Hichem Lassoued

Camille Laurent-Gengoux

SYMPLECTIC RESOLUTION OF THE QUOTIENT OF R 2 BY A NON-FINITE SYMPLECTIC GROUP

We construct smooth symplectic resolutions of the quotient of R 2 under some infinite but discrete sub-group of sl 2 (R). This extends to the setting of smooth real differential geometry the classical Du Val symplectic resolution of C 2 divided by a finite group in algebraic geometry. The first of these infinite groups is G = Z, identified to triangular matrices with spectrum {1}. Smooth functions on the quotient R 2 /G come with a natural Poisson bracket, and R 2 /G is for an arbitrary k ≥ 1 set-isomorphic to the real Du Val singular variety A 2k = {(x, y, z) ∈ R 3 , x 2 + y 2 = z 2k }. We show that each one of the usual minimal resolutions of these Du Val varieties are symplectic resolutions of R 2 /G. The same holds for G = Z⋊Z/2Z (identified to triangular matrices with spectrum {±1}), with one half of D 2k+1 playing the rôle of A 2k .

This article aims to initiate the study of quotients of smooth Poisson manifolds (M, π) by infinite but discrete groups G of Poisson automorphisms of (M, π). We call quotient of M by G and denote by M/G (with an abuse of notation) the quotient of M under the closure of the equivalence relation induced by G, i.e. x ∼ y if and only if there exists sequences (x n ) n∈N , (y n ) n∈N with limits x and y respectively such that x n and y n are in the same G-orbit for all n ∈ N. Such quotients are neither varieties nor algebraic varieties in general, but they are always "Poisson sets" [START_REF] Lassoued | Résolutions symplectiques et de contact de variéetées de Poisson et de Jacobi[END_REF] in the following sense: a real valued function on M/G is said to be smooth if its pull-back to M is smooth on M , i.e. C ∞ (M/G) ≃ C ∞ (M ) G . Since G preserves structure, the algebra C ∞ (M/G) comes equipped with a Poisson bracket (see [START_REF] Laurent-Gengoux | Poisson Structures[END_REF], Section 5.4). Definition 1.1. [START_REF] Lassoued | Dimension preserving resolutions of singular Poisson structures[END_REF] Let (M, π M ) be a Poisson manifold, and C ∞ (M/G) := C ∞ (M ) G . A symplectic realization of M/G is a triplet (Σ, π Σ , ϕ) where (Σ, π Σ ) is a symplectic manifold (maybe with boundary), ϕ : Σ → M/G is a map such that i) ϕ : Σ → M/G is smooth, i.e. ϕ * (F ) is a smooth function on Σ for all F ∈ C ∞ (M/G), ii) the algebra morphism ϕ * : C ∞ (M/G) → C ∞ (Σ) is a Poisson morphism. Such a symplectic realization (Σ, π Σ , ϕ) is called a symplectic resolution of M/G when:

iii) there exists an open dense G-saturated subset U ∈ M on which the G-action is discrete, proper and free1 , iiii) ϕ : ϕ -1 (U/G) → U/G is a diffeomorphism. Definition 1.1 echoes the nowadays classical notion of symplectic realizations of smooth Poisson manifolds [START_REF] Broka | Symplectic Realizations of Holomorphic Poisson Manifolds[END_REF]- [START_REF] Coste | Groupoïdes symplectiques[END_REF]- [START_REF] Dufour | Poisson Structures and Their Normal Forms[END_REF]- [START_REF] Karasev | Analogues of s of the theory of Lie groups for nonlinear Poisson brackets[END_REF]. However, in Definition 1.1, the manifolds Σ and M have the same dimension while in the latter theory, the dimension is doubled. Definition 1.1 is therefore more similar to that of symplectic resolutions of affine Poisson varieties as in [1]- [START_REF] Fu | A survey on symplectic singularities and resolutions[END_REF]- [START_REF] Laurent-Gengoux | From Lie groupoids to resolutions of singularities[END_REF]: indeed, it is an extension of this notion to the context of real differential geometry (as in [START_REF] Laurent-Gengoux | From Lie groupoids to resolutions of singularities[END_REF]). Several examples of symplectic resolutions include quotients of symplectic vector spaces by finite subgroup of the symplectic group, see [START_REF] Bellamy | A new linear quotient of C 4 admitting a symplectic resolution[END_REF]. The most basic examples of such quotients are the very classical Du Val singularities (A n , D n , E 6 , E 7 , E 8 ), see [START_REF] Reid | The Du Val singularities An[END_REF]. It is known that the minimal desingularization of these singularities is a symplectic resolution in the sense of algebraic geometry [1]- [START_REF] Fu | Symplectic Resolutions for Nilpotent Orbits[END_REF]. This implies that it is also a symplectic resolution in the sense of Definition 1.1. As previously said, we wish to investigate quotients of Poisson vector spaces under infinite but discrete groups and their symplectic resolutions, if any. Notice that for G a discrete infinite subgroup of GL(V ), there may be very few non-constant algebraic G-invariant functions, while G-invariant smooth functions C ∞ (V /G) may be still large enough to be worth of study. As we will see, our purpose can not be addressed within algebraic geometry: We have to work with smooth functions. But this is certainly not an easy matter. Our more reasonnable present purpose is to understand in detail two examples that we think to be generic in some sense. Both will turn to be surprisingly subtle, although they are simply the quotients of R 2 by the groups

(1.2) G = 1 n 0 1 n ∈ Z and G ′ = 1 n 0 ǫ ǫ ∈ {±1}, n ∈ Z
which are isomorphic to Z and Z/2Z ⋊ Z respectively. These infinite but discrete groups act on R 2 by Poisson automorphisms for the Poisson structure

(1.3) π R 2 := 1 2π q ∂ ∂p ∧ ∂ ∂q
where (p, q) stands for the canonical coordinates on R 2 . It was proven in [START_REF] Lassoued | Dimension preserving resolutions of singular Poisson structures[END_REF] that the previous Poisson structure does not admit symplectic resolutions. In contarst, the main theorems of the present articles state that the quotients R 2 /G and R 2 /G ′ do admit symplectic resolutions.

Let us describe the content of the article. In Section 2, we study the algebras

C ∞ (R 2 /G) := C ∞ (R 2 ) G and C ∞ (R 2 /G ′ ) := C ∞ (R 2 ) G ′
In the process, we will explain why, unlike in the finite case, real analytic or holomorphic contexts are not relevant here because they admit too little invariant functions. For smooth invariant functions, which form an algebra large enough to be interesting, we give a useful Fourier series expansion. Section 3 is dedicated to reminders on Du Val's symplectic resolutions (in the real case) of A 2k and D 2k+1 .

In section 4, we show that for all k ≥ 2, the quotient space R 2 /G is in bijection (as a set) with the quotient singularity of Du Val

A 2k = (x, y, z) ∈ R 3 | x 2 + y 2 = z 2k .
We then show that this bijection φ k behaves well with respect to Poisson structures.

By "behaves well", we mean that

φ k * : C ∞ (R 3 ) → C ⌊k/2⌋ (R 2 /G) is a Poisson algebra morphism.
We also show similar results for the second group G ′ ≃ Z ⋊ Z/2Z, but it is then the singularities (Du Val) of type D 2k+1 = (x, y, z) ∈ R 3 | zx 2 + y 2 = z 2k+1 which intervenes (more precisely, D 2k+1 ∩ {z ≥ 0}). Section 5 concludes this construction be an extremely surprising result: the (usual) symplectic resolution ϕ : Z k → A 2k of the Du Val singularity A 2k induces a symplectic resolution of the Poisson quotient R 2 /G, when composed with φ k -1 :

A 2k → R 2 /G. In other words, φ -1 • ϕ : Z k → R 2 /
G satisfies the requirements of Definition 1.1. This is not an easy result, and we have to deploy various techniques to achieve this, including Fourier analysis, and delicate consideration on the Du Val resolution.

We also show similar results for the second group G ′ := Z ⋊ Z/2Z. The usual symplectic resolution ϕ :

Z k → D 2k+1 , restricted to ϕ -1 (D 2k+1 ∩ {z ≥ 0}), composed with φ k -1 : D 2k+1 ∩ {z ≥ 0}) → R 2 /G ′ gives a smooth symplectic resolution of the Poisson quotient R 2 /G ′ in this case.
It is of course interesting to see that there are infinitely many symplectic resolutions for both quotient spaces. However, it is unexpected to find A 2k in one case and D 2k+1 in the other. There must be a general phenomena behind this astonishing result.

2. Quotient of R 2 by two infinite groups

We equip R 2 with the coordinates (p, q). Consider the infinite groups G ≃ Z and G ′ ≃ Z ⋊ Z/2Z as in (1.2) and their actions on R 2 by

Z × R 2 → R 2 n • (p, q) → (p + nq, q) and (Z ⋊ Z/2Z) × R 2 → R 2 (n, ǫ) • (p, q) → (p + nq, ǫq) for all (n, ǫ) ∈ Z × {-1, 1} and (p, q) ∈ R 2 .
The closure ∼ of the equivalence relation induced by the action of G = Z above is given by (2.1) (p, q) ∼ (p + nq, q), for any pair (p, q) ∈ R 2 , n ∈ Z and (p, 0) ∼ (p ′ , 0) for all p, p ′ ∈ R. This equivalence relation is closed i.e., (p

1 , q 1 ) × (p 2 , q 2 ) ∈ R 2 × R 2 | (p 1 , q 1 ) ∼ (p 2 , q 2 ) is a closed set of R 2 × R 2 .
For G ′ = Z ⋊ Z/2Z, we still have to identify (p, q) and (p, -q). We denote by ∼ ′ this relation.

Convention 2.2. We denote (with a slight abuse of notation) the equivalence classes of the closed relations ∼ and ∼ ′ by R 2 /G and R 2 /G ′ .

In general, for ∼ be a closed equivalence relation on a manifold M , we call smooth function, real analytic function or function of class C k on M/ ∼ a function whose pull-back on M through the natural projection M → M/ ∼ is smooth, respectively real analytic or of class C k . Equivalently, functions of on of these given types on M/ ∼ are functions on M constant on the equivalences classes.

