Esther Bou

Jean Dolbeault
email: dolbeaul@ceremade.dauphine.fr

INTERPOLATION INEQUALITIES ON THE SPHERE: RIGIDITY, BRANCHES OF SOLUTIONS, AND SYMMETRY BREAKING

Keywords: SBI.tex. 2020 Mathematics Subject Classification. Primary: 58J35, 35B06 Secondary: 26D10, 35K55, 53C21 Interpolation, Gagliardo-Nirenberg-Sobolev inequalities, best constants, heat equation, nonlinear diffusion, entropy methods, carré du champ, bifurcations, symmetry, symmetry breaking, uniqueness, rigidity

This paper is devoted to three Gagliardo-Nirenberg-Sobolev interpolation inequalities on the sphere. We are interested in branches of optimal functions when a scale parameter varies and investigate whether optimal functions are constant, or not. In the latter case, a symmetry breaking phenomenon occurs and our goal is to decide whether the threshold between symmetry and symmetry breaking is determined by a spectral criterion or not, that is, whether it appears as a perturbation of the constants or not. The first inequality is classical while the two other inequalities are variants which reproduce patterns similar to those observed in Caffarelli-Kohn-Nirenberg inequalities, for weighted inequalities on the Euclidean space. In the simpler setting of the sphere, it is possible to implement a parabolic version of the entropy methods associated to nonlinear diffusion equations, which is so far an open question on weighted Euclidean spaces.

INTRODUCTION

In this paper we consider some interpolation inequalities on the sphere which can be written in generic form as

F ∇u 2 L 2 (S d ) , u 2 L 2 (S d ) ≥ u 2 L p (S d ) ∀ u ∈ H 1 (S d , d σ) ( 1
)
where p > 2 has to be specified. Here the function F : (R + ) 2 → R is a two-homogeneous function, i.e.,

F (h a,h b) = h 2 F (a, b) ∀ (h, a,b) ∈ (R + ) 3 ,
and we assume that F (0, 1) = 1. We will not give general assumptions of regularity on F as we shall rather focus on three particular cases that illustrate the variety of possible cases. With standard notations, d σ = S d -1 d v g denotes the uniform probability measure on the unit sphere S d ⊂ R d +1 induced by Lebesgue's measure on R d +1 . The space L q (S d , d σ) with q ∈ [1, +∞) is the standard Lebesgue space with norm • L q (S d ) defined by

u L q (S d ) := S d |u| q d σ 1/q .
The fact that d σ is a probability measure means that u L q (S d ) = 1 if u = 1 a.e. on S d and we also have u L q 1 (S d ) ≤ u L q 2 (S d ) for any u ∈ L q 2 (S d , d σ) as soon as 1 ≤ q 1 ≤ q 2 . The space H 1 (S d , d σ) is obtained by completion of smooth functions on S d with respect to the norm u → ∇u 2 L 2 (S d )

+ u 2 L 2 (S d ) 1/2 .
If [START_REF] Dolbeault | Diffusions hypercontractives[END_REF] is an optimal inequality, and if there is some u ∈ H 1 (S d , d σ) such that equality holds in [START_REF] Dolbeault | Diffusions hypercontractives[END_REF], the question that we address is the symmetry versus symmetry breaking issue: is u necessarily a constant function (symmetry) or not (symmetry breaking). In this last case, there is a manifold of non-constant optimal functions generated by all possible rotations. We also consider limit cases for which compactness is lost and there is no optimal function (or a sequence of functions approximating the equality case is not compact even after normalization): there is symmetry breaking if the optimality case is not achieved by constant functions.

As a side question, let us mention the question of the rigidity in related elliptic equations: up to a normalization by a multiplicative constant, is there at most one nonnegative solution to the Euler-Lagrange equation

-∂ a F (a, b)∆u + ∂ b F (a, b)u = u 2-p L p (S d ) u p-1 with a = ∇u 2 L 2 (S d ) and b = u 2 L 2 (S d )
or not. Here ∆ denotes the Laplace-Beltrami operator on S d with the convention that -∆ is a nonnegative operator. In case of uniqueness and compactness (normalized minimizing sequences for (1) are relatively compact and the limits after extraction of a subsequence solves the equation), then symmetry holds and u = 1 a.e. is the unique solution, while in the symmetry breaking case, there is no rigidity (notice that if u is optimal for (1), then |u| is also optimal). In order to study the symmetry breaking phenomenon, we shall introduce a bifurcation parameter λ > 0 and families of functions (F λ ) λ>0 depending continuously on λ with F λ (0, 1) = λ. Our goal is to characterize the optimal constant µ(λ) in the inequality

F λ ∇u 2 L 2 (S d ) , u 2 L 2 (S d ) ≥ µ(λ) u 2 L p (S d ) ∀ u ∈ H 1 (S d , d σ) (2)
so that symmetry means µ(λ) = λ while symmetry breaking is characterized by µ(λ) < λ. On S d , there is an intrinsic length scale, which compares with λ. The three families of Gagliardo-Nirenberg-Sobolev inequalities on S d studied in this paper are

∇u 2 L 2 (S d ) + λ p -2 u 2 L 2 (S d ) ≥ µ 0 (p, λ) p -2 u 2 L p (S d ) , (3)
∇u 2 L 2 (S d ) + λ p -2 u 2 L 2 (S d ) ≥ µ 1 (p, θ, λ) p -2 u 2 θ L p (S d ) u 2 (1-θ) L 2 (S d ) , ( 4
)
∇u 2 L 2 (S d ) + λ p -2 u 2 L 2 (S d ) θ u 2 (1-θ) L 2 (S d ) ≥ µ 2 (p, θ, λ) p -2 θ u 2 L p (S d ) , (5)
for any u ∈ H 1 (S d , d σ), where λ > 0 is a parameter while µ 0 (p, λ) and µ k (θ, λ) with k = 1, 2 are the optimal constants. Although the case p ∈ [1, 2) makes sense, we shall for simplicity restrict our purpose to p

∈ (2, ∞) if d = 1, 2 and p ∈ (2, 2 * ] if d ≥ 3, with 2 * = 2 d /(d -2).
By convention, we take 2 * = +∞ if d = 1, 2. The exponent

θ := d p -2 2 p (6)
is the exponent in the Euclidean Gagliardo-Nirenberg-Sobolev inequalities

∇u θ L 2 (R d ) u 1-θ L 2 (R d ) ≥ C GNS u 2 L p (R d ) ∀ u ∈ H 1 (R d , d x) (7)
and induces the limitation θ ≥ θ in [START_REF] Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF]. As we shall see later, the regime for λ small corresponds to symmetry while the asymptotic regime as λ → +∞ is related with optimal functions for (7) on the tangent Euclidean space R d , which induce symmetry breaking.

Let us give a brief account of the literature. M.-F. Bidault-Véron and L. Véron proved Inequality (3) in [START_REF] Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF]Corollary 6.1] as a consequence of a rigidity result. Another proof by W. Beckner in [START_REF] Beckner | Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF] relies on spectral methods, the Funk-Hecke formula and a duality result introduced by E. Lieb in [START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF]. An earlier version corresponding to the range p ∈ (2, 2 # ) with 2 # = (2 d 2 + 1)/(d -1) 2 if d ≥ 2 and 2 # = +∞ if d = 1 was established by D. Bakry and M. Emery in [START_REF] Dolbeault | Diffusions hypercontractives[END_REF][START_REF] Émery | Inégalités de Sobolev pour un semi-groupe symétrique[END_REF], using the carré du champ method and the heat flow. We refer to [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF] for a general overview of the carré du champ method in the context of Markov processes or linear diffusion equations, and to [START_REF] Demange | Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature[END_REF][START_REF] Dolbeault | Nonlinear flows and rigidity results on compact manifolds[END_REF] for the extension to nonlinear diffusion equations. Also see Appendix A for further references in the case of the sphere.

Inequalities ( 4) and ( 5) are less standard than (3) and we are not aware of specific references. The case θ = 1 in (4) and ( 5) corresponds to the classical interpolation inequality (3). Inequality (5) appears (with optimal constant) for a special value of θ in [12, Inequality (2.4)] as a consequence of improvements of (3) based on the carré du champ method. For a presentation of entropy methods and improved interpolation inequalities, see [START_REF]Improved interpolation inequalities and stability[END_REF] and references therein.

An important motivation for (5) arises from pure states in Lieb-Thirring estimates and interpolation inequalities for systems studied in [START_REF] Bonheure | Some systems arising from Lieb-Thirring kinetic energy inequalities[END_REF]. We refer to [START_REF]Symmetry and symmetry breaking: rigidity and flows in elliptic PDEs[END_REF] for an overview of functional inequalities and branches of solutions in various frameworks, including the quite similar problem of symmetry breaking in some of the (CKN) Caffarelli-Kohn-Nirenberg inequalities. The sharp threshold for the symmetry range of the parameters in such inequalities is established in [START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF][START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF] in the critical case and in [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF][START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF] in the subcritical case. A complete parabolic proof based on entropy methods is so far missing and results rely on rigidity methods: see [START_REF] Dolbeault | Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization[END_REF][START_REF] Dolbeault | Parabolic methods for ultraspherical interpolation inequalities, Discrete and Continuous Dynamical Systems[END_REF] for partial results based on flows. In the case of the sphere, Inequalities (3), ( 4) and ( 5) have similar properties but by many aspects symmetry versus symmetry breaking issues are simpler. For instance, the regularity of the solution of the nonlinear flow on S d raises no difficulty: see for instance [START_REF] Villani | Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF] for some comments in this direction. Branches of optimizers associated with Inequalities (4) and (5) for θ < 1 have qualitative features which are reminiscent of those corresponding to similar (CKN) inequalities (with also some θ < 1) observed in [START_REF]Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations[END_REF][START_REF] Esteban | A scenario for symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF][START_REF] Dolbeault | Rigidity results with applications to best constants and symmetry of Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities[END_REF]. Beyond numerical observations and a few estimates, these branches for (CKN) are not well understood. Our goal is to state the corresponding properties on the sphere. This paper is organized as follows. In Section 2, we expose results and strategies of proof for Inequality (3). Sections 3 and 4 are devoted to similar properties for Inequalities (4) and ( 5) with a special emphasis on symmetry and symmetry breaking. In the Appendix A, for completeness, we provide some computations based on the carré du champ method applied with a nonlinear flow in the case of the sphere and further bibliographical references for this technique.

CLASSICAL GAGLIARDO-NIRENBERG-SOBOLEV INEQUALITIES ON S d

In this preliminary section, we review some known results for Inequality (3) and give some hints for the proofs which will guide us in the study of the two other inequalities.

Theorem 1. Let d ≥ 1, p ∈ (2, 2 * ) or p = 2 * if d ≥ 3. Inequality (3) holds with µ 0 (p, λ) = λ if λ ≤ d (symmetry case) and µ 0 (p, λ) < λ if λ > d (

symmetry breaking case).

A standard observation is that λ → µ 0 (p, λ) is a concave increasing function as a consequence of

µ 0 (p, λ) = inf u∈H 1 (S d ,d σ)\{0} (p -2) ∇u 2 L 2 (S d ) + λ u 2 L 2 (S d ) u 2 L p (S d )
and that the symmetry breaking point

d = inf λ > 0 : µ 0 (p, λ) < λ is such that d = inf u∈H 1 (S d ,d σ)\R (p -2) ∇u 2 L 2 (S d ) u 2 L p (S d ) -u 2 L 2 (S d )
,

where H 1 (S d , d σ) \ R denotes the set of non-constant functions in H 1 (S d , d σ).

