RPE; one of around 52 kDa corresponding with the immature form of the protein, and a second one of around 28 kDa corresponding with a mature one [START_REF] Erickson | Biosynthesis of a lysosomal enzyme. Partial structure of two transient and functionally distinct NH2-terminal sequences in cathepsin D[END_REF]. It is of interest that the expression of mature cathepsin D was almost completely abolished in the mutants specifically in the RPE (Fig. 7N). Furthermore the expression of LIMP2, a protein required for the biogenesis and maintenance of lysosomes [START_REF] Gonzalez | Lysosomal integral membrane protein-2: a new player in lysosome-related pathology[END_REF] was strongly reduced in the mutants (Fig. 7N). To directly test in vivo protein endocytosis by the RPE we analyzed the uptake of Alexa 488-labeled albumin, an LRP2 ligand and of Alexa 488-labeled transferrin, a protein that may be internalized by both LRP2 and the transferrin receptor. After intravenous eye injection of P60 control mice a definite albumin or transferrin uptake and localization in

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ACCEPTED MANUSCRIPT large vacuoles/lysosomes was observed in the RPE (Fig. 7O) and as previously shown [START_REF] Amsellem | Cubilin is essential for albumin reabsorption in the renal proximal tubule[END_REF] at the renal proximal convoluted tubule. Whereas the renal reabsorption was not affected in Lrp2 FoxG1.cre-KO mutants, no albumin-or transferrin-containing vacuoles were detectable in the mutant RPE (Fig. 7P) strongly suggesting that Lrp2 function is necessary for the formation of the RPE endocytic apparatus.

Impaired endocytosis, loss of lysosomes and decreased LRP2 expression were recently observed in the visceral endoderm (VE) of the Arf4 null mice [START_REF] Follit | Arf4 Is Required for Mammalian Development but Dispensable for Ciliary Assembly[END_REF] that die at mid-gestation most likely due to the impairment of the absorptive VE functions. Western blot analysis showed that ARF4 was readily detected in the retina (Fig. 7Q and [START_REF] Deretic | Molecular assemblies that control rhodopsin transport to the cilia[END_REF]) as well as the RPE of control mice. Furthermore ARF4 and LRP2 were co-immunoprecipitated (Fig. 7R) suggesting that these proteins may physically interact. However, in the LRP2-deficient RPE ARF4 was hardly detectable (Fig. 7Q) suggesting that the lack of LRP2 perturbs not only the endocytic process but possibly also protein trafficking in the mutant RPE. Supporting this idea the expression of the recycling endosome marker Rab11, a known ARF4 partner (Fig. 7Q) required for LRP2 recycling [START_REF] Perez Bay | The fast-recycling receptor Megalin defines the apical recycling pathway of epithelial cells[END_REF] was strongly decreased in the mutants. Although the analysis of the phagocytic function of the LRP2-deficient RPE cells is beyond the scope of this work, the above results support the idea that impaired formation of the RPE barrier and endocytic apparatus most likely contribute to the vitreal modifications and the severe retinopathy observed in the LRP2-deficient eyes.

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Discussion

We here analyzed the impact of Lrp2 ablation on the composition of the vitreous and the structure and function of RPE. We identified several ionic and protein modifications that may, at least partly reflect defective metabolism in the ocular tissues that surround the vitreal cavity as well as signs of RPE cell dedifferentiation. We showed that LRP2 ablation impairs protein endocytosis in the RPE and that the expression of several factors involved in the formation of the endocytic apparatus was strongly reduced. We proposed that these morphological and functional RPE alterations contribute to the progression of the HM and associated retinopathy.

4.1.Vitreous liquefaction and ionic modifications of the LVF in HM

Using the chick model several groups showed that in form deprivation (FD) myopia the axial elongation of the myopic globe is directly related to an increase of the liquid vitreous volume [START_REF] Le Goff | Adult vitreous structure and postnatal changes[END_REF][START_REF] Seko | Disturbance of electrolyte balance in vitreous of chicks with form-deprivation myopia[END_REF][START_REF] Halfter | Regulation of eye size by the retinal basement membrane and vitreous body[END_REF]. Our results also show that the increase in vitreous chamber depth is accompanied by an excessive increase of the LVF supporting its contribution in abnormal eye enlargement. The presence of liquid vitreous may result from vitreous gel liquefaction, i.e. the enzymatic digestion of the vitreous gel known to physiologically occur in aging vitreous [START_REF] Los | Age-related liquefaction of the human vitreous body: LM and TEM evaluation of the role of proteoglycans and collagen[END_REF][START_REF] Van Deemter | Enzymatic breakdown of type II collagen in the human vitreous[END_REF][START_REF] Van Deemter | Trypsin-mediated enzymatic degradation of type II collagen in the human vitreous[END_REF]. In the LRP2-deficient eyes however, the mass of the vitreous gel fraction was not apparently modified arguing against a major contribution of this process in the formation of LVF.