When the equivalence relation is the closure of a relation given by the action of a group G, smooth functions (resp. real analytic or C k -functions) on the quotient are exactly G-invariant smooth functions (resp. real analytic or C k -functions) on M . In the case we are interested with, we therefore have

F [R 2 ] G ≃ F [R 2 /G] and F [R 2 ] G ′ ≃ F [R 2 /G ′ ] for F = C ∞ or C k .
Let us study this algebra. finite groups, the algebra of invariant functions

F [R 2 ] G or F [R 2 ] G ′
may not be finitely generated. We take G = Z and we start with the algebra

C ∞ (R 2 /G) = C ∞ (R 2 ) G . Lemma 2.3. Any function f ∈ C ∞ (R 2 ) G satisfies for all i 1, j 0 ∂ i+j f ∂p i ∂q j (p, 0) = 0, ∀p ∈ R.
Proof. The k th derivative of the relation f (p, q) = f (p + nq, q) with respect to q gives:

∂ k f ∂q k (p, q) = k i=0 k i n k-i ∂ k f ∂p i ∂q k-i (p + nq, q).
For q = 0, we have

k i=0 k i n i ∂ k f ∂p i ∂q k-i (p, 0) = ∂ k f ∂q k (p, 0).
The right hand term coincides with the i = 0 term on left hand. We therefore obtain for all n ∈ Z:

k i=1 k i n i ∂ k f ∂p i ∂q k-i (p, 0) = 0.
The coefficients of a polynomial function that admits infinity many roots are zero.

In particular, the coefficients of the polynomial

P k (X) := k i=1 k i ∂ k f ∂p i ∂q k-i (p, 0) X i
are zero for all k ∈ N * and p ∈ R. This completes the proof.

Proposition 2.4. The algebra C ∞ (R 2 /G) of smooth functions on R 2 /G breaks down as follows C ∞ (R 2 /G) = C ∞ (q) + C, where C ∞ (q)
is the algebra of smooth functions which depends only on the variable q, and C ⊂ C ∞ (R 2 /G) is the subalgebra of G-invariant smooth functions vanishing with all their partial derivatives along the line q = 0.

Proof. Consider a G-invariant smooth function f (p, q). Define smooth functions g, h on R and R 2 respectively by h(q) := g(0, q) and g(p, q) := f (p, q) -h(q). We have f = h + g by construction. It therefore remains to show that g is an element of C.

SYMPLECTIC RESOLUTION OF THE QUOTIENT OF R 2 BY A NON-FINITE SYMPLECTIC GROUP 5

Since the function g is obviously G-invariant, Lemma 2.3 implies that for all i ≥ 1, j ≥ 0:

∂ i+j g ∂p i ∂q j (p, 0) = 0, ∀p ∈ R.
By integration,

∂ j g ∂q j (p, 0) = ∂ j g ∂q j (0, 0) + p t=0 ∂ 1+j g ∂p∂q j (t, 0)dt.
Since g(0, q) = 0 by construction, we have ∂ j g ∂q j (0, 0) = 0, and the first term on the right vanishes. Lemma 2.3 implies the vanishing of the second one. This completes the proof.

There is a similar result for the G ′ -action on R 2 .

Proposition 2.5. The algebra

C ∞ (R 2 /G ′ ) of smooth functions on R 2 /G ′ breaks down as follows C ∞ (R 2 /G) = C ∞ (q 2 ) + C,
where C ∞ (q 2 ) is the algebra of smooth even functions depending only on the variable q, and C ′ ⊂ C ∞ (R 2 /G ′ ) is the subalgebra of G ′ -invariant smooth functions vanishing with all their partial derivatives along the line q = 0.

The following corollary shows that the real analytic case is not interesting: there are too little G-invariant functions.

Corollary 2.6. Real analytic functions on R 2 /G (on R 2 /G ′ ) are the functions (resp. even function) which depend only on the variable q.

Proof. A real analytic function on R 2 which vanishes along a line with all its partial derivatives is zero on R 2 . The result now follows from Propositions 2.4 and 2.5. This corollary also means that it is not possible to make such a study in the holomorphic context and study C 2 /G instead. There is not enough G-invariant holomorphic functions. Let us complete our description of smooth functions on R 2 /G:

Proposition 2.7. Any function f ∈ C ∞ (R 2 ) G ≃ C ∞ (R 2 /G) reads (2.8) f (p, q) = α 0 (q) + n≥1 α n (q) cos 2nπ p q + n≥1 β n (q) sin 2nπ p q ,
where α 0 is a smooth function on R and where (α n (q), β n (q)) n≥1 are smooth functions on R that vanish together with all their derivatives at q = 0. Both above series, moreover, are absolutely convergent on any open ball.

Proof of Proposition 2.7.

Step 1. Let us construct functions α n , β n that satisfy (2.8). For q ∈ R * fixed, the function p → f (p, q) is smooth and q-periodic. Its Fourier transform with respect to the variable p reads

f (p, q) = α 0 (q) + n≥1 α n (q) cos 2nπ p q + n≥1 β n (q) sin 2nπ p q
Let us use the change of variables u = p q to find a friendly expression of the Fourier coefficients:

α 0 (q) = 1 q q/2 -q/2 f (p, q)dp, = 1/2 -1/2 f (uq, q)du ( * 0 ), and
α n (q) = 2 q q/2 -q/2 f (p, q) cos 2nπ p q dp, = 2 1/2 -1/2 f (uq, q) cos (2nπu) du, ( * n ) and β n (q) = 2 q q/2 -q/2 f (p, q) sin 2nπ p q dp, = 2 1/2 -1/2 f (uq, q) sin (2nπu) du. ( * * n )
Formulas ( * 0 , * n , * * n ) still make sense for q = 0, so that the functions α n , β n are defined on R. They are smooth as integrals of smooth functions on a compact interval.

Step 2. Let us show that for all n ≥ 1, the Fourier coefficients α n (q), β n (q) ∈ C ∞ (R, R) are smooth functions which are zero with all their derivatives at q = 0.

A. If the function f (p, q) only depends on q, then α n = β n = 0 for all n 1, while α 0 (q) = f (p, q) and the result is obvious. B. If f (p, q) belongs to C, it vanishes with all its partial derivative along the line q = 0. A simple computation using ( * n , * * n ) then gives the result. By Proposition 2.4, any function in C ∞ (R 2 /G) is a sum of a function of the type A with a function of the second type B. The result therefore holds.

Step 3. Let us show that the series in (2.8) are absolutely convergent on any relatively compact subset U of R 2 . Let us integrate by part the right hand side of ( * n ) twice. Since u → f (uq, q) is periodic of period 1, all boundary term disappear, and we obtain:

(2.9)

α n (q) = -1 n 2 q 2 4π 2 1/2 -1/2 ∂ 2 f (uq, q) ∂q 2 cos(2nπu)du. Let K = B 2 2π 2 N
, where B, N are the greatest possible value of |q| and

| ∂ 2 F (p,q) ∂p 2 | on U. We have |α n (q)| < K n 2 .
The same applies to ( * * n ). Since the series 1/n 2 converges, both series in (2.8) are absolutely convergent.

In fact, the inequality (2.9) can be made much stronger -and it will be needed later.

Lemma 2.10. For every i ≥ 1, k ≥ 1 and M > 0, there exists a contant K that does not depend on n such that:

α (i) n (q) q k ≤ K n k and β (i) n (q) q k ≤ K n k
for all -M < q < M Proof. Using successive integration by part, we obtain:

(2.11)

α (i) n (q) = 1 n k q k (2π) k 1/2 -1/2 ∂ k+i f (uq, q) ∂q k+i cossin k (2nπu)du.
where cossin k is cos, sin, -cos, -sin for k ≡ 0, 1, 2, 3 modulo 4 respectively. Since

|cossin k | ≤ 1, K = 1 (2π) k Max ∂ k+i f (uq, q) ∂q k+i , (u, q) ∈ [0, 1] × [-M, M ]
This maximum exists by compactness: This concludes the proof for the functions

α (i)
n . The proof is similar for the functions

β (i) n .
Here is an immediate consequence of Lemma 2.10:

Proposition 2.12. Both series in Equation (2.8) are uniformly convergent on any compact set, together with all their partial derivatives.

For the group action G ′ , a similar argument leads to:

Proposition 2.13. A function f ∈ C ∞ (R 2 /G ′ ) reads f (p, q) = α 0 (q) + n≥1 α n (q) cos 2nπ p q + n≥1 β n (q) sin 2nπ p q ,
where the functions (α n , β n ) n≥1 are smooth and vanish with all their derivatives at q = 0. Moreover the functions α n are even functions and the functions β n are odd functions for all n ∈ N. The above series, moreover, is absolutely convergent on any relatively compact open set, and so are any of its partial derivatives.

Real Du Val symplectic resolutions

In this section, we mainly recall from [1] and adapt to the real case the construction of the symplectic resolution of the Du Val singularities A 2k and D 2k+1 . First, we define a Poisson structure on the varieties A 2k and D 2k+1 . For any polynomial function F (x, y, z) on R 3 , a Poisson bracket on R 3 is given by [START_REF] Pichereau | Cohomologie de Poisson en dimension trois[END_REF]:

(3.1) {x, y} = ∂F ∂z , {y, z} = ∂F ∂x , {z, x} = ∂F ∂y .
This Poisson bracket admits F as a Casimir, and therefore goes to the quotient to R[x,y,z] (F ) . Unlike in the complex case, the latter may not be easily identified to polynomial functions on the set F = 0 (for instance, for F = x 2 + y 2 + z 2 , the latter set is a point). It only makes sense when the quotient R[x,y,z] (F ) injects itself into the algebra of real valued functions on {F = 0}. This is the case, in particular, for F = x 2 + y 2 -z 2k : We denote by A 2k the set Let us first describe the minimal resolution of A 2k .

A 2k = {(x, y, z) ∈ R 3 | x 2 + y 2 = z 2k }.
(1) The blowup ϕ : Z → A 2k at the origin (0, 0, 0) ∈ A 2k is covered by 3 pieces. (a) In two of them, the strict transform of A 2k is smooth.

(b) In the last one, the z-chart, with coordinates u, v, z, the morphism is given by ϕ 1 : Z 1 → R 3 defined by x = uz, y = vz, z = z, and the inverse image of A 2k is defined by u 2 z 2 + v 2 z 2 = z 2k so that the strict transform is u 2 + v 2 = z 2(k-1) . As a consequence, the strict transform on this chart is isomorphic to A 2k-2 . (2) We then blow-up A 2k-2 , and repeat the procedure . . . (k) Therefore, after k-successive blowups at (0, 0, 0):

ϕ k : Z k → Z k-1 , the
strict transform is given by x 2 + y 2 = 1 with z a free variable. At this point, we obtain a smooth projective variety, called the (real) Du Val resolution of A 2k .