Let us sketch the proof of Theorem 1.

• Symmetry breaking. Let us consider the functional

F λ [u] := ∇u 2 L 2 (S d ) - λ p -2 u 2 L p (S d ) -u 2 L 2 (S d ) . (8)
A Taylor expansion of

F λ [1 + ε ϕ] at order two in ε for a function ϕ such that S d ϕ d σ = 0 shows that lim ε→0 1 ε 2 F λ [1 + ε ϕ] = ∇ϕ 2 L 2 (S d ) -λ ϕ 2 L 2 (S d ) .
Taking for ϕ an eigenfunction of the Laplace-Beltrami operator on S d associated with its lowest positive eigenvalue, i.e., such that -∆ϕ = d ϕ, we find that

∇ϕ 2 L 2 (S d ) -λ ϕ 2 L 2 (S d ) = (d -λ) ϕ 2 L 2 (S d ) < 0 for any λ > d , which shows that µ 0 (p, λ) < λ.
• Symmetry. Let us consider the fast diffusion equation

∂ρ ∂t = ∆ρ m , ρ(t = 0, •) = |u| p (9) for some parameter m ∈ m -(d , p), m + (d , p) where m ± (d , p) := 1 (d + 2) p d p + 2 ± d (p -1) 2 d -(d -2) p . (10)
A computation based on a nonlinear extension of the carré du champ method shows that

d d t F λ [v(t , •)] ≤ -2 β 2 (d -λ) S d ∇v 1/β 2 d σ
where β := 2/ 2-p (1-m) . References are given in Appendix A and the key computation is detailed in Appendix A.1. Hence

d d t F λ [v(t , •)] ≤ 0 and lim t →+∞ F λ [v(t , •)] = 0 if λ ≤ d . This proves that F λ [u] ≥ 0 for an arbitrary u ∈ H 1 (S d , d σ).
See Lemma 14 and final comments at the end of Section A.2 for further justifications.

• Rigidity. As in [START_REF] Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF][START_REF] Erratum | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF][START_REF] Dolbeault | Nonlinear flows and rigidity results on compact manifolds[END_REF], a remarkable consequence of the proof is the following result.

Corollary 2. Let d ≥ 1, p ∈ (2, 2 * ) or p = 2 * if d ≥ 3. If 0 < λ ≤ d , the equation -∆u + λ p -2 u = u p-1 ( 11
)
admits a unique nonnegative solution, which is constant.

The proof is consequence of the carré du champ method based on an idea that goes back to [START_REF] Gidas | Global and local behavior of positive solutions of nonlinear elliptic equations[END_REF].

• Asymptotics. Another piece of information is the asymptotic behavior of the branch λ → µ 0 (p, λ). The following result is taken from [START_REF] Dolbeault | Spectral estimates on the sphere[END_REF]Proposition 10].

Proposition 3. Let d ≥ 1 and p ∈ (2, 2 * ). The optimal constant in (3) is such that lim λ→+∞ µ 0 (p, λ) λ θ = S d -p-2 p θ θ (1 -θ ) 1-θ C GNS . If d ≥ 3 and p = 2 * , then µ 0 (p, λ) = min{λ, d } is not achieved in H 1 (S d , d σ) for any λ > d .
As λ → +∞, the corresponding optimal functions concentrate on points of the sphere. A blow-up analysis shows that the projection on the tangent Euclidean space at a blowup point can be estimated by [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF], and optimal functions for (7) are good test functions, which allows to identify the limit. Notice that concentration is incompatible with symmetry, so that we recover symmetry breaking in the limit as λ → +∞. In the case d ≥ 3 and p = 2 * , loss of compactness occurs and a minimizing sequence behave like concentrating Aubin-Talenti bubbles.

• Reparametrization. The following observation is elementary. Let us consider the optimal constants for Inequalities (4) and ( 5) seen as variational problems, namely

µ 1 (p, θ, λ) = inf u∈H 1 (S d ,d σ)\{0} (p -2) ∇u 2 L 2 (S d ) + λ u 2 L 2 (S d ) u 2 θ L p (S d ) u 2 (1-θ) L 2 (S d ) , ( 12
)
µ 2 (p, θ, λ) = inf u∈H 1 (S d ,d σ)\{0} (p -2) ∇u 2 L 2 (S d ) + λ u 2 L 2 (S d ) u 2 θ -2 L 2 (S d ) u -2 θ L p (S d ) . ( 13
)
As already noted in Section 1, we know for free that

µ 1 (p, 1, λ) = µ 2 (p, 1, λ) = µ 0 (p, λ)
for any λ > 0. The parameter θ is subject to admissibility conditions that will be specified in Sections 3 and 4. Even for θ = 1, Inequalities ( 4) and ( 5) are related with Inequality (3) as follows. Proof. Let us consider the case k = 1. The function U solves the Euler-Lagrange equation

-∆U + 1 p -2 Λ -(1 -θ) µ 1 (p, θ, Λ) X 2 θ U = µ 1 (p, θ, Λ) X 2 θ-2 (p -2) U p-2 L p (S d ) U p-1 with X = U L p (S d ) / U L 2 (S d ) .
The proof is complete with the choice

λ = Λ p -2 -(1 -θ) µ 1 (p, θ, Λ) X 2 θ and κ 2-p = µ 1 (p, θ, Λ) X 2 θ-2 (p -2) U p-2 L p (S d )
.

A similar reparametrization holds for k = 2.

A FIRST FAMILY OF REFINED INTERPOLATION INEQUALITIES ON S d

Let us define

Q (1) p,θ,λ [u] := (p -2) ∇u 2 L 2 (S d ) + λ u 2 L 2 (S d ) u 2 θ L p (S d ) u 2 (1-θ) L 2 (S d )
.

• Admissible parameters. We establish the validity range of the parameters in (4).

Lemma 5. Inequality (4) holds for any

d ≥ 1, θ ∈ [0, 1], p ∈ (2, 2 * ) or p = 2 * if d ≥ 3, with µ 0 (p, λ) ≤ µ 1 (p, θ, λ) ≤ λ ∀ λ > 0 .
Moreover, equality holds and

µ 1 (p, θ, λ) = λ if λ ≤ d .
Proof. By Hölder's inequality, u L 2 (S d ) ≤ u L p (S d ) because d σ is a probability measure. Using [START_REF]Improved interpolation inequalities and stability[END_REF], we learn that

µ 0 (p, λ) = inf u∈H 1 (S d ,d σ)\{0} Q (1) p,1,λ [u] ≤ inf u∈H 1 (S d ,d σ)\{0} Q (1) p,θ,λ [u] = µ 1 (p, θ, λ) ≤ λ ,
where the upper bound is obtained by testing Q (1) p,θ,λ with u = 1 a.e. If λ ≤ d , there is equality because µ 0 (p, λ) = λ (symmetry case) according to Theorem 1.

We read from Lemma 5 that ( 4) is stronger than (3) and this is why we call Inequality (4) a refined Gagliardo-Nirenberg-Sobolev inequality. Inequality (4) can also be seen as an improved inequality compared to (3) or a stability result for (3): see [START_REF]Improved interpolation inequalities and stability[END_REF] for further considerations on these issues and an example corresponding to a special choice of θ in [START_REF] Beckner | Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF].

As in Section 2, let us consider

F (1) λ [u] := ∇u 2 L 2 (S d ) + λ p -2 u 2 L 2 (S d ) - λ p -2 u 2 θ L p (S d ) u 2 (1-θ) L 2 (S d )
.

• Symmetry breaking. As in the case of (3), we obtain an estimate of the symmetry breaking range by Taylor expanding around a constant with a perturbation by a spherical harmonic function.

Lemma 6. Let d ≥ 1, θ ∈ [0, 1], p ∈ (2, 2 * ) or p = 2 * if d ≥ 3. There is symmetry breaking in (4), that is, µ 1 (p, θ, λ) < λ, if λ > d /θ. Proof. Let us consider ϕ ∈ H 1 (S d , d σ) such that S d ϕ d σ = 0, u ε = 1 + ε ϕ, and compute ∇u ε 2 2 = ε 2 ∇ϕ 2 L 2 (S d ) + o(ε 2 ) , u ε 2 2 = 1 + ε 2 ϕ 2 L 2 (S d ) + o(ε 2 ) , u ε 2 p = 1 + ε 2 (p -1) ϕ 2 L 2 (S d ) + o(ε 2 ) . As a consequence, we obtain that lim ε→0 1 ε 2 F (1) λ [u ε ] = ∇ϕ 2 L 2 (S d ) -λ θ ϕ 2 L 2 (S d ) = (d -λ θ) ϕ 2 L 2 (S d ) < 0
if we take a non-trivial function ϕ such that -∆ϕ = d ϕ and λ > d /θ.

• Symmetry. Up to a change in the range of λ, the result and its proof are similar to the case studied in Section 2.

Lemma 7. Let d ≥ 1, θ ∈ [0, 1], p ∈ (2, 2 * ) or p = 2 * if d ≥ 3. There is symmetry in (4), that is, µ 1 (p, θ, λ) = λ, if λ ≤ d /θ.
Proof. If d = 1, one can consider a periodic interval identified with S 1 and refer to [START_REF] Émery | Inégalités de Sobolev pour un semi-groupe symétrique[END_REF] for applying the carré du champ method in this elementary setting. Assume that d ≥ 2 and let us consider again the fast diffusion equation ( 9) with m ∈ m -(d , p), m + (d , p) where m ± (d , p) is defined by [START_REF] Esteban | A scenario for symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF]. The same computation as in the proof of Theorem 1 shows that

d d t F (1) λ [v(t , •)] ≤ -2 β 2 d + λ (1 -θ) x -1 S d ∇v 1/β 2 d σ with x := v 2 θ L p (S d ) v 2 θ L 2 (S d )
.

By a Hölder inequality, we learn that x ≥ 1 and are then back to the same arguments as in the case θ = 1, but now under the condition λ ≤ d /θ.

• Asymptotics. Heuristically, large values of λ correspond to highly concentrated optimal functions, which behave like the functions obtained by a stereographic projection on the tangent Euclidean space at the point of concentration. Let us explain how we define and use the stereographic projection. On R d x, let r = |x| and ω = x/|x| denote spherical coordinates. On the unit sphere

S d ⊂ R d +1 , we consider cylindrical coordinates (ρ ω, z) ∈ R d × (-1, 1) with ρ 2 + z 2 = 1. The stereographic projection S : S d \ {N} → R d , where N ∈ S d is the North Pole defined by z = +1, is such that S(ρ ω, z) = r ω = x where z = r 2 -1 r 2 + 1 and ρ = 2 r 1 + r 2 .
If v is a function on R d , let us consider its counterpart u := S -1 v on S d obtained using the inverse stereographic projection as

u • S -1 (x) = m(r ) d -2 v(x) ∀ x ∈ R d with m(r ) = (1 + r 2 )/2 .
For any q ≥ 1, with δ(q) := 2 dq (d -2), we obtain

S d |u| q d σ = S d -1 R d |v| q m(r ) δ(q) d x , S d |∇u| 2 d σ + 1 4 d (d -2) S d |u| 2 d σ = S d -1 R d |∇v| 2 d x .
Notice that δ(2) = 4. We refer to [12, Theorem 2.1] for a statement concerning the inequality obtained from (3) and a special case of (4) by the stereographic projection.