Electrolyte analysis of the liquid vitreous in FD eyes showed an increased abundance in chloride and decreased concentrations of potassium and phosphorus [START_REF] Seko | Disturbance of electrolyte balance in vitreous of chicks with form-deprivation myopia[END_REF]. In our genetic model of HM the abundance in potassium was also significantly decreased in the LVF, however the concentrations of chloride and phosphorus were decreased or increased respectively. Despite the discrepancy in phosphorus abundance, most likely due to differences

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in animal models and/or experimental designs, it is clear that the electrolytic balance of the liquid vitreous is altered in the highly myopic eyes and that the abundance of potassium, essential for the regulation of retinal cell membrane potential, may be one of the elements involved in the induction/maintenance of the myopic eye growth. Furthermore the significantly increased concentrations of phosphorus and calcium in the myopic LVF may eventually lead to intraocular calcification and/or reflect an inflammatory status, a condition known to be associated with HM [START_REF] Herbort | Myopia and inflammation[END_REF]. Impaired electrolyte composition of the vitreous may at least partly be explained by the dysfunction of Müller cells, one of the cell types involved in transretinal ionic and fluid transport [START_REF] Bringmann | Müller cells in the healthy and diseased retina[END_REF]. This hypothesis is indeed supported by our previous results [START_REF] Cases | Foxg1-Cre Mediated Lrp2 Inactivation in the Developing Mouse Neural Retina, Ciliary and Retinal Pigment Epithelia Models Congenital High Myopia[END_REF] showing an intense glial fibrillary acidic protein staining of the Müller cells in the mutant retinas and extensive gliosis as soon as post-natal day 10 (P10).

Protein modifications of the LVF in HM

Various crystallins and proteins involved in photo-transduction and amacrine-cell

function are significantly decreased in the LVF of the Lrp2-deficient eyes.

Crystallins were originally described as lens-specific structural proteins essential for its refractive properties [START_REF] Graw | Genetics of crystallins: cataract and beyond[END_REF] but their wider expression in ocular and extra-ocular tissues is established [START_REF] Thanos | Role of crystallins in ocular neuroprotection and axonal regeneration[END_REF]. Alpha crystallins belong to the small heat-shock protein family, regulate apoptosis and are thought to be involved in the pathogenesis of HM-induced cataract [START_REF] Zhu | Epigenetic regulation of αAcrystallin in high myopia-induced dark nuclear cataract[END_REF].

The function of beta-and gamma-crystallins is less clear but these proteins are also involved in apoptosis regulation as well as in retinal axon growth regeneration [START_REF] Thanos | Role of crystallins in ocular neuroprotection and axonal regeneration[END_REF]. Recently decreased levels of beta-crystallin S, beta-crystallin B2 and gamma-crystallin C were reported in the aqueous humor of HM patients [START_REF] Ji | Proteomic analysis of aqueous humor proteins associated with cataract development[END_REF]. The same crystallins as well as of four additional family members were also dramatically reduced in the LVF of LRP-2-deficient eyes. This observation is well correlated with the continuous and increased apoptosis observed in the

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retina of the LRP2-deficient eyes [START_REF] Cases | Foxg1-Cre Mediated Lrp2 Inactivation in the Developing Mouse Neural Retina, Ciliary and Retinal Pigment Epithelia Models Congenital High Myopia[END_REF] and suggests that downregulation of these proteins contributes to the pathogenesis of the myopic retinopathy.

In addition, the abundance of recoverin, phosducin, S-arrestin, interphotoreceptorretinoid binding protein, retinoschisin, calbindin, calretinin and calcium-calmodulin dependent protein kinase II was dramatically decreased in the mutant LVF. These proteins are produced by retinal photoreceptors, amacrine, bipolar and/or ganglion cells and required for photoreceptor differentiation and/or function. The reduced expression of these proteins is consistent with the degeneration of the above retinal cell types observed in the LRP2-deficient eyes [START_REF] Cases | Foxg1-Cre Mediated Lrp2 Inactivation in the Developing Mouse Neural Retina, Ciliary and Retinal Pigment Epithelia Models Congenital High Myopia[END_REF] as well as the absence of rod outer segments described here. It may reflect the progression of the disease and/or be associated with the myopic eye growth. The three-fold decreased expression of the amyloid beta precursor-like protein 2, modulator of amacrine cell function, and recently identified myopia-susceptibility gene [START_REF] Tkatchenko | Consortium for Refractive Error and Myopia (CREAM), C. Williams, APLP2 Regulates Refractive Error and Myopia Development in Mice and Humans[END_REF] further supports this hypothesis.