Let us describe the behaviour of the Poisson structure under the successive blowup leading to the Du Val resolution.

( which is symplectic. We have therefore obtained by successive blowing a symplectic resolution of A 2k , which we denote by (Z k , π k ). Let us recapitulate. Proposition 3.4. For all k ≥ 2:

(1) The map φ : Z k is onto, and restricts to a diffeomorphism

Z k \φ -1 ({0}) ≃ A 2k \{0} (2) The pull-back of the Poisson structure (3.2) to the Du Val resolution of ϕ : Z k → A 2k is a symplectic structure. Equivalently, (Z k , ϕ, π k ) is a symplectic resolution of A 2k .
The next point is easy to verify and will be important. Notice that, now, it is only true on R, and not valid on C. Proof. The unique point where the rational function x z k is not defined is x = y = z = 0. Let us see the behaviour under the successive blowups that lead to the Du Val resolution:

(1) For the blowing-up of the singularity A 2k at the origin, the following phenomenon occurs: (a) in the two first affine charts (the x-chart and the y-chart), the pullback is a smooth function, because the pull-back of z can not be equal to 0, (b) in the last one, the restriction to the strict transform (which is isomorphic to A 2k-2 ) of the pull back of x z k is the function x z k-1 . We are therefore brought back to the previous function, upon changing k by k -1. We can then repeat the procedure. . . . (k) At the last step, the strict transform is given by x 2 + y 2 -1 = 0 with z being free, and the back pull of the function x/z k is simply the function x, which is smooth. This completes the proof.

Let us define two sequences (x n ) n≥1 and (y n ) n≥1 of rational functions on A 2k ⊂ R 3 by x 1 (x, y, z) := x, y 1 (x, y, z) := y for all (x, y, z) ∈ A 2k and (3.6) Let us now describe a symplectic resolution of the latter. The procedure is at first very similar.

x n = x1 z k x n-1 -y1 z k y n-1 y n = x1 z k y n-1 + y1 z k x n-1 . From Proposition 3.
(1) The strict transform of the blowing up at zero is covered by the three natural charts. (a) In the x-chart the strict transform is the variety zx -z 2k+1 x 2k-1 = -y 2 , whose unique singular point is (0, 0, 0). (a) The blow up at (1, 0, 0) admits a singular point. Blowing-up at this point, we obtain at last a smooth variety, and the pull-back of the Poisson structure (3.9) is symplectic. (b) The blow up at (-1, 0, 0) admits a singular point. Blowing-up at this point, we obtain at last a smooth variety, and the pull-back of the Poisson structure (3.9) is symplectic. We denote by (Z k+2 , π k+2 ) this symplectic manifold.

Let D + 2k+1 = D 2k+1 ∩ {z > 0} and Z + k+2 := φ -1 ({z > 0}) ⊂ Z k+2 . Z +
k+2 is manifold (with boundary). We have: Proposition 3.10. For all k ≥ 2:

(1) The map ϕ : Z + k+2 → D + 2k+1 is onto, and restricts to a diffeomorphism Proof. The proof is similar to that of the Proposition 3.5. We leave it to the reader.

Z + k+2 \φ -1 ({0}) ≃ D + 2k+1 \{0} (2)
We shall consider the following functions, similar to those in (3.6):

(3.12)

x n = x1 z k+1/2 x n-1 -y1 z k+1/2 y n-1 y n = x1 z k+1/2 y n-1 + y1 z k+1/2 x n-1 .
It follows from all of the above that: Corollary 3.13. The pull back X n := ϕ * x n and Y n := ϕ * y n of functions x n , y n defined in (3.12), by the Du Val resolution ϕ : Z k+2 → D 2k+1 are polynomial functions on Z k+2 .

Quotient Poisson structure

We now introduce the Poisson bracket on the quotient space R 2 /G (resp R 2 /G ′ ), with G and G ′ acting on M = R 2 as in Section 2. We introduce for k ≥ 2 the algebra of functions noted C k,∞ (R 2 ) which are: a) differentiable ⌊k/2)⌋-times on R 2 , b) and smooth outside the straight line q = 0.

We now equip M = R 2 with the following Poisson bracket (4.1) {p, q} = q 2π .

The following lemmas are easily verified. In view of Lemmas 4.2 and 4.3, the following algebras come equipped with the induced Poisson brackets:

C ∞ (M/G) := C ∞ (M ) G , C ∞ (M/G ′ ) := C ∞ (M ) G ′ , C k,∞ (M/G) := C k,∞ (
M ) G , and C k,∞ (M ) G ′ (see Section 2 for notations).

Remark 4.4. The symplectic Poisson bracket {p, q} = 1 is also under the actions of G and G ′ , but does not satisfy Lemma 4.2. Also, since the singular locus of both group actions (i.e. the straight line q = 0) projects to a single point in the quotient space, it makes more sense to choose a Poisson structure on R 2 that vanishes on this straight line.

From now on, we deal with the G-action only: results for the G ′ -action will be stated at the end of the section. Let A be the subalgebra of C k,∞ (R 2 /G) generated by x(p, q) := q k cos 2π p q , y(p, q) := q k sin 2π p q , z(p, q) := q .

Proposition 4.5. The algebra A is stable under the Poisson bracket (4.1).

Proof. A direct computation with this bracket gives:

(4.6)    {x, y} = -2k z 2k-1 ∈ A {y, z} = 2 x ∈ A {z, x} = 2 y ∈ A.
This proves the result.

Let us consider the map R 2 → R 3 defined by φ k : (p, q) → (x(p, q), y(p, q), z(p, q)).

Since x, y, z are G-invariant functions, and are equal to 0 on the straight line q = 0, this map goes to the quotient to define a map:

φ k : R 2 /G → R 3 .
: R 2 /G → A 2k is one-to-one. Its restriction to R * × R/G ≃ A 2k \{0} is a diffeomorphism.
Proof. The relation

x 2 + y 2 = q k cos 2π p q 2 + q k sin 2πk p q 2 = q 2k = z 2k . (4.8)
proves that φ k is valued in A 2k . Hence so is φ k . Injectivity follows from the following two points:

• For any two (p, q), (p ′ , q ′ ) with q, q ′ = 0, x(p, q) = y(p, q) if and only if q = q ′ and p = p ′ + nq for some n ∈ N. • x(p, q) = y(p, q) = z(p, q) = 0 implies q = 0 Hence φ k (p, q) = φ k (p ′ , q ′ ) if and only if (p, q) ∼ (p ′ , q ′ ).

Surjectivity follows easily from the surjectivity of u → (cos u, sin u) from R to S 1 .

Both R 2 /G and A 2k come equipped with Poisson structures. We would like to state that φ k : R 2 /G ≃ A 2k is a Poisson isomorphism. But we have to be careful because there are several algebras of functions on R 2 /G: the one that allows to make this statement is A, as we now state (here

F (A 2k ) = R[x,y,z]
x 2 +y 2 -z 2k stands for the algebra of polynomial functions on A 2k ): Proposition 4.9. The pull-back map of φ k : R 2 /G → A 2k is a Poisson morphism:

C ∞ (R 3 ) → C ⌊k/2⌋ (R 2 /G).

It restricts to a Poisson algebra isomorphism φ

k * : F (A 2k ) ≃ A. Proof. By Equation (4.8), the functions x, y, z ∈ C k,∞ (R 2 ) satisfy x 2 + y 2 = z 2k .
Since there is no other relations between them, save for multiples of this relation, and since the generators x, y, z of F (A 2k ) are also subject to the relation

x 2 + y 2 = z 2k , the algebra morphism φ k * : F (A 2k ) ֒→ A is an algebra isomorphism.
φ k / / P R 3 R 2 /G φ k / / A 2k ? i O O hence R[x, y, z] φ k * / / i * C ⌊k/2⌋ (R 2 ) F (A 2k ) φ k * / / C ⌊k/2⌋ (R 2 /G) ? P * O O this implies that φ k * is a Poisson morphism. 2
For the quotient space R 2 /G ′ , we consider the subalgebra A ′ ⊂ C k (R 2 /G ′ ) generated by

x = q 2k cos 2π p q , y = q 2k+1 sin 2π p q , z = q 2 , for any integer k ≥ 2. view of Lemma 2.10, there exists a constant K such that |α n (q)/q k | < K/n k and |β n (q)/q k | < K/n k for all -M < q < M , so that

α n (Z) Z k < K n k and α n (Z) Z k < K n k
uniformly on C. This proves the uniform convergence on C of the series (5.2) defining H.

To prove that H is smooth, it suffices to prove that the series that define D(H) converge for every partial derivative operator D on Z.

For every partial derivative operator D of orderd, we derive from form the recursion relation (3.6) that:

D(X n+1 ) D(Y n+1 ) = X 1 -Y 1 Y 1 X 1 D(X n ) D(Y n ) + L(X n , Y n ) M (X n , Y n )
where L(X, Y ), M (X, Y ) depend linearly on a finite number of partial derivatives of order ≤ d -1 of X, Y . Hence Equation (5.5) implies then that there exists constants A ′ , B ′ such that U D n ≤ A ′ + B ′ n d .

Since the convergence to zero of the d-first derivatives of |α n (Z)/Z k |, |β n (Z)/Z k | can be made smaller that K n r for an arbitrary r, uniformly on φ -1 (C). This proves the uniform convergence of the series obtained by applying D to all the terms in the series in (5.2). This proves the smoothness of its limits, and in turn the smoothness of H.

We conclude with the main result of this article:

Formula ( 3 . 1 )

 31 equips A 2k with the Poisson structure: (3.2) {x, y} = -2kz 2k-1 , {y, z} = 2x, {z, x} = 2y.Let us study the successive blowing up at the origin (the only singular point) of A 2k and the behavior of the above Poisson structure.