As another observation, we shall use the optimal constant K p,d in the Gagliardo-Nirenberg-Sobolev inequality on R d

∇u 2 L 2 (R d ) + u 2 L 2 (R d ) ≥ K p,d u 2 L p (R d ) ∀ u ∈ H 1 (R d , d x) . ( 14
)
An optimization under scaling (apply the inequality to u h (x) = h d /p (h x) and optimize its left-hand side with respect to h > 0) shows that this inequality is equivalent to (7) and, with θ as in [START_REF] Erratum | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF],

θ θ (1 -θ ) 1-θ K p,d = C GNS .
After these preliminaries, we can state the following result, which extends the results for (3) of [16, Proposition 9 and 10] to Inequality (4).

Lemma 8. Let d ≥ 1, θ ∈ [0, 1], p ∈ (2, 2 * ) or p = 2 * if d ≥ 3. There is an explicit constant µ ∞ 1 (p, θ) > 0 such that µ 1 (p, θ, λ) = µ ∞ 1 (p, θ) λ γ 1 + o(1) as λ → +∞ where γ = 1 -d (p -2) θ/2p = 1 -θ θ .
An explicit expression of µ ∞ 1 can be deduced from the proof.

Proof. We do a blow-up analysis based on a change of variables depending on λ → +∞.

Let u λ := S -1 v λ where

v λ (x) := v 2 (λ-λ ) p-2 x ∀ x ∈ R d .
We take λ := 1 4 d (d -2) (p -2) and v as an optimal function for [START_REF] Dolbeault | Sharp interpolation inequalities on the sphere: New methods and consequences[END_REF]. With r = |x|, we compute

Q (1) p,θ,λ [u λ ] = (p -2) ∇u λ 2 L 2 (S d ) + 1 4 d (d -2) u λ 2 L 2 (S d ) + (λ -λ ) u λ 2 L 2 (S d ) u λ 2 θ L p (S d ) u λ 2 (1-θ) L 2 (S d ) = (p -2) ∇v λ 2 L 2 (R d ) + (λ -λ ) m(r ) -2 v λ 2 L 2 (R d ) |S d | θ (1-2/p) m(r ) -δ(p)/p v λ 2 θ L p (R d ) m(r ) -2 v λ 2 (1-θ) L 2 (R d ) = (λ -λ ) 4 (λ-λ ) p-2 -d p-2 2 p θ 4 ∇v 2 L 2 (R d ) + m(εr ) -2 v 2 L 2 (R d ) |S d | θ (1-2/p) m(εr ) -δ(p)/p v 2 θ L p (R d ) m(εr ) -2 v 2 (1-θ) L 2 (R d )
.

with ε = p-2 4 (λ-λ ) . By taking the limit as λ → +∞ and using Lebesgue's theorem of dominated convergence with m(εr ) → 1/ 2 a.e., and by using the fact that the quotient is proportional to the Gagliardo-Nirenberg-Sobolev inequality up to a numerical constant, we conclude that

Q (1) p,θ,λ [u λ ] = (λ -λ ) 1-θθ µ ∞ 1 (p, θ) 1 + o(1) as λ → +∞, for a constant µ ∞
1 (p, θ) that can be computed explicitly, up to tedious but elementary considerations. This proves that

lim λ→+∞ λ -γ µ 1 (p, θ, λ) ≤ µ ∞ 1 (p, θ).
We have now to prove the reverse inequality. As in [16, Proposition 10], we argue by contradiction. Let (λ n ) n∈N and (u n ) n∈N be such that

u n ∈ H 1 (S d , d σ) with u n L p (S d ) = 1 for any n ∈ N, lim n→+∞ λ n = +∞, Q (1) p,θ,λ n [u n ] = µ 1 (p, θ, λ n ) and lim n→+∞ λ -γ n µ 1 (p, θ, λ n ) ≤ µ ∞ 1 (p, θ) -η ( 15
)
for some η > 0. We learn from the expression of

Q (1) p,θ,λ that λ n u n 2θ L 2 (S d ) ≤ µ ∞ 1 (p, θ) λ γ n
as n → +∞ so that lim n→+∞ u n L 2 (S d ) = 0. Concentration occurs: there exists an eventually finite sequence (y i ) i ∈N of points in S d , sequences of positive numbers (ζ i ) i ∈N and (r i ,n ) i ,n∈N and functions u i ,n ∈ H 1 (S d , d σ) with

u i ,n = u n on S d ∩ B (y i , r i ,n ) and supp u i ,n ⊂ S d ∩ B (y i , 2 r i ,n ) such that, as n → +∞, i ∇u i ,n 2 L 2 (S d ) ∼ ∇u n 2 L 2 (S d ) and i u i ,n 2 L 2 (S d ) ∼ u n 2 L 2 (S d ) ,
with, for all i ,

lim n→+∞ r i ,n = 0 , Σ i ∈N ζ i = 1 and lim n→+∞ S d ∩B (y i ,r i ,n ) |u i ,n | p d σ = ζ i .
With a blow up argument applied to (u i ,n ) n∈N , we prove for all i that

lim n→+∞ λ -γ n (p -2) ∇u i ,n 2 L 2 (S d ) + λ n u i ,n 2 L 2 (S d ) u i ,n -2 (1-θ) L 2 (S d ) ≥ µ ∞ 1 (p, θ) ζ 2 θ/p i .
We may notice that

(p -2) ∇u i ,n 2 L 2 (S d ) + λ n u i ,n 2 L 2 (S d ) u i ,n 2 (1-θ) L 2 (S d ) ≥ (p -2) ∇u i ,n 2 L 2 (S d ) + λ n u i ,n 2 L 2 (S d ) u n 2 (1-θ) L 2 (S d )
.

Let us choose an integer N such that

Σ N i =1 ζ 2 θ/p i ≥ Σ N i =1 ζ i 2 θ/p > 1 - η 2 µ ∞ 1 (p, θ) -1 .
For n large enough, by writing

λ -γ n N i =1 (p -2) ∇u i ,n 2 L 2 (S d ) + λ n u i ,n 2 L 2 (S d ) u i ,n 2 (1-θ) L 2 (S d ) ≥ µ ∞ 1 (p, θ) N i =1 ζ 2 θ/p i 1 + o(1) ≥ µ ∞ 1 (p, θ) - η 2 ,
we obtain a contradiction with [START_REF]Improved interpolation inequalities on the sphere[END_REF]. This concludes the proof.

We collect our results on inequality (4) with optimal constant µ 1 (p, θ, λ) as follows.

(p, θ) > 0, we have µ 1 (p, θ, λ) ∼ µ ∞ 1 (p, θ) λ γ as λ → +∞ with γ = 1 -2 θ θ .

A SECOND FAMILY OF REFINED INTERPOLATION INEQUALITIES ON S d

In this section, we consider [START_REF] Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF]. Let us define

Q (2) p,θ,λ [u] := ∇u 2 L 2 (S d ) + λ p-2 u 2 L 2 (S d ) θ u 2 (1-θ) L 2 (S d ) u 2 L p (S d )
.

• Admissible parameters. We establish the validity range of the parameters in [START_REF] Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF]. We recall that θ = d (p -2)/(2 p) according to the definition [START_REF] Erratum | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF]. Let us define

q := 2 p θ 2 -p (1 -θ) ( 16
)
so that q = 2 if θ = θ and q = p if θ = 1.

Lemma 10. Inequality (5) holds for any d , θ and p such that

d ≥ 1 , θ ∈ [θ , 1] ∩ (1 -2/p, 1] , p ∈ (2, 2 * ) or p = 2 * if d ≥ 3 (17)
with an optimal constant µ 2 (p, θ, λ) such that

µ 0 q, λ q-2 p-2 ≤ µ 2 (p, θ, λ) ≤ λ ∀ λ > 0
where q is given in terms of p and θ by [START_REF] Dolbeault | Spectral estimates on the sphere[END_REF]. Moreover,

µ 2 (p, θ, λ) = λ if λ ≤ d 1 -(1 -θ) p 2 .
Proof. By Hölder's inequality, u L 2 (S d ) ≤ u L p (S d ) because d σ is a probability measure. Using [START_REF]Improved interpolation inequalities and stability[END_REF], we learn that

µ 0 q, λ = inf u∈H 1 (S d ,d σ)\{0} Q (1)
q,1,

λ[u] ≤ inf u∈H 1 (S d ,d σ)\{0} Q (2) p,θ,λ [u] 1/θ ≤ λ with λ := λ q -2 p -2 .
The upper bound is obtained by testing Q (2) p,θ,λ with u = 1 a.e. If λ ≤ d , there is equality because µ 0 q, λ = λ (symmetry case) according to Theorem 1, which gives us a symmetry case. Notice that 1

-(1 -θ) p 2 is positive if θ > 1 -2/p.
As above, let us consider

F (2) λ [u] := ∇u 2 L 2 (S d ) + λ p -2 u 2 L 2 (S d ) θ u 2 (1-θ) L 2 (S d ) - λ p -2 θ u 2 L p (S d ) .
• Symmetry breaking. Again symmetry breaking is obtained by perturbation.

Lemma 11. Assume that d , θ and p satisfy [START_REF] Dolbeault | Nonlinear flows and rigidity results on compact manifolds[END_REF]. There is symmetry breaking in [START_REF] Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF], that is,

µ 2 (p, θ, λ) < λ, if λ > θ d .
Proof. With the same computations as in the proof of Lemma 11 and a non-trivial spherical harmonic function ϕ, we obtain that

lim ε→0 1 ε 2 F (2) λ [u ε ] = λ p -2 θ θ ∇ϕ 2 L 2 (S d ) -λ ϕ 2 L 2 (S d ) = (d θ -λ) ϕ 2 L 2 (S d ) < 0 ,
which concludes the proof.

• Symmetry. We apply the same methods as in Sections 2 and 3.

Lemma 12. Assume that d , θ and p satisfy [START_REF] Dolbeault | Nonlinear flows and rigidity results on compact manifolds[END_REF].

If θ ≤ 1/2, there is symmetry in (5), i.e., µ 2 (p, θ, λ) = λ, if λ ≤ θ d .
It is important to notice that the compatibility of the condition θ ≤ 1/2 with (17) induces restrictions not only on θ but also on

p if d ≥ 2: p < 2 if d = 2 and p ≤ 2 d /(d -1) if d ≥ 3.
Proof. Let us consider an optimal function u ∈ H 1 (S d , d σ) for ( 5 . We assume that u is not a constant so that b < 1 by Hölder's inequality. By optimality, we have that

(p -2) a + λ b = µ 2 (p, θ, λ) b 1 θ -1 < λ b 1 θ -1 .
Let us consider the same flow as in Lemma 12. At t = 0, we know that d d t F (2) λ [v(t , •)] = 0 because u is a critical point, and the computation shows that this amounts to

θ (λ -d ) (p -2) a + λ b + 1 -θ b = 0 .
By Lemma 10 symmetry eventually holds only if λ ≤ θ d < d . Let us assume that our non symmetric minimizer corresponds to the range λ ≤ θ d . Hence λd < 0 and

λ -θ d > θ (λ -d ) + (1 -θ) λ b 1 θ -2 > 0 if θ < 1/2 because b 1 θ -2 < 1.
This is obviously a contradiction. We conclude that there is no non-symmetric optimizer if θ < 1/2 and λ ≤ θ d .

So far, this proof is formal as we assume that a minimizer exists. By standard compactness results, this is always true, except for some limiting values of θ or p. However, one can always approximate an inequality like (5) by another inequality in the same family, with slightly different exponents and pass to the limit. This concludes the proof.

