Bull. of Yamagata Univ., Nat. Sci., Vol.15, No.3, Feb. 2003€¦ · Bull. of Yamagata Univ., Nat....

Bull. of Yamagata Univ., Nat. Sci., Vol.15, No.3, Feb. 2003

Bijections between topologies∗

Yoshinori ITO† and Fuichi UCHIDA‡

(Received May 21, 2002)

Abstract

A homeomorphism induces a natural bijection between the familyof open sets. We consider the inverse problem and obtain an answerand counter examples.

0 Introduction

Let X and Y be topological spaces, f : X → Y a homeomorphism. Then,f induces a natural bijection f∗ : O(X) → O(Y ), where O( ) denotes thefamily of open sets of the topological space. Indeed, f∗(U) is to be the imagef(U) for any open set U of X. Notice that the bijection f∗ preserves theinclusion relation, that is, U ⊂ V ⇐⇒ f∗(U) ⊂ f∗(V ).

Now we consider the inverse problem : “Which bijection ψ : O(X) →O(Y ) is induced by a homeomorphism ?”

Our answer is the following. “Assume the topologies O(X) and O(Y )satisfy the T1-axiom, that is, each one-point set is closed. If a bijectionψ : O(X) → O(Y ) preserves the inclusion relation, then ψ is induced by ahomeomorphism f : X → Y .”

Next, we will give countably many topologies on the set of natural num-bers such that between any two of them, there is a bijection which preservesthe inclusion relation but it is not induced by any homeomorphism.

∗2000 AMS Mathematics Subject Classification. Primary 54A10†Department of Mathematical Sciences, Faculty of Science,Yamagata University, Yama-

gata 990-8560, Japan (e-mail address : [email protected])‡Department of Mathematical Sciences, Faculty of Science,Yamagata University, Yama-

gata 990-8560, Japan (e-mail address : [email protected])

45

Yoshinori ITO and Fuichi UCHIDA

1 Main result

Theorem Let (X,O(X)) and (Y,O(Y )) be T1-topological spaces. Letψ : O(X) → O(Y ) be a bijection which preserves the inclusion relation.Then, ψ is induced by a homeomorphism f : X → Y .

Proof. Let ψ : O(X) → O(Y ) be a bijection which preserves the inclusionrelation. Then, we easily see the following equations :

ψ(X) = Y, ψ(U ∩ V ) = ψ(U) ∩ ψ(V ),ψ(φ) = φ, ψ(∪αUα) = ∪αψ(Uα).

Moreover, we assume the topologies O(X) and O(Y ) satisfy the T1-axiom.Then, we see that each x ∈ X corresponds uniquely to an element y ∈ Ysatisfying the relation :

ψ({x}c) = {y}c.

Here, {x}c denotes the complement of the one-point set {x}.The above relation induces a mapping f : X → Y by ψ({x}c) = {f(x)}c.

By the assumption, we see f is a bijection. Moreover, we see the following :

x ∈ U ⇐⇒ f(x) ∈ ψ(U).

Hence we see

f(U) := {f(x) | x ∈ U} = ψ(U)

for any U ∈ O(X). The equation shows that f is an open map and

f−1(V ) = ψ−1(V )

for any V ∈ O(Y ). The last equation shows that f is continuous. And hencef : X → Y is a homeomorphism.

Therefore, we see ψ = f∗, that is, ψ is induced by the homeomorphismf : X → Y . q.e.d.

2 Counter examples

Denote by N the set of all natural numbers. Put

Ut = {n ∈ N | n > t}, (t = 1, 2, 3, · · · ).Define

Op = {N, φ} ∪ {Ut | t ≥ p}, (p = 1, 2, 3, · · · ).

46

Bijections between topologies

Then we see Op (p = 1, 2, 3, · · · ) is a topology on N.Let p, q be natural numbers such that p ≤ q. Then, the bijection ψ :

Op → Oq is defined by ψ(Ut) = Ut+q−p. Clearly, ψ preserves the inclusionrelation.

On the other hand, there exists the smallest non-empty closed subset inthe topological space (N,Op), which is the set {1, 2, · · · , p} of p elements.Therefore, the topological space (N,Op) is not homeomorphic to the topo-logical space (N,Oq) for any p �= q, and hence the bijection ψ : Op → Oq isnot induced by any homeomorphism, if p �= q.