  ) The Poisson structure (3.2) pulls-back to a Poisson structure on the strict transform of A 2k under the first blow-up A 2k (a) In the two first affine charts, the pulled-back Poisson structure is symplectic (b) In the third one, (where the strict transform is ≃ A 2k-2 ), the pulledback Poisson structure is of the form (3.2) with k being replaced by k -1. (2) The same applies then to the blow-up of A 2k-2 . . . (k) Therefore, after k-successive blowups at (0, 0, 0), the pulled-back Poisson structure, which is still well-defined and has no singularity, is given by (3.3) {x, y} = 0, {y, z} = 2x, {z, x} = 2y.

Proposition 3 . 5 .

 35 The pull back of the rational functions x z k and y z k by the symplectic Du Val resolution Z k → A 2k are smooth functions on Z k .

5 , it follows that Corollary 3 . 7 .

 537 For all n ∈ N the pull back X n := ϕ * x n and Y n = ϕ * y n of the functions x n , y n defined in (3.6) to the Du Val resolution ϕ : Z k → A 2k are smooth functions on Z k . By same procedure, we equip the variety defined by D 2k+1 := { x, y, z) ∈ R 3 | zx 2 + y 2 = z 2k+1 = 0 with the Poisson structure defined as in (3.1) with the help of the function zx 2 + y 2 -z 2k+1 : (3.8) {x, y} = -(2k + 1)z 2k + x 2 , {y, z} = 2zx, {z, x} = 2y.

  (i) The strict transform of the pull-back at the origin of the latter variety is smooth, and the pull-back of the Poisson structure (3.8) is symplectic. (b) In the y-chart the strict transform is smooth and the pull-back of the Poisson structure (3.8) is symplectic. (c) In the z-chart the strict transform is D 2(k-1) , and the pull-back of the Poisson structure (3.8) is the Poisson structure (3.8), with k being replaced by k -1. We are therefore brought back to the previous problem in changing k by k -1 and then repeat the operation. . . . (k) At the k th -step, the situation is quite different from the one of A 2k . The strict transform is the variety: zx 2 + y 2 -z = 0 which admits two singular points: (1, 0, 0) and (-1, 0, 0). The pull-back of the Poisson structure 3.8 is given (as in (3.1)) by (3.9) {x, y} = x 2 -1, {y, z} = 2xz, {z, x} = 2y.

Proposition 3 . 11 .pull back x z k+ 1 2 and y z k+ 1 2

 31121 The pull-back of the Poisson structure (3.8) to the Du Val resolution of ϕ : Z + k+2 → D + 2k+1 is a symplectic structure. Equivalently, (Z + k+2 , ϕ, π k ) is a symplectic resolution of D + 2k+1 . The by the map Z k+2 → D + 2k+1 are smooth functions on Z k+2 .

Lemma 4 . 2 .Lemma 4 . 3 .

 4243 For all k ∈ {2, 3, . . . , +∞}, the algebra C k,∞ (R 2 ) is stable under the Poisson bracket. and The Poisson bracket (4.1) is invariant under the actions of G = Z and G ′ .

  It is still necessary to show that it is a Poisson morphism. A comparison of the Poisson bracket on F (A 2k ) defined in (3.2) with the Poisson brackets of x, y, z ∈ C k,∞ (M ) as computed in (4.6) shows that φ k : R 2 → R 3 is a Poisson map. In view of the following commutative diagram (4.10) R

Proposition 4 . 11 .

 411 The algebra A ′ is stable under the Poisson bracket (4.1).Proof. A direct computation with this bracket gives:   {x, y} = -(2k + 1)z 2k + x 2 ∈ A ′ {y, z} = 2zx ∈ A ′ {z, x} = 2y ∈ A ′ .

  U D n := (D(X n )) 2 + (D(Y n )) 2 satisfies (5.5) U D n+1 = U D n + S D (X n , Y n ) where S D (X, Y ) depend linearly on a finite number of partial derivatives of order ≤ d -1 of X, Y .We claim that for any partial derivative operator D of degree d, there exists constants A, B such that U D n ≤ A + Bn D . For d = 0, Equation (5.4) implies the desired assumption. If this is true for all D of degree ≤ d -1, then in Equation (5.5), there exists A, B such that|S D (X n , Y n )| ≤ A + Bn d-1

Theorem 5 . 6 .

 56 The mapψ k • ϕ : Z k → R 2 /G is a symplectic resolution of (R 2 /G, {•, •}).Proof. Let us check that all four conditions in Definition 1.1 are satisfied. Condition (i) has been established in Proposition 4.9. Condition (ii) follows from Proposition 4.9 and the first item in Proposition 3.4. Conditions (iii) and (iiii) hold in view of Lemma 4.7 and the second item in Proposition 3.4.For the quotient space R 2 /G ′ , the map φk : R 2 /G ′ → D + 2k+1 is bijective by Lemma 4.12. Let us call ψ k : D + 2k+1 → R 2 /G ′ its inverse map. We note ϕ : Z k+2 → D + 2k+1 the usual symplectic resolution of D + 2k+1 , see Section 3.

  Since x 2 +y 2 = z 2k , this map takes values in the Du Val singular surface A 2k (see Section 2).

	Lemma 4.7. The map φ k

This implies that U/G is a manifold (and makes sense of Condition (iiii)).

Let us consider the map R 2 → R 3 defined by φ k : (p, q) → (x(p, q), y(p, q), z(p, q)).

Since x, y, z are G ′ -invariant functions, and are equal to 0 on the straight line q = 0, this map goes to the quotient to define a map: φ k : R 2 /G ′ → R 3 . Since zx 2 + y 2 = z 2k+1 , this map takes values in the Du Val singular surface D 2k+1 (see Section 2). More precisely, it takes values in D + 2k+1 = D 2k+1 ∩ {z ≥ 0} Lemma 4.12. The map φk : R 2 /G ′ → D + 2k+1 is one-to-one Proof. The relation

proves that φ k is valued in D 2k+1 . Since q 2 ≥ 0, it takes indeed values in D + 2k+1 . Hence so is φk . It remains to prove that φk is bijective: The proof is identical to that of the Lemma 4.7.

As for R 2 /G and A 2n , since R 2 /G ′ and D + 2k+1 are in bijection, and since they are also Poisson sets, we would like to assert that φk is a Poisson isomorphism. As for R 2 /G and A 2n , the precise statement will use the subalgebra

Proposition 4.14. The pull-back map of φk :

It restrict to a Poisson algebra isomorphism φk

The proof of this Proposition is identical to that of the Proposition 4.9.

Symplectic resolutions of the quotients

We deal with the case of G first, and use the notations of the previous section. The map φ k : R 2 /G → A 2k is bijective by Lemma 4.7. Let us denote by ψ k : A 2k → R 2 /G its inverse map. We note ϕ : Z k → A 2k the usual symplectic resolution of A 2k , see Section 3. Corollary 3.7 will now be used to show the very surprising following result:

. By Proposition 2.7, we have

for some smooth functions (α n ) n≥1 and (β n ) n≥1 on R that vanish at 0 together with all their derivatives, and some smooth function α 0 (all unique). We wish to define a function H on Z k by:

(5.2)

where the functions X n , Y n ∈ C ∞ (Z k ) are as in Corollary 3.7 and Z = ϕ * z. This needs some justification.

(1) All terms of both series defining H (see (5.2)) are smooth: (a) Corollary 3.7 states that X n , Y n are smooth functions for all n ≥ 1.

(b) Z = ϕ * z is also a smooth function.

(c) Last, since α n , β n vanish at 0 together with all their derivatives, the quotients α n (z)/z n , β n (z)/z n are smooth functions on R. In view of (b), α n (Z)/Z n , β n (Z)/Z n are also smooth functions.

(2) Both series are convergent, together with all their partial derivatives. We will come back to this point at the end the proof. Now that the existence and the smoothness of the function H is well-established, let us check that

It suffices to check that the pull-back through (ψ k • ϕ) * of each one of the terms of the series defining F are the terms of both series defining H. Consider the following smooth functions on R 3 :

z k y n where the functions x n , y n are as in (3.6). We have:

Z n X n by definition of X n (see Cor. 3.7)

Summing up these identities, we obtain (5.3).

To complete the proof, we need to prove uniform convergence of the series (5.2) defining H, and uniform convergence of all the series obtained by applying a given partial differential operator to all its terms. Let C ⊂ Z k be a compact neighborhood of φ -1 (0). Pulling-back the relation:

. This is a neighborhood of the horizontal line in R 2 , contained into a band B of the form -M < q < M for some M ∈ R. In Proposition 5.7. The map

Proof. The proof of this result is identical to that of Proposition 5.1. We simply use Proposition 2.13 instead of Proposition 2.7, and Corollary 3.13 instead of Corollary 3.7.


Leia Colin

Antoine Chevallier

Satoru Tsugawa

Florian Gacon

Christophe Godin

Virgile Viasnoff

Timothy E Saunders

Olivier Hamant
email: olivier.hamant@ens-lyon.fr.

Cortical tension overrides geometrical cues to orient microtubules in confined protoplasts

In plant cells, cortical microtubules (CMTs) generally control morphogenesis by guiding cellulose synthesis. CMT alignment has been proposed to depend on geometrical cues, with microtubules aligning with the cell long axis in silico and in vitro. Yet, CMTs are usually transverse in vivo, i.e., along predicted maximal tension, which is transverse for cylindrical pressurized vessels. Here, we adapted a microwell setup to test these predictions in a single-cell system. We confined protoplasts laterally to impose a curvature ratio and modulated pressurization through osmotic changes. We find that CMTs can be longitudinal or transverse in wallless protoplasts and that the switch in CMT orientation depends on pressurization. In particular, longitudinal CMTs become transverse when cortical tension increases. This explains the dual behavior of CMTs in planta: CMTs become longitudinal when stress levels become low, while stable transverse CMT alignments in tissues result from their autonomous response to tensile stress fluctuations.