Let us conclude by a few remarks. The issue of symmetry versus symmetry breaking is far from fully answered in case of inequality [START_REF] Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF], which is not unexpected in view of similar difficulties in corresponding Caffarelli-Kohn-Nirenberg inequalities. This is clearly an open and interesting direction of research. A good point is that the carré du champ method at least provides some partial answers, which are not available in similar inequalities on the Euclidean space with weights. There are many other questions which are pending and will be the subject of future work: asymptotics as λ → +∞ for (5), reparametrization and numerical computation of the branches, expansion of the branches near the bifurcation point, energy estimates in relation with symmetry breaking in the asymptotic regime corresponding to p -2 > 0, small, etc.

APPENDIX A. NONLINEAR CARRÉ DU CHAMP METHOD ON THE SPHERE

For completeness, we give a summary of the carré du champ method in integral form applied to the sphere with entropy methods and a nonlinear diffusion equation of fast diffusion or porous medium type. More details can be found in a series of earlier papers on manifolds (see [START_REF] Demange | Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature[END_REF][START_REF] Dolbeault | Nonlinear flows and rigidity results on compact manifolds[END_REF], [START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF]Section 5.2] or [START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF]) with similar computations done in the perspective of uniqueness for positive solutions of elliptic equations (rigidity) that can be found in [START_REF] Gidas | Global and local behavior of positive solutions of nonlinear elliptic equations[END_REF][START_REF] Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF]. In the specific case of the sphere, it is possible to reduce the problem to a computation based on the ultraspherical operator (see [START_REF] Dolbeault | Sharp interpolation inequalities on the sphere: New methods and consequences[END_REF][START_REF]Improved interpolation inequalities on the sphere[END_REF][START_REF]Interpolation inequalities on the sphere: linear vs. nonlinear flows (inégalités d'interpolation sur la sphère : flots non-linéaires vs. flots linéaires)[END_REF][START_REF]Improved interpolation inequalities and stability[END_REF][START_REF] Dolbeault | Parabolic methods for ultraspherical interpolation inequalities, Discrete and Continuous Dynamical Systems[END_REF]). The method in the d = 1 case is slightly different (see for instance [START_REF]Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF]Section 5.3]) and results do not require the use of nonlinear diffusion equations as it is known for a long time from [START_REF] Dolbeault | Diffusions hypercontractives[END_REF][START_REF] Émery | Inégalités de Sobolev pour un semi-groupe symétrique[END_REF], so this will not be discussed here. There are exceptional values of the exponent (p = 6 if d = 3, or p = 3 if d = 4) which require a specific treatment that will not be detailed here, but the result can easily be recovered by considering the limit as p → 6 if d = 3 and p → 3 if d = 4 in the inequalities: see for instance [START_REF]Interpolation inequalities on the sphere: linear vs. nonlinear flows (inégalités d'interpolation sur la sphère : flots non-linéaires vs. flots linéaires)[END_REF]. Here we adopt the presentation of [START_REF] Dolbeault | Nonlinear flows and rigidity results on compact manifolds[END_REF] and go along the lines of the proof of [START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF]Lemma 4.3]. We adopt simplified notations and write Id instead of using the standard metric, or simply write the Laplace-Beltrami operator as -∆. We split the method in two steps, with a first algebraic part closely related with the computations of [START_REF] Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF] and entropy methods applied to a parabolic equation in order to establish the inequalities. Because of the regularizing effects of the parabolic equation, there is no regularity issue in considering positive solutions and solutions will be considered as smooth without further notice. An approximation scheme has of course to be done for initial data in H 1 (S d , d σ): see for instance [START_REF] Dolbeault | Parabolic methods for ultraspherical interpolation inequalities, Discrete and Continuous Dynamical Systems[END_REF] for details in the case of the ultraspherical operator. We also consider the following trace free tensor

Mu := ∇u ⊗ ∇u u - 1 d -1 |∇u| 2 u Id ,
where

∇u ⊗ ∇u := ∂ i u ∂ j u i j and ∇u ⊗ ∇u 2 = |∇u| 4 = i j ∂ i u 2 ∂ j u 2 .
Using L : Id = M : Id = 0 and

Lu 2 = Hu 2 - 1 d (∆u) 2 , Mu 2 = ∇u ⊗ ∇u u 2 - 1 d |∇u| 4 u 2 = d -1 d |∇u| 4 u 2 ,
we deduce from

S d ∆u |∇u| 2 u d σ = S d |∇u| 4 u 2 d σ -2 S d Hu : ∇u ⊗ ∇u u d σ = d -1 d -2 S d Mu 2 d σ -2 S d Lu : ∇u ⊗ ∇u u d σ - 2 d S d ∆u |∇u| 2 u d σ that S d ∆u |∇u| 2 u d σ = d d + 2 d d -1 S d Mu 2 d σ -2 S d Lu : ∇u ⊗ ∇u u d σ .
This provides us with a first identity,

S d ∆u |∇u| 2 u d σ = d d + 2 d d -1 S d Mu 2 d σ -2 S d Lu : Mu d σ . ( 18
)
The Bochner-Lichnerowicz-Weitzenböck formula on S d takes the simple form where the last term, i.e., Ric(∇u, ∇u) = (d -1) |∇u| 2 , accounts for the Ricci curvature tensor contracted with ∇u ⊗ ∇u. An integration of this formula on S d shows a second identity,

S d (∆u) 2 d σ = d d -1 S d Lu 2 d σ + d S d |∇u| 2 d σ . ( 19
)
Hence for some parameters β and κ to be fixed later, the functional

The function ρ(t , x) = w(m t , x) β p ∀ (t , x) ∈ R + × S d with κ and β as in [START_REF]Symmetry and symmetry breaking: rigidity and flows in elliptic PDEs[END_REF] solves the standard fast-diffusion equation [START_REF] Demange | Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature[END_REF] Proof. These results are easily proved using a few integrations by parts and Lemma 13.

A standard consequence is, for instance, the fact that The limit as t → +∞ can be identified by analyzing k[w] = 0 and proving that this means that w is then a constant if p < 2 * . In the case p = 2 * , identifying w as a constant can be done only by considering w as a limit as t → +∞ of a solution of [START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF].

Proposition 4 .

 4 Let d ≥ 1, p ∈ (2, 2 * ) and Λ > 0, given. If for some admissible θ and either k = 1 or k = 2, U is a non-trivial minimizer for µ k (θ, Λ), then for an appropriate choice of κ > 0, the function u = κU solves[START_REF]Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations[END_REF] for some λ > 0. The two parameters κ and λ are explicitly given in terms of µ k (θ, Λ), U L 2 (S d ) and U L p (S d ) .

  ) such that u L p (S d ) = 1 and let a := ∇u 2 L 2 (S d ) and b := u 2 L 2 (S d )

A. 1 .

 1 A purely algebraic computation. Let Hu denote the Hessian of a C 2 function u on S d and define the trace free Hessian by Lu := Hu -1 d -1 (∆u) Id .

1 2 ∆

 2 (|∇u| 2 ) = Hu 2 + ∇(∆u) • ∇u + (d -1) |∇u| 2

1 a 2 ,Lemma 13 .A. 2 .

 12132 2 d σ + (κ + β -1) Lu 2 -2 bLu : Mu + c Mu 2 and a = 1 , b = (κ + β -1)d -1 d + c = (κ + β -1) d d + 2 + κ (β -1) .(21)Let us consider the choice of the parameters κ = β (p -2) + 1 and m = 1 + 2 by definition[START_REF] Esteban | A scenario for symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF], we havem ± (d , p) := 1 (d + 2) p d p + 2 ± d (p -1) 2 d -(d -2) p . Let d ≥ 2, p ≥ 1 if d = 2, and p ∈ [1, 2 * ] if d ≥ 3. If m -(d , p) ≤ m ≤ m + (d , p),then q is a positive quadratic form if κ and β are given in terms m and p by[START_REF]Symmetry and symmetry breaking: rigidity and flows in elliptic PDEs[END_REF]. As a consequence, we havek[u] ≥ d S d |∇u| 2 d σ ∀ u ∈ C 2 (S d ) .Proof. It follows from the reduced discriminant condition b 2 -ac < 0 by tedious but elementary computations. In the cases p = 6 if d = 3 and p = 3 if d = 4, computations have to be done directly with m, without using the parameter β (see below).In dimension d = 1, we have a similar result with m = β = 1 using the Poincaré inequality on the circle S 1 . Entropy methods and nonlinear flows. Let us introduce the parabolic evolution setting corresponding to the nonlinear diffusion equation ∂w ∂t = w 2-2β ∆w + κ |∇w| 2 w .

  λ w β ≤ -2 β 2 (d -λ) S d |∇w| 2 d σ if w solves[START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF] and F λ is defined by[START_REF] Bonheure | Some systems arising from Lieb-Thirring kinetic energy inequalities[END_REF]. Hence if λ ≤ d and (23) is supplemented with the initial datum w(0, •) = u 1/β , thenF λ [u] ≥ F λ w β (t , •) = lim s→+∞ F λ w β (s, •) = 0 ∀ t ≥ 0 .

  Let d ≥ 2, p ≥ 1 if d = 2, m > 0 and p ∈ [1, 2 * ] if d ≥ 3.With κ and β as in[START_REF]Symmetry and symmetry breaking: rigidity and flows in elliptic PDEs[END_REF], any smooth positive solution of[START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF] is such thatd d t S d w 2 β d σ = 2 β 2 (p -2) ∇w β 2 d σ = -2 β 2 k[w].

					. From this obser-
	vation, it is clear that
	d d t S d	w β p d σ =	d d t S d	ρ d σ = 0 .
	Lemma 14. S d	|∇w| 2 d σ
	and, with k defined by (20),
	d
	d t S d

As a consequence, if m

-(d , p) ≤ m ≤ m + (d , p), then we have d d t S d ∇w β 2 d σ ≤ -2 β 2 d S d |∇w| 2 d σ

© 2022 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

Acknowledgment: This work has been partially supported by the Project EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR).


Marion Jasnin

Jordan Hervy

Stéphanie Balor

Anaïs Bouissou

Amsha Proag

Raphaël Voituriez

Jonathan Schneider

Thomas Mangeat

Isabelle Maridonneau-Parini

Wolfgang Baumeister

Javad Najafi

Serge Dmitrieff

Nicolas Minc
email: nicolas.minc@ijm.fr

Elasticity of podosome actin networks produces nanonewton protrusive forces

Keywords: Cytoplasm, Hydrodynamics, Cell mechanics, viscoelasticity, confinement, cell cortex

INTRODUCTION

The cytoplasm is a complex heterogenous medium crowded with macromolecules and entangled cytoskeleton networks [START_REF] Luby-Phelps | Cytoarchitecture and physical properties of cytoplasm: volume, viscosity, diffusion, intracellular surface area[END_REF]. These define rheological properties that may influence molecular processes ranging from molecular diffusion to reaction kinetics and protein folding [START_REF] Stagg | Macromolecular Crowding Tunes Folding Landscape of Parallel α/β Protein, Apoflavodoxin[END_REF][START_REF] Molines | Physical properties of the cytoplasm modulate the rates of microtubule polymerization and depolymerization[END_REF][START_REF] Regner | Anomalous Diffusion of Single Particles in Cytoplasm[END_REF][START_REF] Konopka | Crowding and confinement effects on protein diffusion in vivo[END_REF]. However, the cytoplasm also hosts the motion of much larger elements closer to the cellular scale, for which the impact of cytoplasm properties remain much less understood. For instance, during critical events such as fertilization, cell polarization or asymmetric divisions, cells actively displace large nuclei, microtubule asters or mitotic spindles across their cytoplasm [START_REF] Tanimoto | Shape-motion relationships of centering microtubule asters[END_REF][START_REF] Reinsch | Mechanisms of nuclear positioning[END_REF]. These organelles are moved by forces generated from active polar cytoskeletal networks such as actomyosin bundles or microtubules and associated motors, that need to overcome mechanical resistance from the cytoplasm medium [START_REF] Xie | Cytoskeleton Force Exertion in Bulk Cytoplasm[END_REF]. Therefore, addressing the nature and magnitude of frictional interactions of large components with the cytoplasm remains an important endeavor for cellular spatial organization.