References

[1] N.Bourbaki : Elements of Mathematics, General Topology, Chapters 1-4,Springer-Verlag, 2nd printing 1989.

[2] J.L.Kelley : General Topology, Graduate Texts in Mathematics 27,Springer-Verlag 1955.

47


On the construction of smoothSL(m, H) × SL(n, H)-actions on S4(m+n)−1∗

Shintaro KUROKI†


Abstract

In this paper,we shall constract some smooth SL(m, H)×SL(n, H)-actions on (4m+4n-1)-sphere S4(m+n)−1. To constract such an action,we use an R

2-action on S7. This idea was introduced by F.Uchida [5].

1 Introduction

Let H be the field of quaternion numbers. Let Mn(H) be the set of all(n,n)-quaternion matrices. Suppose that we operate the scalar on the right.Define f : Mn(H) −→ M2n(C) by

f(A + Bj) =

(A −BB A

),

where A,B ∈ Mn(C), A means a conjugate complex matrix of A, and i, j,kare the canonical quaternion basis.

Define SL(n, H) = {A ∈ Mn(H); det(f(A)) = 1}. Let A∗ be a conjugateand transpose matrix of A, we define Sp(n) = {A ∈ Mn(H); AA∗ = In =A∗A}. Put G = SL(m, H) × SL(n, H), and K = Sp(m) × Sp(n).

Consider the standard K-action on S4(m+n)−1 ⊂ Hm⊕Hn, that is, (A,B) ·u ⊕ v = Au ⊕ Bv for (A,B) ∈ K and u ⊕ v ∈ S4(m+n)−1.

We see this action has codimension one principal orbits and just twosingular orbits.

∗2000AMS Mathematics Subject Classification. Primary 57S20.†Department of Mathematical Sciences, Faculty of Science, Yamagata University,

Yamagata 990-8560, Japan (e-mail address: [email protected])

49

Shintaro KUROKI

In theorem 4.1, we shall show that the R2-action on S7 characterizes theG-action. In section 5, we constract a G-action from R

2-action on S7 but thisaction may not be smooth. In section 7, we constract some smooth G-actionsfrom the vector field on S7.

My hearty thanks go on F.Uchida and Y.Ito for informing me of usefulreferences and for their comments.

2 Twisted linear actions

For h, h′ ∈ H;Re(h) > 0, Re(h′) > 0, there is the G-action Φ(h,h′) onS4(m+n)−1 defined by

Φ(h,h′)((X,Y ),x ⊕ y) = Xx exp (θh) ⊕ Y y exp (θh′)

for some θ ∈ R. This action is called the twisted linear action which isintroduced by F.Uchida[4].

The action Φ(h,h′) has just three orbits and one of them is an open orbitand the others are compact orbits. Each restricted Sp(m)×Sp(n)-actions oftwisted linear actions are standard action.

3 Certain subgroups of SL(m,H) × SL(n,H)

In this section we shall prove Lemma 3.1 to prove Lemma 4.1.Let L(n), L∗(n), N(n), N∗(n) denote the closed connected subgroups of

SL(n,H) consisting of matrices in the form

1 ∗ . . . ∗0... ∗0

,

1 0 . . . 0∗... ∗∗

,

x ∗ . . . ∗0... ∗0

and

x 0 . . . 0∗... ∗∗

respectively, where x is a non-zero quaternion.

Lemma 3.1 Suppose m ≥ 4, n ≥ 4. Let S be a closed subgroup of G. If thenatural K-action on the homogeneous space G/S has at most three isotropytypes Sp(m− 1) × Sp(n− 1), Sp(m) × Sp(n− 1) and Sp(m− 1) × Sp(n).Then

1. If S contains Sp(m− 1) × Sp(n),

then L(m) × SL(n,H) ⊂ S ⊂ N(m) × SL(n,H).

50

On the construction of smooth SL(m, H ) × SL(n, H )-actions on S4(m+n)−1

2. If S contains Sp(m) × Sp(n− 1),

then SL(m,H) × L(n) ⊂ S ⊂ SL(m,H) ×N(n).

3. If S contains Sp(m−1)×Sp(n−1) but S doesn’t contain Sp(m−1)×Sp(n) nor Sp(m) × Sp(n− 1),

then L(m) × L(n) ⊂ S ⊂ N(m) ×N(n).