microtubule | cell geometry | cortical tension | protoplast | Arabidopsis T he relative contribution of cell geometry and forces in cyto- skeleton organization has been a long-lasting debate in cell and developmental biology, particularly given the implications for cell division plane determination [START_REF] Sachs | Über die Anordnung der Zellen in jüngsten[END_REF][START_REF] Errera | Sur une condition fondamentale d'e´quilibre des cellules vivantes[END_REF][START_REF] Kny | Über den Einfluß von Zug und Druck auf die Richtung der Scheidewände in sich teilenden Pflanzenzellen[END_REF][START_REF] Guérin | Forces behind plant cell division[END_REF]. In plants, the organization of the cytoskeleton has a direct impact on cell expansion and morphogenesis [START_REF] Baskin | Anisotropic expansion of the plant cell wall[END_REF]. Contrary to animal cells, which are often soft and contractile, plant cells are stiff and pressurized. Their mechanics rely on two parameters: turgor pressure (of osmotic origin) and a stiff cell wall which physically constrains the cell [START_REF] Cosgrove ; Colin | Growth of the plant cell wall[END_REF]. Cellulose microfibrils are the main load-bearing components of the wall. Their deposition occurs at the plasma membrane through the activity of cellulose synthases. While the trajectory of cellulose synthase can be biased by preexisting cellulose microfibrils [START_REF] Chan | Interaction between autonomous and microtubule guidance systems controls cellulose synthase trajectories[END_REF], the deposition of new cellulose microfibrils is usually guided by cortical microtubules (CMTs) [START_REF] Paredez | Visualization of cellulose synthase demonstrates functional association with microtubules[END_REF] In silico simulations of microtubule self-organization suggest that, because of their high persistence length, microtubules should align along the cell long axis [START_REF] Mirabet | The self-organization of plant microtubules inside the cell volume yields their cortical localization, stable alignment, and sensitivity to external cues[END_REF]. Consistently, microtubules growing in confined environments in vitro (microwells) tend to align with the longitudinal axis of the well [START_REF] Lagomarsino | Microtubule organization in three-dimensional confined geometries: Evaluating the role of elasticity through a combined in vitro and modeling approach[END_REF]. More recently, wallless plant cells, also called protoplasts, have been confined in agar microwells and hyperosmotic conditions, and CMTs were found to align with the long axis of the protoplast [START_REF] Durand-Smet | Cytoskeletal organization in isolated plant cells under geometry control[END_REF]. Although such alignments can be observed in nongrowing plant cells in vivo, CMTs are usually transverse in growing cells in vivo [START_REF] Baskin | Anisotropic expansion of the plant cell wall[END_REF]. To explain this discrepancy, additional factors thus need to be considered.

First, molecular factors at cell edges, like CLASP proteins, could allow CMTs to bend as they grow and, thus, bias the final orientation of the CMT arrays [START_REF] Ambrose | A CLASP-modulated cell edge barrier mechanism drives cell-wide cortical microtubule organization in Arabidopsis[END_REF]. However, microtubules can have transverse and longitudinal orientations in adjacent cells at the shoot apical meristem [START_REF] Burian | A correlative microscopy approach relates microtubule behaviour, local organ geometry, and cell growth at the Arabidopsis shoot apical meristem[END_REF]. Furthermore, microtubules can rapidly shift from transverse to longitudinal orientation upon light-induced growth arrest without any change in cell geometry [START_REF] Lindeboom | A mechanism for reorientation of cortical microtubule arrays driven by microtubule severing[END_REF]. Last, local ablations can induce reorientations of CMTs, without major change in cell geometry [START_REF] Hamant | Developmental patterning by mechanical signals in Arabidopsis[END_REF]. Therefore, additional factors must be involved, e.g., to modulate CLASP localization or CMT dynamics. A three-dimensional (3D) computational model managed to recapitulate CMT behavior in the first stages of Arabidopsis embryo development by including CLASP-dependent microtubule stability at cell edges and auxin regulation of microtubule dynamics [START_REF] Chakrabortty | A plausible microtubule-based mechanism for cell division orientation in plant embryogenesis[END_REF].

Although molecular regulators at cell edges are likely to play a key role in CMT orientation, the transverse orientation of CMTs could also be explained without a contribution of cell edges. Indeed, CMTs may indirectly align with maximal tensile stress direction in the cell wall, i.e., the transverse direction for a pressurized elongated cylinder [START_REF] Green | A mechanism for the origin of specifically oriented textures in development with special reference to Nitella wall texture[END_REF][START_REF] Hamant | Developmental patterning by mechanical signals in Arabidopsis[END_REF]. This stress-based scenario is supported by the frequent correlation between predicted tensile stress direction and CMT orientation in several tissues, with or without mechanical perturbations [START_REF] Hamant | Are microtubules tension sensors?[END_REF]. It also provides a mechanism for plant cells to resist tensile stress, as in Wolff's law where bones remodel themselves to resist the load. However, the exact pattern of stress remains debatable, because of the additional complexity brought about by the tissue context. Furthermore, how the CMTs sense the tensile status of the cell wall remains an open problem. In a recent perspective, we proposed that microtubules may be able to autonomously align with changes

Significance

In plants, microtubules largely determine the direction of cell expansion and the orientation of cell division planes. However, what processes orient the microtubules has remained debated. Here, we used microfabricated wells to confine and deform wallless plant cells in a controlled way to analyze the response of microtubules to cell geometry and surface tension. We demonstrate that microtubules align with cell geometry by default, whereas when surface tension increases (e.g. when turgor pressure increases), they align with the direction of maximal tension. Not only does this explain many observations in plant tissues, but it also provides a simple mechanism at the core of plant morphogenesis, in which microtubules can spontaneously align with tension, in a typical self-organized system.

in maximal tension [START_REF] Hamant | Are microtubules tension sensors?[END_REF]. Here, we adapted a single-cell microwellbased method on wallless Arabidopsis protoplast to test this prediction.

Here, we find that CMT orientation depends on curvature anisotropy. CMTs shift their orientation from curvature-derived steric constraints to curvature-derived maximal tension depending on pressurization. Because this behavior does not rely on the presence of a cell wall, this also provides further support for an autonomous microtubule mechanosensing mechanism.

Results

Cortical Microtubules Primarily Align in the Transverse Direction in Confined and Pressurized Protoplasts. CMTs are sensitive to curvature-derived steric constraints due to their high bending stiffness. In elongated cells, they are expected to align longitudinally, i.e., along the direction of lower curvature (or the flattest direction) (Fig. 1A). For a pressure vessel, tension and curvature are related by Laplace-Young law, ΔP = 2γ/R, with P for pressure, γ for surface tension, and R for radius of the vessel. In short, at equilibrium, maximal tension is in the direction of minimal radius or maximal curvature (SI Appendix, Text and Fig. S1). In an elongated pressure vessel like a plant cell, maximal tension is transverse to the long axis (Fig. 1A). If CMTs are sensitive to curvature-derived tension, they are predicted to align transversely. To test the relative contribution of these opposing curvature-derived cues, we took inspiration from a method initially developed to mimic the cellular microenvironment for animal cells [START_REF] Gao | Artificial microniche array with spatially structured biochemical cues[END_REF]. To be able to rapidly deform the cell cortex and obtain simple global geometries, we used wallless plant cells (spherical protoplasts) and confined them in rectangular wells. We then modified their pressurization using solutions of different osmolarity (Fig. 1B; see Materials and Methods; SI Appendix, Fig. S2A). Note that there was no detectable wall synthesis within the timeframe of the experiment (SI Appendix, Fig. S3).

To confirm the shape and pressurized status of protoplasts, we monitored protoplast deformation, using the pUbQ10::LTI6b-TdTomato membrane marker reporter line (Fig. 1 C-F). When viewed from the bottom, confined protoplasts appeared elongated in rectangular wells. Orthogonal sections and 3D reconstruction confirmed their pressurized status, with hemispherical top and bottom membranes (Fig. 1 C-F and Movies S1 and S2). The protoplasts only appeared to be geometrically deformed laterally, as if they were confined between two plates. Importantly, such confinement did not induce sharp edges, as the protoplast contour remained almost tangential to the microwell walls (Fig. 1 C-F and SI Appendix, Fig. S2B). Thus, the cell-edge contribution to CMT behavior is not explicitly studied in this work. Using the measured aspect-ratio (length/width between 1.13 and 1.17), we confirmed that predicted maximal tensile stress is transverse in confined protoplasts (Fig. 1G and SI Appendix, Text).

To follow microtubule dynamics, we used the microtubule reporter line p35S::GFP-MBD (Fig. 1H) (Microtubule Binding Domain of rat MAP4; ref. [START_REF] Hamant | Developmental patterning by mechanical signals in Arabidopsis[END_REF]. Note that protoplasts were still alive after 17 h in the wells, confirming that this environment is not toxic for the protoplast, as already shown for animal cells (Fig. 1I). To test whether CMTs exhibit a bias in their orientation in these conditions, we used 15 × 20 μm microwells, and we observed the GFP-MBD signal 2 h after transfer from 600 mOsmol/ L to 280 mOsmol/L mannitol. Qualitatively, we observed a wide diversity of CMT behaviors in confined and pressurized protoplasts (Fig. 1J). To quantify this pattern, we measured the orientation of microtubules on small regions of interest (ROIs), thus providing a local quantification of each microtubule segment, using the subcellular fibrillar tool (SFT hereafter, Fig. 1K; ref. [START_REF] Tsugawa | Extracting subcellular fibrillar alignment with error estimation: Application to microtubules[END_REF]. Typically, we obtained a distribution of orientations per protoplast. This method revealed that CMTs in confined and pressurized protoplasts indeed exhibit a bias in the transverse orientation (n = 126 protoplasts, P < 10 -3 [Rayleigh's test for uniformity], Fig. 1L). This conclusion was independent on the analysis method, as a similar trend was obtained using the ImageJ plugin Ori-entationJ (SI Appendix, Fig. S4).

We also checked whether local curvature could affect the CMT orientation globally. To do so, we analyzed a subset of confined protoplasts that were large enough to touch four walls of the wells, instead of just two, when confined (SI Appendix, Fig. S2C). Our analysis revealed the same transverse bias in CMT orientation in this population of protoplasts, consistent with a response of CMTs to global curvature anisotropy in our setup (SI Appendix, Fig. S2D). Altogether, this suggests that CMTs can align with maximal tension. This also suggests that such alignment does not depend on the cell wall.

Challenging the Transverse Orientation of CMTs in Confined and

Pressurized Protoplasts. To challenge our finding, we performed three additional tests. First, in principle, the transverse bias could be caused by the adhesion of the protoplast to the well edges. Yet, the tangential curvature suggests very weak adhesion to the well sides (Fig. 1D and SI Appendix, Fig. S2B). To check this formally, we pretreated the microwells with pluronic acid, which is classically used in cell cultures to inhibit surface-tissue adhesion [START_REF] Li | Extracellular matrix scaffolding guides lumen elongation by inducing anisotropic intercellular mechanical tension[END_REF]. In these conditions, we still observed a transverse bias in CMT orientation (n = 29 protoplasts, P < 10 -3 [Rayleigh's test for uniformity], Fig. 2 A andB).