Many past studies have measured rheological properties of bulk cytoplasm, using either extracts in vitro, or by actuating endogenous or foreign probes in living cells. These have shown that cytoplasm response will depend on time-scale, force amplitude or component size [START_REF] Hurst | Intracellular softening and increased viscoelastic fluidity during division[END_REF][START_REF] Moeendarbary | The cytoplasm of living cells behaves as a poroelastic material[END_REF][START_REF] Valentine | Mechanical properties of Xenopus egg cytoplasmic extracts[END_REF][START_REF] Hiramoto | Rheological properties of echinoderm eggs during cell division[END_REF][START_REF] Tseng | Micromechanical Mapping of Live Cells by Multiple-Particle-Tracking Microrheology[END_REF]. The question of object size has notably raised important notions for cytoplasm mechanics, as for components smaller or larger than a typical mesh size, the cytoplasm may exhibit different rheological signatures ranging from fluid, to viscoelastic, poroelastic, or even glassy materials [START_REF] Hu | Size-and speed-dependent mechanical behavior in living mammalian cytoplasm[END_REF][START_REF] Parry | The Bacterial Cytoplasm Has Glass-like Properties and Is Fluidized by Metabolic Activity[END_REF][START_REF] Śmigiel | Protein diffusion in Escherichia coli cytoplasm scales with the mass of the complexes and is location dependent[END_REF][START_REF] Niwayama | Hydrodynamic property of the cytoplasm is sufficient to mediate cytoplasmic streaming in the Caenorhabiditis elegans embryo[END_REF][START_REF] Klughammer | Cytoplasmic flows in starfish oocytes are fully determined by cortical contractions[END_REF][START_REF] Berret | Local viscoelasticity of living cells measured by rotational magnetic spectroscopy[END_REF]. However, these studies were restricted to regimes of relatively small objects typically below the micron size, as well as low forces and displacement amplitudes. As objects reach closer to the cell size and move across longer distances, they are predicted to drag and shear large portions of the cytoplasm fluid [START_REF] Xie | Contribution of cytoplasm viscoelastic properties to mitotic spindle positioning[END_REF][START_REF] Nazockdast | Cytoplasmic flows as signatures for the mechanics of mitotic positioning[END_REF][START_REF] Wu | Laser Ablation and Fluid Flows Show a Single Force Mechanism Governs Spindle Positioning[END_REF]. In such regime, cytoplasm resistance may be influenced by boundary conditions at the static cell surface, through so-called wall effects. These have been well documented in fluid mechanics, and emerge from hydrodynamic interactions that couple a moving object with a static wall. They are predicted to enhance frictional coefficients by a significant amount if the object contained by the walls approaches the size of the container [START_REF] Happel | Low Reynolds number hydrodynamics: with special applications to particulate media[END_REF]. To date, however, direct evidence that these effects are relevant to living cells is still lacking, in part given the technical difficulty of applying calibrated forces onto large components in vivo.

Here, we employed calibrated in vivo magnetic tweezers to move passive components of sizes ranging from ~1% to ~50% of the cell diameter across the cytoplasm of living cells. By tracking resultant cytoplasm shear flows and using finite element hydrodynamic simulations, we demonstrate how object size disproportionally restrain its mobility as a result of hydrodynamic interactions with the cell surface. We also find that these interactions can yield to position-dependent cytoplasm resistance with objects initially closer to the cell surface becoming harder to pull or push. This work suggests that large organelles may be coupled to the cell surface without any direct cytoskeletal connections, and highlights the underappreciated impact of confinement by cell boundaries to organelle motion.

RESULTS

Probing bulk cytoplasm rheology at multiple length scales.

We sought to probe the rheology of bulk cytoplasm in response to the typical translational motion of relatively large organelles, such as e.g. micrometric acidic organelles or even larger nuclei or mitotic spindles. These objects are commonly moved by molecular motors with directional speeds ranging from fractions up to tens of µm/s, across distances in the scale of few to few tens of micron [START_REF] Rogers | Membrane trafficking, organelle transport, and the cytoskeleton[END_REF]. We used sea urchin unfertilized eggs as model cell types.

These are large cells, with a diameter of 95 µm, amenable to quantitative injection, that are arrested in a G0-or G1-like state of the cell cycle before fertilization [START_REF] Voronina | Cyclin B synthesis is required for sea urchin oocyte maturation[END_REF]. Importantly, in contrast to fertilized eggs, unfertilized eggs do not feature any large-scale cytoskeletal organization, with F-actin and microtubule filaments distributed throughout the cytoplasm, and no detectable thick actin cortex (Fig. S1A-B) [START_REF] Heil-Chapdelaine | Relative changes in F-actin during the first cell cycle: Evidence for two distinct pools of F-actin in the sea urchin egg[END_REF]. In addition, Particle Imaging Velocimetry (PIV) of relatively large granules, did not reveal any persistent cytoplasm flows, suggesting that the cytoplasm material may be considered at rest in these cells (Fig. S1C).

To compute the viscoelastic response of the cytoplasm at different length-scales, we injected single magnetic beads of 1 µm in diameter, as well as bead aggregates of variant sizes from few up to ten microns [START_REF] Hiramoto | Mechanical properties of the protoplasm of the sea urchin egg. I. Unfertilized egg[END_REF][START_REF] Tanimoto | Physical Forces Determining the Persistency and Centering Precision of Microtubule Asters[END_REF]. To reach component sizes closer to cell size, we injected a suspension of soya oil mixed with hydrophobic ~1 µm magnetic beads inside eggs. Upon injection, this suspension formed large magnetized oil droplets that ranged in size from 18 to 45 µm (corresponding to ~20-48% of the cell diameter) [START_REF] Xie | Contribution of cytoplasm viscoelastic properties to mitotic spindle positioning[END_REF]. We applied calibrated forces, by approaching a magnet tip close to the cell surface. This caused components to translate through the cytoplasm along the magnetic gradient (Fig. 1A) [START_REF] Tanimoto | Physical Forces Determining the Persistency and Centering Precision of Microtubule Asters[END_REF]. Magnetic forces and their dependence on bead type, magnet-bead distance and aggregate size were calibrated in vitro using test viscous fluids (Figs. S1D-F).

Interestingly, all these objects exhibited similar creep responses to an applied force. First, objects moved at constant speed, indicating a viscous behavior. Then the displacement-time curve inflected reflecting elastic responses. At longer time-scales the behavior was linear again indicating a fluidization of elastic elements [START_REF] Xie | Contribution of cytoplasm viscoelastic properties to mitotic spindle positioning[END_REF][START_REF] Hiramoto | Mechanical properties of the protoplasm of the sea urchin egg. I. Unfertilized egg[END_REF][START_REF] Crick | The physical properties of cytoplasm: A study by means of the magnetic particle method Part I[END_REF] (Fig. 1B). Accordingly, when the force was released, objects recoiled back towards their initial positions with partial recoils that reflected a dissipation of stored elastic energy. However, during recoils, small beads were more prone to random forces and space exploration, exhibiting a much less directional recoiling behavior (Fig. 1C and Movie S1).

To quantify cytoplasm viscoelastic parameters, we fitted both creep and recovery curves with a 1D Jeffreys' model, consisting of a Kelvin-Voigt viscoelastic element in series with a dashpot which represents the dissipation of the material in the long asymptotic time [START_REF] Hiramoto | Mechanical properties of the protoplasm of the sea urchin egg. I. Unfertilized egg[END_REF][START_REF] Barnes | An introduction to rheology[END_REF] (Fig. 1B inset). This allowed to compute a short-term viscous drag of the translating object, γ 1 , a restoring stiffness k, and thus the viscoelastic relaxation time-scale of cytoplasm material τ 1 = γ 1 /k. Mean values of viscous drags increased with object size, from 11 ± 6 pN.s/µm (mean ± std) for small 1 µm beads, to 33 ± 49 pN.s/µm for larger aggregates and up to 660 ± 312 pN.s/µm for very large oil droplets (Fig. 1D). Similarly, the restoring stiffness increased from 4 ± 3 pN/µm, to 12 ± 13 pN/µm and 191 ± 112 pN/µm, respectively (Fig. 1E).

Importantly, however, the relaxation time-scales was mostly independent of object size, and was around ~4 s during force loading and ~7 s during relaxation (Figs. 1F-G). These results suggest similar general rheological responses of the cytoplasm to translating objects ranging in size from few up to 50% of cell size.

To confirm that viscoelastic responses and parameters reflected bulk cytoplasm properties, we next injected magnetic droplets and rinsed eggs in diluted (50%) or concentrated (120%) artificial sea water (ASW) to vary extracellular osmolarity and, respectfully decrease or increase macromolecular crowding [START_REF] Molines | Physical properties of the cytoplasm modulate the rates of microtubule polymerization and depolymerization[END_REF][START_REF] Giese | Resistance of the Purple Sea Urchin to Osmotic Stress[END_REF][START_REF] Miermont | Severe osmotic compression triggers a slowdown of intracellular signaling, which can be explained by molecular crowding[END_REF] (Figs. 2A-B and Movie S2). Both creep and relaxation responses had similar shapes as in controls, but viscous drag and restoring stiffness values were markedly different (Fig. 2C). In hypoosmotic conditions (50% ASW), they were reduced by ~4.2X and 3.8X, respectively, and increased by 1.6 and 2.8X in hyperosmotic conditions (120% ASW). Accordingly, characteristic viscoelastic time-scales during the pulling phase were similar in control and hypoosmotic conditions (4.1 ± 2.8 s and 3.9 ± 2.4 s), but slightly reduced to 2.2 ± 1.5 s in hyperosmotic conditions. These results support that the measured viscoelastic resistance to these large translating objects directly relate to cytoplasm crowding and mechanical properties.

Cytoplasm viscoelastic shear flows associated to the motion of large objects.

To understand how the motion of different sized objects impact cytoplasm organization, we mapped cytoplasm flows, by tracking granules visible in bright field with PIV. The movement of small beads, did not create notable large-scale flows, reflecting the high viscosity of the cytoplasm that screens perturbations at long length-scales (Fig. S2A). In contrast, the motion of very large droplets, yielded notable and reproducible cytoplasm flow fields. The maximum cytoplasm flow speed was on the order of droplet speed at the front and back of the droplet along the pulling direction and dropped rapidly away from the droplet. Moreover, two vortices were generated at the upper and lower parts of the pulling axis, often asymmetric in size when the droplet and force axis were, for instance, off-center (Figs. 3B-D and Movie S3). This suggests that large translating objects create significant shear in the cytoplasm with magnitudes and organization that may depend on object size and its distance to cell boundaries.