Here, the results are described up to an equivalece under automorphisms ofSL(m,H) × SL(n,H) which leave invariant the subgroup Sp(m) × Sp(n).

Proof Let F be a closed subgroup of SL(m,H). We see that if isotropy typesof the standard action of Sp(m) on SL(m,H)/F are Sp(m) or Sp(m − 1),then F = SL(m,H) or L(m) ⊂ F ⊂ N(m) or L∗(m) ⊂ F ⊂ N∗(m). (byUchida[3] Lemma 2.1)

We prove 3. Let i1 : SL(m,H) −→ G and i2 : SL(n,H) −→ G beinclusions, p1 : G −→ SL(m,H) and p2 : G −→ SL(n,H) be projections.Put S(1) = i−1

1 (S), S(2) = i−12 (S), S1 = p1(S) and S2 = p2(S). Then we see

Sp(m− 1) ⊂ S(1) ⊂ S1 and Sp(n− 1) ⊂ S(2) ⊂ S2 by the assumption whichS ⊃ Sp(m − 1) × Sp(n − 1). We consider the standard action of Sp(m) onSL(m,H)/S1. Let p : G/S −→ SL(m,H)/S1 be the induced mapping of p1

(i.e. p(gS) = p1(g)S1). Put Kx the isotropy group at x ∈ G/S. Then we seep1(Kx) ⊂ Sp(m)p(x). So p1(Kx) is conjugate to Sp(m) or Sp(m − 1) by theassumption on isotropy types of K-action on G/S.

Consequently isotropy types of the standard Sp(m)-action on SL(m,H)/S1

is Sp(m) or Sp(m − 1). Similarly isotropy types of the standard action ofSp(n) on SL(n,H)/S2 is Sp(n) or Sp(n− 1). So we see that S1 = SL(m,H)or L(m) ⊂ S1 ⊂ N(m) or L∗(m) ⊂ S1 ⊂ N∗(m) and S2 = SL(m,H) orL(n) ⊂ S2 ⊂ N(n) or L∗(n) ⊂ S2 ⊂ N∗(n).

By the similar way, S(1) and S(2) are the same relation of inclusion re-spectively. If S1 = SL(m,H), then S ⊃ Sp(m) × Sp(n − 1). It contra-dicts the assumption. So S1 �= SL(m,H). Similarly, S2 �= SL(n,H). Thenwe see L(m) ⊂ S(1) ⊂ S1 ⊂ N(m) or L∗(m) ⊂ S(1) ⊂ S1 ⊂ N∗(m) andL(n) ⊂ S(2) ⊂ S2 ⊂ N(n) or L∗(n) ⊂ S(2) ⊂ S2 ⊂ N∗(n).

Consequently we see 3. And we can prove 1,2 similary. q.e.d.

4 Smooth SL(m,H)×SL(n,H)-actions on S4(m+n)−1

Denote by e1, · · · , em, em+1, · · · , em+n the orthonormal standard basis ofH

m ⊕ Hn. If m ≥ 4, n ≥ 4, then the isotropy subgroup at e1u ⊕ em+1v ∈

51

Shintaro KUROKI

S4(m+n)−1 of the standard K-action is isomorphic to H;

H = { (

(1 00 A

),

(1 00 B

)) ∈ K;A ∈ Sp(m− 1), B ∈ Sp(n− 1)}.

PutF = { e1u⊕ emv;u ∈ H, v ∈ H, |u|2 + |v|2 = 1} (∼= S7).

Then the set F is the fixed point set of the standard H-action on S4(m+n)−1.Let Φ : G×S4(m+n)−1 −→ S4(m+n)−1 be a smooth action whose restricted

K-action is standard. Denote by F (S) the fixed point set of restricted S-action of Φ on S4(m+n)−1 (S is a subgroup of G). Then we see F = F (H).

Lemma 4.1 Put L1 = L(m) × L(n), L2 = L∗(m) × L(n), L3 = L(m) ×L∗(n)andL4 = L∗(m) × L∗(n). Then,

F = F (L1) , F (L2) , F (L3) or F (L4).