Second, if the bias in orientation truly depends on curvature anisotropy, it should be lost in square-shaped wells. To test this, we used 14 × 14 μm microwells. In these wells, as expected, we did not observe any significant bias in CMT orientation (n = 74 protoplasts, P = 0.11 [Rayleigh's test for uniformity], Fig. 2 C and D).

Last, if CMTs align with the direction of maximal tension at the protoplast cortex, they should do so for any rectangular geometry, with the bias increasing when curvature anisotropy increases, in accordance with the Laplace-Young law. To test that prediction, we used 12 × 40 μm microwells. As expected, we found a significant bias of CMT orientation in the transverse direction (n = 99 protoplasts, P < 10 -3 [Rayleigh's test for uniformity], Fig. 2 E andF). Interestingly, the bias in the transverse direction was indeed stronger in the 12 × 40 μm microwells than in the 15 × 20 μm microwells (P < 10 -3 , null hypothesis: mean angle of 12 × 40 > mean angle of 15 × 20) permutation test). Altogether, this supports a scenario in which CMTs orient with predicted maximal tension in confined and pressurized protoplasts.

Pressurization Determines CMT Orientation in Confined Microwells. If CMTs align with the pattern of tension for a given geometry, this alignment should become less pronounced when tension decreases. Recently, the analysis of protoplasts confined in agar wells in hyperosmotic conditions has shown that CMTs align with the long axis of the protoplast, consistent with this prediction [START_REF] Durand-Smet | Cytoskeletal organization in isolated plant cells under geometry control[END_REF]. To test this in our setup, protoplasts were introduced in 12 × 40 μm microwells in a hypertonic solution (600 mOsmol/L mannitol) as previously, and then the protoplasts were exposed to an even more hypertonic solution (800 mOsmol/L mannitol, compared with 280 mOsmol/L mannitol). As expected, because of reduced pressurization, protoplasts were more deformable (Fig. 3A and SI Appendix, Fig. S4). In these conditions, the transverse CMT bias was lost in all replicates 2 h after the addition of the hypertonic solution. Instead, a global bias in the longitudinal distribution was observed (n = 97 protoplasts, P < 10 -3 [Rayleigh's test for uniformity], Fig. 3 B andC).

Consistent with their reduced pressure-dependent stiffness and increased deformability, several protoplasts exhibited a very elongated shape (Fig. 3B). This increased range of shapes allowed us to check the correlation between protoplast aspect-ratio and CMT orientation. We first calculated a mean CMT orientation per protoplast, with the OrientationJ, Dominant Direction plugin, and second, we calculated a deformation ratio (Materials and Methods). We found that average CMT orientation is longitudinal for the most elongated protoplasts (Fig. 3 D andF). Yet, such a bias was significantly reduced in protoplasts with an aspect-ratio below 2.5; in such protoplasts, CMT orientation was more isotropic (p <2.5 = 0.052 vs. p >2.5 = 0.020 [Rayleigh's test for uniformity], Fig. 3 D andE). Altogether, this confirms that the transverse orientation of CMTs depends on the degree of protoplast pressurization. Our results also suggest that CMTs align with geometry when surface tension is low, i.e., the long axis of the protoplasts.

Microtubule Orientation Respond to Deviations in Cortical Tension.

So far, our results are consistent with CMTs aligning with predicted maximal tension at the protoplast cortex. Results obtained in vitro suggest that the microtubule response to tension is transient [START_REF] Inoue | Sensing surface mechanical deformation using active probes driven by motor proteins[END_REF]. Taking advantage of time-lapse imaging within our setup, we next investigated the transient nature of the microtubule response in vivo.

When observing a confined protoplast over a long time, changes in CMT orientations could be detected, further confirming that the protoplast is alive. Note that CMT reorientations appeared slow when compared to in vivo data. In the example shown in SI Appendix, Fig. S5 A and B, subtle changes were observed (see also Movie S3): Upon transfer from 800 mOsomol/L to 400 mOsmol/ L, the initial bias in the longitudinal orientation was progressively lost in favor of a more uniform, and even transverse, orientation and then back to longitudinal. As shown in SI Appendix, Fig. S5 C-E, CMTs could completely change their orientations while being confined over long time periods. Within the same geometry, protoplasts can thus acquire new CMT orientation depending on pressure.

The use of 800 mOsm/L mannitol conditions can generate highly deformed protoplasts and is perfectly adapted to show the contribution of cell geometry, as an aspect-ratio superior to 2.5 was required to induce a significant bias in CMT orientation along the long axis of the protoplast (SI Appendix, Fig. 3 D and E, and ref . 11). However, such high osmolarity may also induce artifacts, or even trigger slowed life mechanisms, as in dormant seeds for instance [START_REF] Yan | Microtubule self-organisation during seed germination in Arabidopsis[END_REF]. In fact, high osmolarity has been proposed to slow microtubule dynamics in yeast because of cytoplasmic crowding [START_REF] Molines | Physical properties of the cytoplasm modulate the rates of microtubule growth and shrinkage[END_REF]. We thus focused on the comparison between 600 and 280 mOsm/L mannitol, even if, in these conditions, we could not induce such high aspect-ratios. Long-term kinetics showed changes in CMT orientation during the switch from hyperosmotic to hypoosmotic conditions and vice versa (SI Appendix, Figs. S6-S8 and Movies S4-S6). CMT arrays appear to retain their ability to modify orientation when osmotic conditions fluctuate. Strikingly, as we followed CMT behavior in protoplasts being transferred from 600 to 280 mOsmol/L mannitol, we observed new microtubules appearing in the transverse . This is consistent with a scenario in which transverse tension promotes the polymerization of microtubules in that direction.

To test whether CMTs are able to maintain their stress-derived transverse orientation in the long term, without any imposed change in osmolarity, we examined CMT orientations in protoplasts 16 h after the transfer from 600 to 280 mOsmol/L mannitol (Fig. 4B). In such conditions, protoplast shape appeared stable. It is possible that many aspects of the protoplast change in this timeframe (e.g., plasma membrane composition, metabolism), yet transverse curvature remains higher than longitudinal curvature. After 16 h, the transverse CMT bias was lost in 12 × 40 μm microwells (n = 69 protoplasts, P <10 -3 (null hypothesis: mean angle in 12 × 40 μm microwells at 16 h < mean angle in 12 × 40 μm microwells at 2 h), Fig. 4 C andD). This is consistent with a scenario in which CMTs align with changes in maximal tension and usually return to their geometry-derived orientation if no more changes in tension direction occur.

Discussion

We have demonstrated that CMTs can change their orientation in confined protoplasts without cell walls or sharp edges. This provides a hierarchical scenario: 1) CMTs align with the long axis of the cell by default, because of their high flexural rigidity (also consistent with refs. 9 and 11); 2) tension promotes alignment of CMTs with maximal tension, which is transverse for an elongated cell; and 3) although cell edges are dispensable for CMT self-organization and reorientation, the associated molecular regulators can modulate the final CMT orientation in vivo.

The microtubule response to mechanical stress is supported by many studies from different groups, on different plant parts, and in response to different mechanical perturbations [START_REF] Green | A mechanism for the origin of specifically oriented textures in development with special reference to Nitella wall texture[END_REF][START_REF] Hejnowicz | Tensile tissue stress affects the orientation of cortical microtubules in the epidermis of sunflower hypocotyl[END_REF][START_REF] Hamant | Developmental patterning by mechanical signals in Arabidopsis[END_REF][START_REF] Robinson | Global compression reorients cortical microtubules in Arabidopsis hypocotyl epidermis and promotes growth[END_REF]. Yet, most of the obtained results so far relate to CMT behavior in tissues, in which the exact stress pattern can be difficult to predict. Here, using a confined protoplast protocol, we not only demonstrate that a full cell wall is not required for the microtubule response to stress, but we also show that microtubules are able to perceive both cell geometry and cortical tension, depending on a threshold in cortical tension. These results are consistent with the proposal that microtubules may spontaneously orient with maximal tension [START_REF] Hamant | Are microtubules tension sensors?[END_REF], building on in vitro data on stretched microtubules [START_REF] Franck | Tension applied through the Dam1 complex promotes microtubule elongation providing a direct mechanism for length control in mitosis[END_REF] and on the predicted weakness of the geometrical cue for CMT self-organization [START_REF] Mirabet | The self-organization of plant microtubules inside the cell volume yields their cortical localization, stable alignment, and sensitivity to external cues[END_REF].

Here, we have focused on the role of cell geometry and cortical tension in determining CMT behavior. Yet, cell edges are also important players to explain CMT orientations in vivo [START_REF] Ambrose | A CLASP-modulated cell edge barrier mechanism drives cell-wide cortical microtubule organization in Arabidopsis[END_REF][START_REF] Chakrabortty | A plausible microtubule-based mechanism for cell division orientation in plant embryogenesis[END_REF]. Interestingly, the three contributions-cell geometry, cortical tension, cell edges-are highly interconnected. In particular, curvature depends on cell geometry, prescribes tension stress pattern, and can locally become an obstacle for CMTs at cell edges. One avenue for future studies will be to check whether cell edge components are sensitive to shape or tension and, in turn, whether changes in tension can modify global shape and cell edge geometry. At this stage, we also cannot completely exclude the possibility that specific regulators may be recruited at cell edges, independent from cell geometry and tension.

The MBD construct induces known artifacts, such as extra microtubule bundling and twisted organs [START_REF] Celler | Microtubules in plant cells: Strategies and methods for immunofluorescence, transmission electron microscopy, and live cell imaging[END_REF]. One cause of our observed transverse CMT bias in pressurized protoplasts may be the MBD domain. However, longitudinal CMTs were observed in protoplasts confined in agar wells using the same GFP-MBD line [START_REF] Durand-Smet | Cytoskeletal organization in isolated plant cells under geometry control[END_REF]. Furthermore, as protoplast curvature and aspect-ratio do not change over 16 h, and CMT orientation can change, the MBD domain does not prevent CMT reorientation, independent of curvature or protoplast geometry.