To test if experimental flows and viscoelastic responses could correspond to a simple interaction of the moving object with the surrounding cytoplasm fluid, we set up finiteelement hydrodynamic simulations. We represented the oil droplet as a solid spherical object and displaced it with a constant force inside a confined Jeffreys' viscoelastic medium (implemented as an equivalent Oldroyd-B model) and then released the force. As inputs of this model, we used the two viscosities and the second viscoelastic time-scale measured from experiments. The resulting simulations accounted for both flow fields organization and speed magnitude at various steps of the pulling and relaxation phases observed in experiments (Fig. 3F and Movie S3). We validated the simulations by computing the consistency between input and output values of material properties, which we measured from the simulated timedisplacement curves following a 1D Jeffreys' model, as in experiments. We found that the quantitative creep and recovery curves of the experiments were consistent with the simulation output, asserting that the 1D Jeffreys' model was sufficient to describe the 3D viscoelastic response of the cytoplasm (Figs. 3E andS2B-F).

Viscoelastic drag and restoring stiffness depend on confinement.

Flow analysis suggested that the displacement of large objects generate shear flows from hydrodynamic interactions between the object and cell boundaries. We sought to test if these interactions could affect object mobility and notably enhance drag and/or restoring stiffness.

For a sphere located at the exact center of a compartment filled with a Newtonian fluid, the viscous drag of the object is predicted to follow a modified Stokes' law γ 1 (λ) = 6πηrC(λ)

where η is the fluid viscosity, r, the object radius and C a wall correction factor: ܥሺߣሻ ൌ ସ൫ଵିఒ ఱ ൯ ସିଽఒାଵఒ య ିଽఒ ఱ ାସఒ ల with λ = r/R the ratio of the sphere radius to that of the container [START_REF] Happel | Low Reynolds number hydrodynamics: with special applications to particulate media[END_REF]. Therefore, such correction amounts to an increase in the effective viscosity felt by the object: η*(λ) = ηC(λ). The correction factor asymptotically approaches 1 for small and weakly bounded object, is ~2.5 for objects ~30% of container size, reaches up to ~7 when objects are ~50% of container size, and diverges to infinity when object size approaches that of the container (Fig. 4A). By running 3D simulations, we confirmed the validity of this formula in the context of our particular pulling assay, and also predicted that confinement should have a similar impact on both viscous and elastic responses of the cytoplasm.

To test the relevance of confinement to experimental results, we next plotted measured viscous drags as a function of λ for the different sized objects (Fig. 4A). We found that experimental drags were in good agreement with the results of both analytical formula and 3D hydrodynamic simulations in the large intervals of confinement ratio. However, we noted some deviations from experiments for beads aggregates, in which drags were under theoretical curves. We interpret this deviation as a result of a plausibly more porous nature of aggregates and their non-perfect circular shapes. Importantly, for large oil droplets, the deviation from linear Stokes' relationship became pronounced as the confinement ratio λ exceeded ~0.1-0.2, with correction factors that ranged from ~1 to ~9. In addition, as predicted by simulations, we found that the effect was similar for restoring stiffness, directly demonstrating that the proximity of boundaries can enhance both viscous and elastic cytoplasm resistance (Fig. 4A).

To better establish the relevance of this effect in dose-dependence, we next performed 3D simulations in the size range of spherical oil droplets, and compared predicted drags and restoring stiffness to individual experimental results. For this, we inputted average experimental values of cytoplasm viscosities and second viscoelastic time-scale, and varied droplet and egg size and initial position (see below) according to experiments. The agreement between experiments and simulations was in general very good, suggesting that experiments are close to the theoretical limit (Figs. 4B-C). Together, these results strongly suggest that hydrodynamic interactions between a large moving object and the cell surface, effectively enhances cytoplasm viscoelasticity at this scale.

Slippage on boundaries reduces the effect of confinement on organelle mobility.

As organelles and cell surfaces may come with different properties like rugosity or hydrophobicity, that may influence how the cytoplasm fluid interacts with these surfaces, we sought to evaluate the impact of stick or slip conditions on the confinement effects. In experiments, it was technically challenging to modulate slippage. Therefore, we performed a series of simulations to test how boundary conditions impact flow fields and size dependency.

We considered three different scenarios: a no-slip boundary condition over the oil droplet and cortex surface, slippage only over the oil surface, and finally slippage over both oil and cortex surfaces. Interestingly, the general flow patterns were independent of boundary conditions chosen (Figs. 5A-C). However, the impact of dissipation for a sphere of a given size was reduced when the fluid slips on surfaces, yielding to a net increase in average fluid speed (Fig. 5D). Under the same applied force, the fluid speed drops as object size increases, but boundary conditions affect the speed in the same manner for spheres of different sizes (Fig. 5D). Accordingly, both viscous drag and restoring stiffness increased in a non-linear manner as a function of the confinement ratio λ under all boundary conditions, but the effect was more pronounced when the fluid could adhere to all surfaces (Figs. 5E-F). We conclude that the impact of confinement on organelle mobility may still be significant independently of the type of boundary conditions, but more pronounced for non-slip conditions.

Heterogeneity and anisotropy of drags and restoring stiffness.

As organelles may adopt different locations within the cytoplasm, being e.g. initially closer to the cell surface, we sought to test if hydrodynamic interactions affect mechanical resistance of the cytoplasm depending on object's initial position. We first performed a range of simulations. We considered a sphere with a confinement ratio of λ = 0.2 in a cell filled with Jeffreys' fluid, and placed it at different positions along the force axis. We found that both drags and restoring stiffness increased as the object was closer to the cell surface, reaching an enhancement of ~2.5X at an offset distance of ~30% of cell size closer to the surface.

Simulating a much smaller sphere of λ = 0.04 as control, showed that this local effect is only relevant to relatively large objects (Figs. 6A-B). Interestingly, because of the linearity of the viscoelastic-model used, this position-dependent enhancement was perfectly symmetric, whether the object was pushed against or pulled away from the surface. Accordingly, analysis of fluid flows revealed symmetric flow maps with respect to the boundary, indicating a similar interaction of the fluid with the container surface in both cases (Fig. 6C). Finally, these symmetric enhanced effects were also observed for restoring stiffnesses (Fig. 6D). In a second sets of simulations, we also changed object position along an axis orthogonal to the force axis. In this situation, boundary conditions also enhanced cytoplasm resistance, but the effect was less pronounced than when objects moved orthogonal to the cell surface (Fig. 6A).

Therefore, hydrodynamic interactions with cell boundaries may yield to anisotropic cytoplasm resistance facilitating the motion of large objects parallel as compared to orthogonal to the cell surface.

To test these predictions in real cells, we pulled oil droplets with different initial positions. To minimize variability arising from different droplets sizes, we performed successive pulls of the same droplets, spaced by sufficient time for the material to relax, and varied their initial position with magnetic tweezers. The average experimental values exhibited a minimum at the cell center, and increased in a symmetric manner as the droplet was placed closer to the cell surface, with a good alignment on simulations data (Fig. 6D). Interestingly, the positiondependent enhancement, for both drag and restoring stiffness was symmetric, as predicted from the linear model, suggesting that both cytoplasm compression and extension can result in viscous and elastic stresses. Together, these simulations and experiments demonstrate that cytoplasm viscoelasticity felt by large objects in a cell may be position-dependent and anisotropic.

DISCUSSION

Here we used magnetic tweezers to displace large passive components of different sizes in the cytoplasm of living cells with unidirectional calibrated forces. Our results suggest that objects floating in bulk cytoplasm become coupled to the cell surface through the fluid, with no need for any direct cytoskeletal connections. This hydrodynamic coupling can occur over distances of up to tens of microns and enhance the viscoelastic resistance of the cytoplasm, thereby reducing organelle mobility. The strength of the coupling increases as the distance between the object and the surface reduces, so that larger components or components closer to the cell surface, become disproportionally harder to displace. Size-and positiondependency had previously been reported for the diffusion of smaller components in different cells. They were attributed to properties such as the hierarchy of cytoplasm pore sizes [START_REF] Luby-Phelps | Probing the structure of cytoplasm[END_REF], glass-like transitions of bulk material [START_REF] Parry | The Bacterial Cytoplasm Has Glass-like Properties and Is Fluidized by Metabolic Activity[END_REF], or to heterogeneities across cells [START_REF] Śmigiel | Protein diffusion in Escherichia coli cytoplasm scales with the mass of the complexes and is location dependent[END_REF]. Our data for large objects, support a different physical origin, in which both size and position dependency can emerge solely from hydrodynamic interactions with the surface. As such, previously proposed alterations in bulk material properties or gradients in these properties, although not required here, could add up to the impact of hydrodynamic interactions in different cell types.

Our data suggest that the impact of hydrodynamic coupling becomes pronounced for objects over ~20-30% of cell size, which is the typical size ratio for organelles such as nuclei, mitotic spindles or microtubule asters. Accordingly, in vivo direct or indirect force measurements reported spindle, nuclei and aster drags that largely exceeded Stokes drags calculated from the size of these organelles and cytoplasm viscosity [START_REF] Xie | Contribution of cytoplasm viscoelastic properties to mitotic spindle positioning[END_REF][START_REF] Tanimoto | Physical Forces Determining the Persistency and Centering Precision of Microtubule Asters[END_REF][START_REF] Simone | Uncovering the balance of forces driving microtubule aster migration in C. elegans zygotes[END_REF]. Other relevant parameters that could impact cytoplasm resistance include organelle shape and porosity especially when considering fibrous-like structures such as asters and spindles [START_REF] Nazockdast | Cytoplasmic flows as signatures for the mechanics of mitotic positioning[END_REF][START_REF] Nazockdast | A fast platform for simulating semiflexible fiber suspensions applied to cell mechanics[END_REF]. Furthermore, as demonstrated here, the relative proximity of organelles to boundaries, may also greatly impact its mobility. As a consequence, we expect cell geometry to also influence organelle mobility. For instance, a nucleus moving in a tube-like cell such as fission yeast or a columnar epithelial cell [START_REF] Saleh | Length-limitation of astral microtubules orients cell divisions in intestinal crypts[END_REF][START_REF] Tran | A mechanism for nuclear positioning in fission yeast based on microtubule pushing[END_REF], is predicted to face very large resistance from the cytoplasm, given the little space left for the fluid to flow around the organelle [START_REF] Happel | Low Reynolds number hydrodynamics: with special applications to particulate media[END_REF]. Although the role of the cytoplasm has been generally omitted in standard models for organelle positioning [START_REF] Reinsch | Mechanisms of nuclear positioning[END_REF][START_REF] Lele | Mechanical principles of nuclear shaping and positioning[END_REF], we anticipate they could impact our current appreciation of the mechanics of nuclear or spindle positioning, as well as that of shape changes.