Finally, retinaldehyde dehydrogenase 2 (RALDH2), an enzyme essential for the synthesis of all-trans-retinoic acid (RA), strongly expressed in the human RPE [START_REF] Harper | Identification of active retinaldehyde dehydrogenase isoforms in the postnatal human eye[END_REF] and essential in visually-guided normal and myopic eye growth [START_REF] Rada | Identification of RALDH2 as a visually regulated retinoic acid synthesizing enzyme in the chick choroid[END_REF], was not detectable in the LVF of the LRP2-deficient eyes. Our finding is in full agreement with the de-differentiation of the LRP2-deficient cells, the essential role of RALDH2 in myopic eye growth and may suggest a direct and/or indirect role of LRP2 in RA homeostasis in the eye.

Various Lrp2 ligands as well as proteins of the complement, coagulation and immune systems are significantly increased in the LVF of the Lrp2-deficient eyes

The level of 74 proteins was two-to 14-fold higher in the mutant LVF. LRP2 is clearly associated with the uptake and cellular handling of numerous of these proteins in neuronal and non-neuronal tissues [START_REF] Kozyraki | Multiligand endocytosis and congenital defects: roles of cubilin, megalin and amnionless[END_REF][START_REF] Fleming | Transthyretin internalization by sensory neurons is megalin mediated and necessary for its neuritogenic activity[END_REF]. Furthermore the LRP2-ligands albumin,

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transthyrretin and vitamin-D binding protein were previously reported to be upregulated in the ocular fluids of HM patients [START_REF] Duan | Proteomic analysis of aqueous humor from patients with myopia[END_REF][START_REF] Shao | Functional analysis of misfolded transthyretin extracted from abnormal vitreous with high myopia related ocular pathologies[END_REF]. Although the specific function of these proteins in the ocular tissues remains unclear the above observations support their significance as potential myopia biomarkers or pathogenic factors and may indirectly involve LRP2 in the regulation of eye growth.

The levels of components of the complement, kinin-kallikrein and immune systems, or several inflammation-associated proteins such as alpha1-antitrypsin and transferrin were significantly increased in the mutant LVF. These modifications are consistent with the presence of an inflammatory microenvironment in the myopic LRP2-deficient eye and in agreement with the idea that HM is an inflammation-related disease [START_REF] Herbort | Myopia and inflammation[END_REF].

Thrombin is a multifunctional protein with both proteinase and growth-factor like activities. Thrombin activity is physiologically detected in the vitreous liquid [START_REF] Bertelmann | Thrombin activity in normal vitreous liquid[END_REF] where it may be needed to avoid bleeding events. Increased thrombin activity was previously reported in patients with rhegmatogenous retinal detachment [START_REF] Bastiaans | The role of thrombin in proliferative vitreoretinopathy[END_REF]. The expression of factors of the thrombin pathway including alpha-2 antiplasmin, alpha-2 macroglobulin, antithrombin III and plasminogen was significantly increased in the LRP2-deficient LVF suggesting an association with the evolution of the disease.

Morphological and functional defects of the LRP2-deficient RPE cells may contribute to the vitreous modifications and myopic retinopathy

A significantly higher abundance of several proteins of the complement, coagulation or immune systems was found in the vitreous of patients with diabetic retinopathy, age-related macular degeneration or proliferative vitreoretinopathy [START_REF] Koss | Proteomics of vitreous humor of patients with exudative age-related macular degeneration[END_REF][START_REF] Angi | Proteomic analyses of the vitreous humour[END_REF][START_REF] Hernández | Identification of new pathogenic candidates for diabetic macular edema using fluorescence-based difference gel electrophoresis analysis[END_REF][START_REF] Yu | Elucidation of the pathogenic mechanism of rhegmatogenous retinal detachment with proliferative vitreoretinopathy by proteomic analysis[END_REF]. In these cases however the increased vitreal protein concentration was most probably linked to a systemic dysregulation and an intraocular leakage of serum proteins due to the breakdown of the ocular

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barriers [START_REF] Hernández | Identification of new pathogenic candidates for diabetic macular edema using fluorescence-based difference gel electrophoresis analysis[END_REF][START_REF] Chowers | The iron carrier transferrin is upregulated in retinas from patients with age-related macular degeneration[END_REF]. The modifications that we observe in the LRP2-deficient LVF are unlikely to be systemic as indicated by the similar ionic composition of the serum in control and mutant mice as well as the physiological levels of the major serum components albumin and transferrin in the mutants.