More interestingly, our results support a scenario in which CMTs respond to changes in tension, rather than the pattern of tension at steady state. This observation is consistent with in vitro data on transiently stretched microtubule networks [START_REF] Inoue | Sensing surface mechanical deformation using active probes driven by motor proteins[END_REF]. This allows us to revisit the role of cell geometry in defining microtubule orientation. As our system imposes a stable geometry, it allows us to reveal that the microtubule response to stress requires fluctuations in shape and stress pattern (cycles of pressurization and depressurization in our experiments). Fluctuations are often viewed as extra noise on top of an average situation. Here, we provide clear evidence that, instead, fluctuations allow cells to perceive stress, to which they respond. In other words, the average orientation of microtubules in cells seem to be the result of stress fluctuation rather than a product of cell geometry alone.

Is there evidence of such stress fluctuations in planta? Low frequency fluctuations could easily be explained by circadian turgor rhythms. Nastic movement of leaves and stems further suggest that variations in the mechanical status of the plant are subcircadian [START_REF] Derr | Fluttering of growing leaves as a way to reach flatness: Experimental evidence on Persea americana[END_REF]. The analysis of hypocotyl growth also suggests that shorter period fluctuations exist in Arabidopsis seedlings, from which our protoplasts were extracted [START_REF] Takatani | Microtubule response to tensile stress is curbed by NEK6 to buffer growth variation in the Arabidopsis hypocotyl[END_REF]. Last, the calcium waves at the shoot apical meristem, every 10-30 min, may indicate that mechanical fluctuations in plant tissues are underestimated [START_REF] Li | Calcium signals are necessary to establish auxin transporter polarity in a plant stem cell niche[END_REF]. These different systems suggest that such fluctuations may be the norm, rather than the exception.

Our setup opens avenues for other future research questions. In particular, such a single cell system offers the opportunity to revisit the molecular factors involved in the microtubule-cell wall continuum, which remains largely ill-described. This may also prove useful for the analysis of other cortical factors like auxin transporters. Our protocol may fill a missing link in the study of plant mechanotransduction, playing a similar role as in vitro studies of fibroblasts in animal mechanotransduction studies, i.e., to provide a context in which mechanical parameters are better controlled and in which conclusions can then be tested in a multicellular context.

Materials and Methods

Experiments were conducted in two laboratories (Mechanobiology Institute [MBI] in Singapore and Laboratoire de Reproduction et Développement des Plantes [RDP] in Lyon, France).

Plant Growth Conditions (MBI and RDP). At MBI, Arabidopsis thaliana plants were grown at room temperature in long-day conditions (16 h/8 h light/dark cycle), on medium containing 11.94 g/L of Arabidopsis medium (Duchefa) supplemented with 2 mM Ca(NO 3 ) 2 , pH 5.8 for 7-15 d. At RDP, A. thaliana seeds were sown on MS (Murashige and Skoog) medium and grown in vitro in long-day conditions (16 h/8 h light/dark period) for 7-15 d. The microtubule reporter line (p35S::GFP-MBD) and the plasma membrane reporter line (pUbq10::LTI6b-TdTomato) were previously described [START_REF] Hamant | Developmental patterning by mechanical signals in Arabidopsis[END_REF].

Protoplast Generation (MBI and RDP). Protoplasts were obtained from 7-to 15-d-old seedlings. Cell wall was digested through seedling immersion in 1.1 mL of solution A (D-Mannitol [600 mOsmol/L], CaCl 2 [2 mM], MgCl 2 [2 mM], L-ascorbic acid [1 mM], BSA [3.5 mg/mL], MES [10 mM at pH 5.5]) supplemented with wall digesting enzymes (Cellulysine [Calbiochem] [17 mg/mL], Cellulase RS [17 mg/mL], Pectolyase Y23 [0.4 mg/mL]). Ca. 10 seedlings were incubated in the enzymatic solution. At RDP, the incubation lasted for 2 h under slow agitation on a rotating wheel (12 rpm) at room temperature (i.e., ca. 21-25 °C). At MBI, the incubation lasted overnight under agitation (40 rpm) at room temperature (i.e., ca. 20 °C). After incubation, the seedling suspension was aspired up and down five or six times using a 1-mL pipette to break the tissue and free the protoplasts. Samples were centrifuged at 1,000 rpm for 4 min, the supernatant was discarded and protoplasts were washed for 1 min with 1 mL of solution A. A second centrifugation (1,000 rpm, 4 min) was performed, and most of the supernatant was discarded, keeping ca. 200 μL of a suspension of protoplasts in solution A. Protoplasts were then placed at 4 °C for 30 min. We did not select for a specific type of protoplasts (e.g., with different size or different number of chloroplasts, SI Appendix, Fig. S2A) to ensure that our observations were robust with respect to the diversity of protoplasts. The digestion of the cell wall was confirmed by the absence of calcofluor (cellulose dye) and propidium iodide (pectin dye) staining (SI Appendix, Fig. S3).

Microwell Fabrication (MBI).

A PDMS (90% silicone elastomer base, Sylgard + 10% silicone elastomer curing agent, Sylgard) stamp was first fabricated using lithography (SU8 method) and silanized (Trichloro(1H,1H,2H,2H-perfluorooctyl)-silane, Aldrich) for 2 h under vacuum. PDMS was then poured on the mold and degassed 15 min at 10 mBar before curing 2 h at 80 °C. After removing, the stamp was cut into pieces of about 1 cm × 0.5 cm and silanized 2 h under vacuum. Microwells were obtained by pressing PDMS molds into a drop of ultraviolet (UV) curable monomers of Norland Optical Adhesive 73 (NOA73), beforehand deposited in ibidi dish (ibidi glass bottom dish 35 mm). At MBI, UV curing was performed with UV-KUB 9 (Kloé SA, 200 mW/cm 2 ) for 30 s. At RDP, UV curing was performed with UV-KUB 2 (Kloé SA, 40 mW/cm 2 ) for 3 min. Ibidi dish were filled with 3 mL of water, and the PDMS mold was carefully removed with tweezers to avoid formation of air bubbles in the microwells. For a more detailed protocol, see ref. [START_REF] Gao | Artificial microniche array with spatially structured biochemical cues[END_REF].

Surface Passivation with Pluronic Acid. To test the contribution of protoplast adhesion to CMT behavior, microwells were treated for 30 min at 37 °C with 2% (vol/vol) pluronic acid (a nonionic surfactant). The negative impact of pluronic acid on protoplast adhesion was confirmed with the following control experiment: We added 300 μL from the same protoplast's suspension on Ibidi dish either pretreated for 30 min at 37 °C with 2% (vol/vol) pluronic acid or incubated with water in the same conditions. Protoplasts were let to sediment for 2 h. Then, the Ibidis were tilted in a similar way (being attached to the same support) to let the protoplast suspension flow down and the number of remaining protoplasts (i.e., still attached to the Ibid's coverslip) was counted. Many protoplasts were still attached in the control (water) Ibidi (n = 6,216 protoplasts), whereas, in the pluronic-treated Ibidi, fewer protoplasts were present (n = 2,003 protoplasts) and most of them were located at the coverslip-plastic edge of the Ibidi. This confirms that the pluronic acid surfactant hinders adhesion of Arabidopsis protoplasts.

Protoplast Confinement (MBI and RDP). Microwells in Ibidi dishes were washed with solution A before protoplast confinement. After removing solution A from the Ibidi dish, 50 μL of protoplast solution was deposited on microwells and let to sediment at room temperature for 5 min. The Ibidi dish was then centrifuged at 300 × g for 3 min (using a plate holder). The step of sedimentation increases the number of confined protoplasts into the microwells. After centrifugation, 3 mL of solution B (identical to solution A except for D-Mannitol [280 m to 800 mOsmol/L]) were added to induce hypoosmotic (D-Mannitol 280 mOsmol/L), mild hypoosmotic (D-Mannitol 400 mOsmol/L) or hyperosmotic (D-Mannitol 800 mOsmol/L) shock.

Microscopy and Imaging (MBI and RDP).

For the experiments reported in Figs. 123, live imaging started 2 h after the addition of the hypotonic or hypertonic solution and stopped ∼2 h later. For the experiments reported in Fig. 4, kinetics was performed immediately after the addition of solution B, and late time points (t = 16 h) were also considered.

At MBI, images were acquired on a spinning-disk confocal microscope (Yokogawa W1 on Nikon T1E) and an inverted confocal LSM710 microscope (Zeiss) using 100× lens (oil immersion, N.A.= 1.45 [Nikon] and 1.46 [Zeiss]). At RDP, images were acquired on LSM800 inverted confocal microscope (Zeiss) using 63x lens (oil immersion, N.A. = 1.4).

Time-lapse experiments: Protoplasts were prepared as described above and were confined in 12 × 40μm microwells in hypertonic solution (Movie S1: 800 mOsmol/L mannitol, and Movies S2-S4: 600 mOsmol/L mannitol). The Ibidi dishes were installed under the microscope, and specific microwell positions on plate were recorded. Movie S3 was performed on widefield inverted microscope (Zeiss) using 63× lens (oil immersion, N.A. = 1.4) and Movies S1, S2, and S4-S6 were performed on LSM800 inverted confocal (Zeiss), using 63× lens (oil immersion, N.A. = 1.4). For Movie S3, the 800 mOsmol/L mannitol solution was replaced by a 400 mOsmol/L mannitol solution, and imaging started immediately after the addition of the 400 mOsmol/L mannitol solution. Images were taken every 30 min for 8 h. For Movies S4-S6, imaging started in 600 mOsmol mannitol solution, and at t = 0, the 600 mOsm/L mannitol solution was replaced by the 280 mOsm/L mannitol solution. Further changes in solutions were performed every 2 h, as shown in the figures. Images were taken every 20 min until protoplast's death.

Cell wall staining: p35S::GFP-MBD seedlings (positive control for the dyes) and p35S::GFP-MBD protoplasts (obtained after 2 h of enzymatic digestion at RDP) were immersed in solution A containing either 0.3 mg/mL of propidium iodide, which stains demethyl-esterified pectins [START_REF] Rounds | Propidium iodide competes with Ca(2+) to label pectin in pollen tubes and Arabidopsis root hairs[END_REF], for 10 min or 0.01% of calcofluor, which stains cellulose microfibrils [START_REF] Herth | The fluorochrome, calcofluor white, binds oriented to structural polysaccharide fibrils[END_REF], for 15 min. Seedlings and protoplasts were then imaged on LSM800 inverted confocal microscope as described above. The absence of cell wall was tested immediately after digestion and 22 h after digestion.