Finally, our study also highlights large shear flows that form as a direct result of organelle motion under force, and cause the cytoplasm to recirculate over the scale of the whole cell [START_REF] Xie | Contribution of cytoplasm viscoelastic properties to mitotic spindle positioning[END_REF]. Cytoplasm flows can organize processes ranging from cortical polarity [START_REF] Mittasch | Non-invasive perturbations of intracellular flow reveal physical principles of cell organization[END_REF], RNA localization [START_REF] Forrest | Live Imaging of Endogenous RNA Reveals a Diffusion and Entrapment Mechanism for nanos mRNA Localization in Drosophila[END_REF], to nuclear positioning and internal organization [START_REF] Jord | Cytoplasmic forces functionally reorganize nuclear condensates in oocytes[END_REF][START_REF] Deneke | Self-Organized Nuclear Positioning Synchronizes the Cell Cycle in Drosophila Embryos[END_REF]. They typically emerge from cortical contractile acto-myosin flows that generate surface stresses that propagate through the cytoplasm [START_REF] Niwayama | Hydrodynamic property of the cytoplasm is sufficient to mediate cytoplasmic streaming in the Caenorhabiditis elegans embryo[END_REF][START_REF] Klughammer | Cytoplasmic flows in starfish oocytes are fully determined by cortical contractions[END_REF][START_REF] Deneke | Self-Organized Nuclear Positioning Synchronizes the Cell Cycle in Drosophila Embryos[END_REF][START_REF] Yi | Dynamic maintenance of asymmetric meiotic spindle position through Arp2/3-complex-driven cytoplasmic streaming in mouse oocytes[END_REF], or from the activity of bulk cytoskeleton and associated motors [START_REF] Serbus | Dynein and the actin cytoskeleton control kinesin-driven cytoplasmic streaming in Drosophila oocytes[END_REF][START_REF] Shamipour | Bulk Actin Dynamics Drive Phase Segregation in Zebrafish Oocytes[END_REF]. Based on our observations, we propose that the natural motion or rotation of large organelles during e.g. asymmetric division or nuclear translocation could create large scale flows, that may impact the polarization of both cytoplasmic and cortical elements. Further work on how cytoplasm and cortical material properties and forces are integrated to control organelle mobility, will enlarge our understanding of cellular organization.

ACKNOWLEDGMENT

We thank all members of the Minc lab for discussion and technical help. J.

Material and methods

Sea urchin gametes

Purple sea urchins (Paracentrotus lividus) were obtained from Roscoff Marine station (France) and maintained at 16 °C in aquariums of artificial seawater (ASW; Reef Crystals, Instant Ocean). Gametes were collected by intracoelomic injection of 0.5 M KCl. Eggs were rinsed twice with ASW, kept at 16 °C, and used on the day of collection.

Injection

Unfertilized eggs were placed on protamine-coated 50 mm glass-bottom dishes (MatTek Corporation) after removing the jelly coat through an 80-μm Nitex mesh (Genesee Scientific). The bead suspensions were injected using a micro-injection system (FemtoJet 4;

Eppendorf) and a micro-manipulator (Injectman 4; Eppendorf). Injection pipettes were prepared from siliconized (Sigmacote) borosilicate glass capillaries (1 mm diameter). Glass capillaries were pulled using a needle puller (P-1000; Sutter Instrument) and ground with a 40° angle on a diamond grinder (EG-40; Narishige) to obtain a 10 μm aperture. Injection pipettes were back-loaded with ~2 μl of bead suspension before each experiment, and were not re-used.

Immunostaining

Immunostaining was performed using procedures described previously [START_REF] Foe | Stable and dynamic microtubules coordinately shape the myosin activation zone during cytokinetic furrow formation[END_REF]. Samples were fixed for 70 minutes in 100 mM Hepes, pH 6.9, 50 mM EGTA, 10 mM MgSO4, 2% formaldehyde, 0.2% glutaraldehyde, 0.2% Triton X-100, and 400 mM glucose. To reduce autofluorescence, eggs were then rinsed in PBS and placed in 0.1% NaBH4 in PBS freshly prepared, for 30 min. Eggs were rinsed with PBS and PBT (PBS + 0.1% TritonX) and blocked in PBT supplemented with 5% goat serum and 0.1% bovine serum albumin (BSA) for 30 minutes. Samples were rinsed with PBT before adding primary antibodies. For microtubule staining, cells were incubated for 48 h at 4 °C with a mouse anti-α-tubulin (DM1A; Sigma-Aldrich) primary antibody at 1:5,000 in PBT, rinsed 3 times in PBT and incubated for 4 hours at room temperature with anti-mouse secondary antibody coupled to Dylight 488 (Thermo Fisher Scientific) at 1:1,000 in PBT for 4-5 hours. To stain F-actin, samples were also incubated together with secondary antibodies in a solution of Rhodamine Phalloidin at 4 U/ml in PBT. Eggs were washed three times in PBT then twice in PBS, transferred in 50% glycerol in PBS, and finally transferred into mounting medium (90% glycerol and 0.5% N-propyl gallate in PBS).

Hypoosmotic and hyperosmotic shocks

To elevate cytoplasm crowding with hyperosmotic shocks, ASW was prepared to 120% of its normal content by adjusting the amount of DI water to salts mixtures. The eggs laid down on protamine-coated dishes inside the hyperosmotic water and then injected. Higher concentrations of 150% were tested, but droplets could not be pulled in such cytoplasm. To reduce cytoplasm density with hypoosmotic shocks, the eggs were first injected and then ASW was diluted by the same volume of DI water equal to ASW inside the dishes, to reach an ASW at 50% of its normal content.

Magnet force calibration

Magnetic forces were calibrated in vitro following procedures described previously [START_REF] Xie | Contribution of cytoplasm viscoelastic properties to mitotic spindle positioning[END_REF][START_REF] Tanimoto | Physical Forces Determining the Persistency and Centering Precision of Microtubule Asters[END_REF][START_REF] Salle | Asymmetric division through a reduction of microtubule centering forces[END_REF]. The magnetic force field created by the magnet tip was first characterized by pulling super-paramagnetic 1 µm Dynabeads (MyOne Streptavidin C1; Thermofisher) in a viscous test fluid (80% glycerol; viscosity 8.0×10 -2 Pa.s at 22 °C) along the principal axis of the magnet tip. Small motion of the fluid was subtracted by tracking 0.5 μm non-magnetic fluorescent tracers (Molecular probes; Invitrogen) in the same suspension. The speed of magnetic beads V was computed as a function of the distance to the magnet, to obtain and trace the decay function of the magnetic force, which was fitted using a double exponential function (Fig. S1E).

To calculate the dependence of the force on aggregate size, bead aggregates from the same beads as those used in vivo in sizes ranging from 2 to 7 μm, similar to that observed in oil droplets, were pulled in the same fluid as above. The speed V a was measured and transformed into a force using Stokes' law F = 6πηRV a , where η is the viscosity of the test fluid, R the aggregate effective radius defined as the geometric mean of the longest and shortest axes of the aggregate ܴ ൌ ଵ ଶ ඥܮ ଵ ܮ ଶ . In live cell experiments the size of the aggregates inside the oil droplets was measured at three different positions in the bright field and averaged (Fig. S1D).

The force-size relationship at a fixed distance from the magnet was well represented and fitted by a cubic function (Fig. S1F). These speed-distance and force-size relationships were combined to compute the magnetic forces applied to the beads and aggregates as a function of time, from the size of aggregates and their distance to the magnet tip.

Magnetic force application

Magnetic tweezers were implemented as described previously [START_REF] Xie | Contribution of cytoplasm viscoelastic properties to mitotic spindle positioning[END_REF][START_REF] Tanimoto | Physical Forces Determining the Persistency and Centering Precision of Microtubule Asters[END_REF][START_REF] Salle | Asymmetric division through a reduction of microtubule centering forces[END_REF]. The magnet probe used for force applications in vivo was built from three rod-shaped strong neodymium magnets (diameter 4 mm; height 10 mm; S-04-10-AN; Supermagnet) prolonged by a sharpened steel piece with a tip radius of ~60 μm to create a magnetic gradient. The surface of the steel tip was electro-coated with gold to prevent oxidation. The probe was controlled with a micromanipulator (Injectman 4; Eppendorf) and mounted on an inverted epifluorescent microscope. Large aggregates could not usually cross the interface due to the oil surface tension, aggregate size, and hydrophobic properties of the beads, and were used to pull oil droplets in the cytoplasm. Upon force application, the magnet was quickly retracted when the oil distance to the cell cortex was ~10 μm, which usually caused aggregate detachment from the oil-cytoplasm interface and backward relaxation of the droplet along the initial pulling force axis. The initial position-dependent experiments with the oil droplets were done similarly as for aggregates. Oil droplets were first pulled toward the cortex on one side of the egg.

However, the droplets could not stay very close to the cortex because of the viscoelastic response of the cytoplasm. Then, the dish was rotated and the oil droplet was pulled from the opposite side along the diagonal direction. Droplets were displaced slowly to change the offset and suppress the oil droplet from recoiling toward its initial position [START_REF] Xie | Contribution of cytoplasm viscoelastic properties to mitotic spindle positioning[END_REF]. The droplet was pulled again in the same direction after at least a 5 minutes delay to ensure a complete relaxation of the droplet.

Imaging

Time-lapses of oil droplets, aggregates, and beads moving under magnetic force were recorded on two inverted microscope set-ups equipped with a micromanipulator for magnetic equipped with a Yokogawa CSU-X1FW spinning head, and a Prime BSI camera (Photometrics), using a 60X water-immersion objective (Apo; NA 1.2; Nikon).

Tracking bead and droplet positions

Magnet tip position was recorded in the bright field or fluorescent images through the fluorescent beads attracted onto the magnet. Oil droplets and aggregates time-lapse images were rotated to align their displacement vector antiparallel to the horizontal x-axis meaning that they all move from the right to the left. The position of beads and aggregates were tracked from their fluorescence signal using the TrackMate plugin in Fiji. The trajectories of the beads were rotated to align their displacement from the right to the left. Bright-field images of oil droplets were segmented in Fiji from the contrast at the periphery of the droplet and tracked using the TrackMate plugin. Displacement of the oil droplet was corrected when the egg had moved during the pulling.

Viscoelastic parameter calculation

Oil droplet and bead displacements were fitted with a Jeffreys' model using a custom written code in Matlab (Mathwork) to compute viscoelastic parameters (Figs. 1B-C). For the rising phase, the position was fitted using:

݀ݔሺݐሻ ݂ሺݐሻ ൌ 1 ߢ 1 െ ݁ ି ఊ భ ௧ ൨ ݐ ߛ ଶ , .ݍܧ 1
where dx is the displacement along the x-axis and f is the magnetic force. This re-scaling of the displacement by force allows to compensate for variations in force amplitude during each pull, and implicates that we assume that viscoelastic responses are mostly linear. These fits allowed to compute the restoring stiffness, κ, and the viscoelastic drags ߛ 1 and ߛ 2 on oil droplets and beads, allowing to compute viscoelastic time-scales as

߬ 1,2 = ߛ 1,2/ κ.
Trajectories of beads during the relaxation phase were influenced by the intrinsic cytoplasm fluctuations and beads started random motions after moving a few steps backward (Fig. 1A).

It was assumed the viscoelastic response of beads was ended and that the Brownian motion started when the angle between two successive steps exceeded 90 degrees. Relaxation phases of beads, aggregates, and oil droplets were fitted using:

ௗ௫ሺ௧ሻ ௗ௫ሺሻ ൌ ሺ1 െ ܽሻ݁ ష ഓ ܽ, .ݍܧ 2
where t = 0 corresponds to the time of the end of force application, to compute the decay time-scale ߬ r .

Fit to the data were obtained for each single experiment using optimization method in Matlab in which the results are less sensitive than Nonlinear least squares method to the initial values.