Ultrastructural analysis of the outer BRB forming RPE cells showed that the apical morphology of the RPE cells was severely affected in the mutants. Furthermore, the formation of the TJs, crucial for the epithelial barrier function was compromised and the mutant RPE cells abnormally expressed mesenchymal markers including aSMA, collagen and fibronectin, well described signs of RPE fibrosis [START_REF] Kimura | Attenuation of EMT in RPE cells and subretinal fibrosis by an RAR-γ agonist[END_REF][START_REF] Takahashi | Effects of mechanical stress and vitreous samples in retinal pigment epithelial cells[END_REF] and often observed after rhegmatogenous retinal detachment. These observations suggested that RPE cell dysfunction, the most likely ensuing impaired transretinal and ionic fluid transport [START_REF] Bringmann | Müller cells in the healthy and diseased retina[END_REF] and disruption of the BRB barriers may be associated events and contribute to the mutant LVF modifications. They furthermore suggested that the LRP2-deficient RPE cells might be less protected against LVF components with pathogenic potential.

Defective serum protein endocytosis and apical dedifferentiation of the epithelial proximal tubule cells has previously been associated with loss of LRP2 in the cystotic mouse kidney [START_REF] Gaide Chevronnay | Time course of pathogenic and adaptation mechanisms in cystinotic mouse kidneys[END_REF]. In line with a central role of LRP2 in protein uptake in the RPE, its inactivation interfered with endocytic uptake of albumin and transferrin specifically in this tissue

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LRP2 activity has previously been observed in mice lacking ARF4 [START_REF] Follit | Arf4 Is Required for Mammalian Development but Dispensable for Ciliary Assembly[END_REF], a protein thought to play a key role in transmembrane protein sorting and to be possibly involved in human diseases such as retinal degeneration. In line with a potential implication of LRP2 in the formation of a proper endocytic machinery, ARF4 interacts with LRP2 in the control RPE whereas its expression is strongly reduced in the mutant. Disruption of lysosomal formation affects a multitude of cellular processes in the RPE, including the phagocytosis of photoreceptor outer segments tips and is thought to contribute to chronic retinal degenerations [START_REF] Guha | Approaches for detecting lysosomal alkalinization and impaired degradation in fresh and cultured RPE cells: evidence for a role in retinal degenerations[END_REF]76]. We propose that the myopic retinopathy observed in the LRP2 mutants [START_REF] Cases | Foxg1-Cre Mediated Lrp2 Inactivation in the Developing Mouse Neural Retina, Ciliary and Retinal Pigment Epithelia Models Congenital High Myopia[END_REF] is directly associated with defective lysosomal formation in the RPE.

In sum our work provides the first differential analysis of the ionic and protein content of the myopic vitreous, uncovers novel potential biomarkers of HM and identifies the RPE cell layer as a pathogenic target/actor of HM associated pathologies. It strongly suggests that LRP2 may be an essential component of the protein network that regulates eye growth and provides a framework for detailed approaches to dissect the molecular components involved in the development and progression of HM. The protein-protein interaction network was constructed using String software (version 10, www.string-db.org). The nodes of the network represent the proteins, the lines between the marbles represent the predicted functional associations between the proteins. Lrp2 (megalin), A2M (alpha2-macroglobulin), Afm (afamin), Agt (angiotensinogen), Alb (Albumin), Apoa4

(apolipoprotein A-IV), ApoE (apolipoprotein E), Clu (clusterin), Gc (vitamin D binding protein), Sepp1 (selenoprotein P) and Serping1 (plasma protease C1 inhibitor).

A

  C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT water flux into and out of the retina. It has an inner component provided by the inter endothelial tight junctions (TJs) of the intra-retinal vasculature and an outer one provided by

  demonstrating the conservation of this pathway in the ocular tissues. Moreover the mutant RPE lacks the lysosomal structures normally present in the control RPE and the expression of proteins involved in various steps of the endocytic process, including EEA1, Rab11, cathepsin D and LIMP2, is strongly reduced. These findings might reflect a preponderant implication of LRP2 in protein endocytosis in the RPE. Alternatively LRP2 might be directly required for the establishment of an endocytic apparatus in the RPE, a hypothesis compatible with the lack of detectable endocytic structures in the kidney of the LRP2 null mutants [75]. The absence of A C C E P T E D M A N U S C R I P T

FundingTable 1 .