Image Processing. The first five slices of the confocal stack were used to make a maximal projection. This removed most of the cytoplasmic signal in the images, including cytoplasmic microtubules. Then, all projections corresponding to one technical replicate were assembled as a stack, projections were aligned with the ImageJ plugin AlignSlice, and all images were cropped to keep the central part of the protoplast (to avoid artifacts on the sides of the cells due to the projection). Finally, a Gaussian blur (radius: 2.00 pixels) was performed on the images. Then, microtubule orientation analysis was performed using SFT [START_REF] Tsugawa | Extracting subcellular fibrillar alignment with error estimation: Application to microtubules[END_REF]. This method is based on nematic tensors and provide orientation and anisotropy of the microtubule arrays. Each protoplast is divided in ROIs, and the orientation of each ROI is then analyzed. On average, this represented 394 ROIs (i.e., CMT fragment angles) per protoplast. The orientation is measured from an imaginary horizontal line, and the angle between this line and the ROIs is calculated. A polar histogram was constructed, displaying the global probability for each orientation to belong to a class of angle, ranging from 0°to 90°. Thus, 10 classes of angles of 9°w ere formed, from 0°-9°to 81°-90°. We used the probability density function where (probability) = sum of (probability density function) × (bin width) over bins. When 10 classes of angles (0-90°) are defined, bin width is (π/2)/10 ∼ 0.157. In the figures, we plotted the probability in each bin, which is (probability density function) × (bin width). The probability in each bin is about 0.1, and the sum of all histogram area is 1 with 10 bins.

To confirm the results obtained with SFT, the plugin OrientationJ, Distribution in the ImageJ software was used [START_REF] Rezakhaniha | Experimental investigation of collagen waviness and orientation in the arterial adventitia using confocal laser scanning microscopy[END_REF] on the images previously assembled (with the same crop and Gaussian blur). As in SFT, protoplasts were divided in several ROIs (one ROI = 8 pixels). The orientation of CMTs in each ROI was measured. For each technical replicate, all of the orientations were then distributed into six classes of angles, ranging from 0°to 90°, each class measuring 15°. Data were normalized and histograms were constructed, displaying the proportion of ROIs in each class of angles. Both of these methods (i.e., SFT and OrientationJ plugin) allow a very local analysis of each microtubule orientation.

Correlation between protoplast deformation and CMT orientation was obtained with the OrientationJ, Dominant direction plugin. This plugin measures the mean orientation of microtubule in each cell. The cell deformation was estimated by drawing an ellipse around each protoplast with the ImageJ software and calculating a ratio between the major and the minor axis of that ellipse. The scatterplot represents the average orientation of the microtubule for each protoplast according to their aspect-ratio after confinement.

Statistical Analysis. Statistical tests were conducted with R software. Statistical analysis of the results (sample size [n], P value, type of test) is presented in Results.

Data Availability. All study data are included in the article and supporting information.

Fig. 1 .

 1 Fig. 1. Cortical microtubule response to protoplast confinement. (A) Main hypothesis: In a confined protoplast, transverse curvature is higher than longitudinal curvature, resulting in a competition between steric and mechanical cues; lower curvature would accommodate microtubule high bending stiffness, while high curvature would prescribe high tension to which microtubules may align with. (B) Schematic summary of the protocol used to confine and image protoplasts in microwells. (C) pUbQ1O::LTI6b-TdTomato (membrane marker) protoplast. Transmitted light channel (Left) and TdTomato fluorescence channel (Right). (D) pUBQ10::LTI6b-tdTomato images of a deformed and turgid (280 mOsmol/L mannitol) protoplast. Transmitted light channel (Left) and fluorescence of TdTomato channel (Right). Orthogonal views are presented at Right. (E) FEM representation of an unconfined protoplast (sphere, Upper) and confined protoplast (Lower). (F) Three-dimensional reconstruction from confocal stacks for an unconfined (Upper) and confined (Lower) protoplast expressing pUBQ10::LTI6b-tdTomato (280 mOsmol/L mannitol). (G) FEM simulation for a confined ellipsoid with an aspect-ratio (length/width) of 1.15, as measured in our experiments (1.13-1.17). Maximal tensile stress is predicted to be transverse, following Laplace-Young law. (H) p35S::GFP-MBD protoplast. (Left) Transmitted light channel. (Right) GFP fluorescence channel. Red, autofluorescence of chloroplasts; green, microtubule signal. (I, Left) Transmitted light channel and GFP fluorescence channel, 2 h after protoplast's insertion in the microwell. (I, Right) Transmitted light channel and GFP fluorescence channel, 16 h after protoplast's insertion in the microwell. Two different protoplasts are shown. (J) Examples of p35S::GFP-MBD protoplast images, illustrating the diversity of microtubule orientations in 15 × 20 μm microwells in pressurized (280 mOsmol/L mannitol) and confined protoplasts. Protoplasts were imaged 2 h after their transfer into hypotonic solution. (K) Analysis of CMT orientation with SFT. The cropped zone is delineated with the dotted red lines. The orientation of CMTs in each ROI is color coded. Polar histograms represent the CMT angle distribution for the protoplast. Each bar corresponds to an angle range of 9°. (L) Polar histograms of the probability of CMT orientations (classes of 9°angles) in all protoplasts, confined in 15 × 20 μm microwells, imaged 2 h after their transfer in 280 mOsmol/L mannitol (n = 126 protoplasts, three independent replicates). (Scale bars, 5 μm.)

Fig. 2 .

 2 Fig. 2. Challenging the transverse orientation of CMTs in confined and pressurized protoplasts. (A) Schematic representation of the stress pattern in 15 × 20 μm microwell. p35S::GFP-MBD pressurized protoplast, confined in 15 × 20 μm microwell, was pretreated with pluronic acid to reduce protoplasts adhesion to the well sides, and in 280 mOsmol/L mannitol. (Lower Left)Transmitted light channel. (Lower Right) GFP fluorescence channel. (B) Polar histograms representing the probability of CMT orientations (classes of 9°angles) in protoplasts confined in 15 × 20 μm passivated microwells and imaged 2 h after their transfer to 280 mOsmol/L mannitol (n = 29 protoplasts). (C) Schematic representation of protoplast curvature in 14 × 14 μm square-shaped wells. p35S::GFP-MBD protoplast was confined in 14 × 14 μm microwell in 280 mOsmol/L mannitol. (Lower Left)Transmitted light channel. (Lower Right) GFP fluorescence channel. (D) Polar histograms representing the probability of CMT orientations (classes of 9°angles) in protoplasts confined in 14 × 14 μm microwells and imaged 2 h after their transfer to 280 mOsmol/L mannitol (n = 74 protoplasts, three independent replicates). (E) Schematic representation of protoplast curvature in 12 × 40 μm wells. p35S::GFP-MBD protoplast was confined in 12 × 40 μm microwell in 280 mOsmol/L mannitol. (Lower Left) Transmitted light channel. (Lower Right) GFP fluorescence channel. (F) Polar histograms representing the probability of CMT orientations (classes of 9°angles) in protoplasts confined in 12 × 40 μm microwells, imaged 2 h after their transfer to 280 mOsmol/L mannitol (n = 99 protoplasts, four independent replicates). (Scale bars, 5 μm.)

Fig. 3 .

 3 Fig. 3. Pressurization determines CMT orientation in confined microwells. (A) p35S::GFP-MBD-confined protoplast in 12 × 40 μm microwell in hypertonic solution (800 mOsmol/L mannitol). (Left) Schematic representation. (Center) Transmitted light channel. (Right) GFP fluorescence channel. (B) Examples of highly deformed p35S::GFP-MBD protoplasts in 800 mOsmol/L mannitol. (C ) Polar histograms representing the probability of CMT orientations (classes of 9°angles) in protoplasts in hypertonic solution and imaged 2 h after their transfer to 800 mOsmol/L mannitol (n = 102 protoplasts, six independent replicates). (D) Scatterplot representing the dominant orientation of CMTs in each protoplast (OrientationJ, Dominant direction plugin analysis) according to the protoplast aspect-ratio after confinement (each dot represents a protoplast, n = 97 protoplasts, six independent replicates). (E and F ) CMT orientations (classes of 15°angles) for protoplasts with an aspect ratio inferior to 2.5 (n = 90 protoplasts) (E ) and superior to 2.5 (n = 12 protoplasts) (F ). (Scale bars, 5 μm.)

Fig. 4 .

 4 Fig. 4. Pressurization determines CMT orientation in confined microwells. (A) Time-lapse imaging of a p35S::GFP-MBD-confined protoplast in a 12 × 40 μm microwell as it is transferred from 600 to 280 mOsmol/L mannitol (at t = 0 min). New transverse CMTs progressively appear upon pressurization (red arrows). (B) Schematic representation of the experiment: Protoplasts were isolated in 600 mOsmol/L mannitol and then confined 12 × 40 μm microwells in the presence of 280 mOsmol/L mannitol for 16 h. (C ) p35S::GFP-MBD-confined protoplast in a 12 × 40 μm microwell, imaged 16 h after transfer to 280 mOsmol/L mannitol. (Left) Transmitted light channel. (Right) GFP fluorescence channel. (D) Polar histograms representing the probability of CMT orientations (classes of 9°angles) in protoplasts in 12 × 40 μm microwells and imaged 16 h after transfer to 280 mOsmol/L mannitol (n = 72 protoplasts, three independent replicates). (Scale bars, 5 μm.)

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of 8 | www.pnas.org/cgi/doi/10.1073/pnas.2008895117 Colin et al. Downloaded by guest on December 15, 2020

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ACKNOWLEDGMENTS. We thank the MBI, Singapore, for hosting A.C. and O.H. for 6 months, where the original experiments were performed. We also thank our colleagues for constructive discussions about this project and for critical reading of the manuscript. We thank Platim for help with imaging and Gianluca Grenci for the design of the microwell casts at MBI. This work was supported by European Research Council Grant 615739 "MechanoDevo" (to O.H.), Ministry of Education, Culture, Sports, Science and Technology KAKENHI Grant-in-Aid for Scientific Research on Innovative Areas "Plant-Structure Optimization Strategy" GrantJP18H05484) (to S.T.) and MBI seed grant support (to V.V. and T.E.S.).

This article is a PNAS Direct Submission.