Proof First we prove F = F (L1) ∪ F (L2) ∪ F (L3) ∪ F (L4).(⊃) is trivial.If x ∈ F , then Gx ⊃ Kx = Sp(m) × Sp(n − 1) or Sp(m − 1) × Sp(n) or

Sp(m− 1) × Sp(n− 1). By Lemma 2.1 Gx ⊃ L1, L2, L3, orL4.Next we put

F0 = { e1u + em+1v ∈ F ;uv �= 0} , F0(S) = F0 ∩ F (S).

Then we see F0(L1), F0(L2), F0(L3), and F0(L4) are mutually disjoint closedsubsets of F0, and F0 is connected. Consequently F0 = F0(Li) for somei = 1, 2, 3, 4.

F = F0 ∪ {e1u, em+1v;u, v ∈ Sp(1)}= F0(Li) ∪ {e1u, em+1v;u, v ∈ Sp(1)} = F (Li)

for some i = 1, 2, 3, 4. q.e.d.

By Lemma 4.1 and the automorphism σ(A) =tA−1, we can assume F =F (L1) for the smooth action Φ without loss of generality.

Let CA(B) be the centralizer of B in A. We define M to be the identitycomponent of CG(H). Then we see F is M -invarant.

An element (X,Y ) ∈ M can be denoted by (X,Y ) = (D1(θ1), D2(θ2)) forsome (θ1, θ2) ∈ H

2. Here we denote

D1(θ1) =

exp θ1 0 · · · 00... xIm−1

0

, D2(θ2) =

exp θ2 0 · · · 00... yIn−1

0

52


for x, y ∈ R;x1−m = | exp(θ1)|, y1−n = | exp(θ2)|.Consider Φ|M×F : M × F −→ F . Let a1, a2 ∈ R and h1, h2 be pure

quaternions.

Φ|M×F ((D1(a1 + h1), D2(a2 + h2)), e1u⊕ em+1v)

= (D1(h1), D2(h2)) · Φ|M×F ((D1(a1), D2(a2)), e1u⊕ em+1v))

By identifing e1u ⊕ em+1v ∈ S4(m+n)−1 with (u, v) ∈ F , we can defineR

2-action ΦM on F ;

ΦM((a1, a2), (u, v)) = Φ|M×F ((D1(a1), D2(a2)), e1u, em+1v)

Then for some C∞ function a : R2 × F −→ H and b : R

2 × F −→ H, wecan describe

ΦM((a1, a2), (u, v)) = (a(a1, a2, u, v), b(a1, a2, u, v)) = (a, b)

where |a|2 + |b|2 = 1.Put u = r1 exp h′

1, v = r2 exph′2 (where h′

1, h′2 are pure quaternions,

r1, r2 ∈ R),then

Φ|M×F ((D1(a1), D2(a2)), (u, v))

= Φ|M×F{(D1(a1), D2(a2)),Φ|M×F ((D1(h′1), D2(h

′2)), (r1, r2))}

= (exph′1, exp h′

2) · ΦM((a1, a2), (r1, r2))

= (exph′1, exp h′

2) · (a(a1, a2, r1, r2), b(a1, a2, r1, r2)).

And we put

j1 =

−11

. . .

1

∈ Sp(m), j2 =

−11

. . .

1

∈ Sp(n).

By j1, j2, we see

a(a1, a2,−u, v) = −a(a1, a2, u, v), a(a1, a2, u,−v) = a(a1, a2, u, v)b(a1, a2, u,−v) = −b(a1, a2, u, v), b(a1, a2,−u, v) = b(a1, a2, u, v).

If we consider that a, b are functions of (a1, a2, r1, r2), then a is an oddfunction for r1 and an even function for r2, and b is an even function for r1

and an odd function for r2. So

a(a1, a2, r1, r2) = r1α(a1, a2, r1, r2), b(a1, a2, r1, r2) = r2β(a1, a2, r1, r2)

53

Shintaro KUROKI

where α, β : R2 × S1 −→ H are even C∞-function for r1 and r2.Consequentry

a(a1, a2, u, v) = uα(a1, a2, r1, r2),b(a1, a2, u, v) = vβ(a1, a2, r1, r2).

Theorem 4.1 Let Φ,Φ′ : G × S4(m+n)−1 −→ S4(m+n)−1 be smooth actionswhose restricted Sp(m)×Sp(n)-action is standard and F = F (L(m)×L(n)).If ΦM = Φ′

M , then Φ = Φ′.