 1 provided by RK: Institut National sur la Recherche Médicale ANR-10-LABX-65, ANR-11-IDEX-004-02, and Sanofi-Ophtalmology. Sanofi provided support but did not have any additional role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.A C C E P T E D M A N U S C R I P TACCEPTED MANUSCRIPT Ionic composition of control and mutant liquid vitreous fraction (LVF) at P90. The number of eyes (n) used for one analysis is indicated. The number of analysis (m) for each condition is indicated.

Figure 1 :

 1 Figure 1: (A-B) Analysis of the vitreous gel. (A) SDS-Page/silver staining of vitreous gel

Figure 2 :

 2 Figure 2: General traits of protein identification. Linear regression obtained by plotting the

Figure 3 :

 3 Figure 3: Interactome of proteins under-expressed in the mutant LVF. The protein-

Figure 4 :

 4 Figure 4: Decreased expression of crystallins in mutants. (A) Histograms represent the

Figure 5 :Figure 6 :

 56 Figure 5: Interactome of the proteins over-expressed in the mutant LVF. (A) The

Figure 7 :

 7 Figure 7: Lrp2 deficiency or dysfunction affects epithelial cell morphology. (A-B)

ACCEPTED MANUSCRIPT 2.7.6. Immunoprecipitation

	Normal RPE cell lysates (n = 3) were applied on Sepharose A or Sepharose G (GE
	Healthcare) beads coupled with anti-Lrp2 or anti-Arf4 antibodies. Immunoprecipitation
	15 g of vitreous gel, retina or RPE were separated in 4-12% SDS-PAGE gels (Invitrogen) and transferred to nitrocellulose was performed using standard procedures. Control immunoprecipitation was performed with sheep or mouse pre-immune serum. All precipitated proteins were analyzed by reducing SDS-PAGE and were detected by autoradiography using ECL reagents as M A N U S C R I P T described by the manufacturer (GE Healthcare).
	membranes. After blocking the membranes were incubated with the following primary antibodies: mouse anti-EAA1 (1/250; 610457, BD-transduction lab) and anti-collagen II (1/250; M2193, Santa Cruz), and rabbit anti-alpha smooth muscle actin (1/100; ab5694, Abcam), anti-fibrillin (1/500; sc-20084, Santa Cruz), anti-phospho Erk1-2 (Thr202-204; 1/1,000; 4376, Cell Signaling technology), anti-Erk1-2 (1/1,000; 4695, Cell Signaling technology), anti-LIMP2 (1/100; PA5-20540, ThermoFisher), anti-rab11 (1/200; sc-9020, Santa Cruz) and anti-ZO-1 (1/250; sc-10804, Santa Cruz), and goat anti-collagen alpha 1 A C C E P T E D
	(1/250; sc-8784, Santa Cruz) anti-fibronectin (1/250; sc-6952, Santa Cruz). Mouse anti-
	GAPDH (1/500; SC-166545, Santa Cruz) was used as an internal control. Immunoblots
	were washed in TBS-0.02% Tween 20 (TBST), incubated for 1 h with the appropriate
	horseradish peroxidase-coupled IgG (diluted 1:5000), washed in TBS-0.02% Tween 20,
	revealed using Super Signal (Pierce, Rockford, IL), and exposed to Hyperfilm ECL
	(Amersham Biosciences). Silver staining using the Biorad Silver Stain Plus kit was
	performed according to the manufacturer's instructions.

Table 2 .

 2 Pathways down-regulated in the mutant vitreous fluid.The table lists the number of proteins (N) participating in each pathway, the associated pvalue and the mean fold change of protein abundance.

	Pathways		N	p-value	ratio
	Metabolism
	Glycerolipid	and	21	1.75 10 -20	3.67
	glycolysis
	Citric acid cycle		9	3.67 10 -8	3.97
	Regulation
	Cytochrome P-450		11	2.04 10 -7	3.23
	Phototransduction		6	3.73 10 -5	529
	Proteasome		10	4.37 10 -8	13.5
	Neurotrophin		10	2.66 10 -4	5.58
	Protein processing		17	5.99 10 -8	8.99
	Actin cytoskeleton		17	2.03 10 -6	4.3