Flow analysis

The recorded oil droplet images in DIC/bright field were analyzed using the particle image velocimetry PIVlab tool in Matlab. The exterior of the egg and oil droplet surface at each frame were masked to be excluded from the analysis. Contrast limited adaptive histogram equalization (CLAHE) and two-dimensional Wiener filter with accordingly windows of 20 and 3 pixels widths were applied on the images in the pre-processing steps for noise reduction. Image sequences were investigated in the Fourier space by four interrogation windows with 64, 32, 16 and 8 pixels widths and 50% overlapped area. The spline method was used for the window deformation and subpixel resolution obtained by two-dimensional Gaussian fits. The distribution of velocity components of vectors for each set was visually inspected and restricted to remove outliers in the post-processing stage. Moreover, two other filters based on the standard deviation and local median of velocity vectors were applied to validate the vector fields. The output vector fields after smoothing were used for further analysis and plotting flow maps and streamlines in MATLAB.

Finite element hydrodynamic simulations using COMSOL

Pulling of the object in the cytoplasm was modeled as a time dependent problem using the finite element software COMSOL, using an Oldroyd-B viscoelastic model. Oldroyd-B is a three element viscoelastic fluid model consisting of a Maxwell element in parallel with a viscous element represented by the constitutive equation ߪ˙ ߢ

ఙ ఎ మ ൌ ߟ ଵ ߳¨ ሺఎ భ ାఎ మ ሻ ఎ మ
߳˙, where σ and denote stress and strain and dot represents temporal derivative [START_REF] Bland | The Theory of Linear Viscoelasticity[END_REF]. The parameters

ߢ ൌ ߢ ൬ ߛ ଶ ߛ ଵ ߛ ଶ ൰ ଶ , .ݍܧ 3 ߛ ଶ ൌ ߛ ଶ ଶ ߛ ଵ ߛ ଶ , .ݍܧ 4 ߛ ଵ ൌ ߛ ଵ ߛ ଶ ߛ ଵ ߛ ଶ . .ݍܧ 5
The viscous drag γ 2 o is associated to the dashpot in series with the elastic element describing the reorganization of cytoplasm. The drag γ 1 o is connected to the dashpot parallel to the elastic element serving for the viscosity of cytoplasm, and κ o is the cytoplasm stiffness. We could obtain the viscosity through the Stokes' law γ = 6πηr. However, we also had to take into account the correction factor from confinement C(λ), yielding γ(λ) = 6πηrC(λ), with:

ܥሺߣሻ ൌ 4ሺ1 െ ߣ ହ ሻ 4 െ 9ߣ 10ߣ ଷ െ 9ߣ ହ 4ߣ , .ݍܧ 6
where λ = r/R was the ratio of oil droplet radius to egg radius. Considering the simulation results (Fig. S3B), we assumed the viscoelastic time-scale τ 1 = γ 1 /κ as a constant material property which allowed us to apply the same confinement correction factor C(λ) for calculating stiffness inside the cell.

For simplicity, we modeled the oil droplet as a spherical elastic sphere with different radii r, and set its initial position in the center of the coordinate system except for the one-by-one and position dependent simulations (Figs. 3E-F, 4-6, and S2). The egg was represented as a large non-deformable sphere of radius R. The cytoplasm was interacting with the pulled object by including viscoelastic flow and solid mechanics modules in COMSOL through stresses at the boundaries. A Heaviside step function was used to apply a constant force as a body load with an amplitude equals to the average magnetic force in the experiments and the same pulling time. A no-slip wall condition was set as a default on the boundaries and changed to slip condition on the walls of droplet and cell to examine its impact on the results (Fig. 5).

We first verified the lack of impact of the elasticity of the pulled objects by varying this parameter over a very wide range. The elasticity had an overall little impact, lesser than 20%, on the viscosities. The viscoelastic time-scale τ 2 was differing from the actual cytoplasm properties by less than ~44% (Fig. S2B). As COMSOL is a finite-element simulation software in which elements are discretized by a mesh, we also tested several mesh sizes including fine, normal, coarse, and coarser and observed the results from different mesh size were not significantly different (Fig. S2C). The simulations were performed using a coarse mesh size and the size of the pulled object was reduced to the extent that re-meshing was feasible for the combination of used parameters and simulation time was reasonable. We then needed to ensure that bulk viscosities and viscoelastic time-scales, τ 2 = γ 2 /κ could be correctly evaluated from the drags and spring constants by fitting the one-dimensional Jeffreys' model to the simulated displacement curves. In the COMSOL model, we varied input cytoplasm properties η 1 , η 2 , and τ 2 around the average experimental values and measured the simulated displacements. We next fitted the one-dimensional Jeffreys' model to the scaled pulling curves, extracted and converted the output bulk properties to Oldroyd-B model which were close to the input values (Figs. S2D-F). The maximum deviation between the input and output viscosities in the range of values that were tried was less than ~20%. These results show that the one-dimensional model is sufficient to describe bulk cytoplasm properties in three dimensions.

Heat maps, flow fields, and streamlines of the simulations were plotted using COMSOL and the rest of the analysis were done using custom written codes in MATLAB similar to the experimental data. The input parameters of the simulations for the Figs. 4-6 are summarized in Table S1.

  N. is supported by a fellowship from the ARC foundation (PDF20191209818) and is an EMBO Non-Stipendiary Fellow (ALTF 881-2019). S.D. acknowledges support from ITMO Cancer of Aviesan on funds managed by Inserm. This work was supported by the Centre National de la Recherche Scientifique (CNRS), the Université Paris Cité, and grants from La Ligue Contre le Cancer (EL2021.LNCC/ NiM), the Agence Nationale pour la Recherche (ANR, "TiMecaDev"), the Fondation Bettencourt Schueller ("Coup d'élan"), and the European Research Council (ERC CoG "Forcaster" no. 647073) to N.M.

Super-paramagnetic 1

 1 µm Dynabeads (MyOne Streptavidin C1; Thermofisher) were pulled inside the cytoplasm as single beads. To prepare beads for injection, a 1 µl bead suspension was diluted in 100 µl washing solution (1 M NaCl with 1% Tween-20) and sonicated for 20 minutes. Beads were then separated by a magnet, rinsed in 100 µl PBS BSA 1%, and sonicated for 20 minutes after 5 minutes of incubation. The beads were then re-suspended in 20 µl of 2 µg/ml Atto565-biotin (Sigma-Aldrich), incubated for 5 minutes, and finally resuspended in 20 µl PBS.Super-paramagnetic 800 nm particles (NanoLink; Solulink) were used from their propensity to form large aggregates inside the eggs. A solution of 10 μl of undiluted streptavidin beads was first washed in 100 μl of washing solution and sonicated for 1 hour. The beads were then incubated for 15 minutes in 100 μl of 2 μg/ml Atto565-biotin (Sigma-Aldrich), rinsed in 100 μl PBS, and finally re-suspended in 20 μl. To form aggregates in cells, beads first were pulled by the magnet on one side of the egg to form an aggregate close to the cortex gathering all the beads inside the cytoplasm. Then, the dish was rotated and the aggregate was pulled from the other side of the egg. The aggregates often stretched along the pulling direction which could potentially reduce their viscous drag.To pull oil droplets in the cytoplasm, a suspension of 10 μl of 1.2 μm hydrophobic superparamagnetic beads (magtivio; MagSi-proteomics C18) was washed in 100 μl of 30, 50, and 70 percent ethanol solution. It was then dried in vacuum for ~20 minutes and resuspended in ~10 μl soybean oil (Naissance; Huile de soja). All the bead suspensions were kept at 4 °C until use. All oil-injected eggs in each sample were surveyed to identify oil droplets with a sufficient amount of beads needed for pulling. After approaching the magnet, beads inside the oil progressively formed an aggregate and slowly moved toward the magnet while the oil droplet was stationary until the aggregate contacted the oil-cytoplasm interface on the side facing the magnet tip.

  Figs. S1A-B were imaged on a spinning-disk confocal microscope (TI-Eclipse; Nikon)

  η o 1 respectively indicate stiffness, viscosity of the dashpot in series with the spring, and viscosity of the dashpot parallel to the spring. This viscoelastic model is equivalent to Jeffreys' model as one can write the Oldroyd-B model parameters as a function of the Jeffreys' parameters measured experimentally:

Figure 1 .

 1 Figure 1. Probing cytoplasm rheology at large length-scales (A) Sea urchin unfertilized eggs injected with single 1 µm beads, beads aggregate with sizes

Figure 2 .

 2 Figure 2. Viscoelastic properties depend on cytoplasm crowding. (A) Schematic of

Figure 3 .

 3 Figure 3. Experimental and theoretical mapping of shear cytoplasm flows created by

Figure 4 .

 4 Figure 4. Cell confinement enhances cytoplasm viscoelastic resistance in a non-linear

Figure 5 .

 5 Figure 5. Impact of slip or stick boundary conditions for the influence of cell

Figure 6 .

 6 Figure 6. Cytoplasm viscoelastic properties are heterogeneous and anisotropic

Figure S1 .

 S1 Figure S1. Cytoplasm organization in unfertilized eggs, and magnetic force calibration. (A) Confocal images of an unfertilized sea urchin egg fixed and stained for Microtubules and F-actin. (B) Confocal images of a fertilized sea urchin egg fixed and stained for MTs and Factin. Images are maximum intensity projections of a mid-slice egg stack. (C, top) Vector field for internal flows in an unfertilized egg overlaid on DIC images. (Bottom) Heat map of the flow speed and streamlines for the same frame as above. Scale bars in A-C, 30 µm. (D) Intensity profile along the two main axes of hydrophilic bead aggregate in eggs imaged in fluorescence, and of hydrophobic bead aggregate inside the oil droplet imaged in bright field. These profiles are used to compute the mean size of aggregates. Scale bars in the inset, 10 µm. (E) Speed of 1 µm magnetic beads moved in a test viscous fluid, plotted as a function of their distance to the magnet tip, used to calibrate the magnet. The red curve is a double exponential fit. (F) Magnetic force at 70 µm from the magnet tip, plotted as a function of aggregate size to calibrate how magnetic forces evolve as a function of aggregate size. The red curve is a cubic fit.

Figure S2 .

 S2 Figure S2. Validation of finite element 3D hydrodynamic simulations. (A) Vector field of cytoplasm flows as a 1 µm bead is pulled through the cell. The moving direction of the bead is indicated by the red arrow. (B) Normalized simulation output values of the two viscosities and time-scale in the Oldroyd-B model plotted as a functions of oil elastic modulus. (C) Displacement curve of pulling and release phase for the same sphere of 20 µm in diameter using different discretization mesh sizes in the simulation. (D, top) Output values of viscosity η 1 computed from the 3D simulation results using the Jeffreys' model plotted as a function of the input value given to the simulation. (Middle) Plot of the output/input ratio of the second viscosity η 2 in the Oldroyd-B model as a function of different input values for η 1. (Bottom) Plot of the output/input ratio of the time-scale τ 2 =k/η 2 in the Oldroyd-B model , as a function of different input values for η 1. (E) Output values of η 2 , output/input ratio for η 1 , and output/input ratio of τ 2 plotted as functions of input values for η 2 . (F) Output values of τ 2 , output/input ratio for η 1 , and output/input ratio of η 2 plotted as functions of input values for

τ 2 .

 2 Dashed red lines guide the eyes and correspond to equal values for input and output. Scale bar, 30 µm.

Figure S3 .

 S3 Figure S3. Initial position of droplets and viscoelastic time-scales at different aspect ratio. (A) Distribution of normalized initial positions of oil droplets in experiments presented in Fig. 4. (B) Viscoelastic time-scale computed from the simulation as a function of confinement ratio.

Supplemental Information

The supplemental information contains 1 supplemental table, 3 supplemental figures and 3 supplemental movie legends.

Parameters