Proof There is a decomposition SL(m,H) = Sp(m) · D1(R) · L(m), whereD1(R) = {D1(θ); θ ∈ R}. Let (g1, g2) be an element of G. Then, there are(k1, k2) ∈ K, (a1, a2) ∈ R2 and (l1, l2) ∈ L(m)×L(n) satisfying the equations;

gs = ks ·Ds(as) · ls (s = 1, 2).

Then we obtain

Φ((g1, g2), e1u⊕ em+1v) = k1e1uα⊕ k2em+1vβ.

By the above decomposition of gs, we obtain (for u, v �= 0)

a1 = log‖g1e1u‖‖e1u‖ , a2 = log

‖g2em+1v‖‖em+1v‖ .

Thus we obtain

Φ((g1, g2), e1u⊕em+1v) = g1e1u· ‖e1u‖‖g1e1u‖α⊕g2em+1v· ‖em+1v‖

‖g2em+1v‖β

for u, v �= 0. Here

α = α(log‖g1e1u‖‖e1u‖ , log

‖g2em+1v‖‖em+1v‖ , ‖e1u‖, ‖em+1v‖),

β = β(log‖g1e1u‖‖e1u‖ , log

‖g2em+1v‖‖em+1v‖ , ‖e1u‖, ‖em+1v‖).

Moreover, when u = 0 or v = 0, we obtain

Φ((g1, g2), 0 ⊕ em+1v) = 0 ⊕ g2em+1v

‖g2em+1v‖β,

Φ((g1, g2), e1u⊕ 0) =g1e1u

‖g1e1u‖α⊕ 0

where

α = α(log ‖g1e1u‖, log ‖g2em+1‖, 1, 0), β = β(log ‖g1e1‖, log ‖g2em+1v‖, 0, 1).

Let u⊕v be an element of S4(m+n)−1. Then, u = k′1e1u,v = k′

2em+1v forsome (k′

1, k′2) ∈ K,u = ‖u‖, v = ‖v‖.

So we obtain when u, v �= 0

54


Φ((g1, g2),u ⊕ v) = g1u‖u‖‖g1u‖α⊕ g2v

‖v‖‖g2v‖β

where

α = α(log‖g1u‖‖u‖ , log

‖g2v‖‖v‖ , ‖u‖, ‖v‖),

β = β(log‖g1u‖‖u‖ , log

‖g2v‖‖v‖ , ‖u‖, ‖v‖).

And

Φ((g1, g2),u ⊕ 0) =g1u

‖g1u‖α⊕ 0,

Φ((g1, g2), 0 ⊕ v) = 0 ⊕ g2v

‖g2v‖β

where

α = α(log ‖g1u‖, log ‖g2em+1‖, 1, 0), β = β(log ‖g1e1‖, log ‖g2v‖, 0, 1).

If ΦM = Φ′M , then

ΦM((a1, a2), (u, v)) = (uα, vβ) = Φ′M((a1, a2), (u, v)) = (uα′, vβ′).

We see α = α′, β = β′. Consequentry Φ = Φ′. q.e.d.

5 Induced smooth R2-action on F

In this section we study the converse of section 4. That is we shall con-stract a G-action on S4(m+n)−1 from the R

2-action ΦM of F .Denote ΦM : R2 × F −→ F by

ΦM((a1, a2), (u, v)) = (uα(a1, a2, r1, r2), vβ(a1, a2, r1, r2))

for some C∞-functions α, β : R2 × S1 −→ H which are even functions with

respect to variables r1 and r2, where r1 = ±|u|, r2 = ±|v|, and r21 + r2

2 = 1.If ΦM is an R2-action on F, then α, β are satisfied the following equations;

α(0, X) = 1 = β(0, X),

α(Θ′ + Θ, X) = α(Θ, X) · α(Θ′, (uα(Θ, X), vβ(Θ, X))),

β(Θ′ + Θ, X) = β(Θ, X) · β(Θ′, (uα(Θ, X), vβ(Θ, X))),

α, β �= 0

55

Shintaro KUROKI

where Θ,Θ′ ∈ R2, X ∈ S1.Now we define a mapping Φ : G× S4(m+n)−1 −→ S4(m+n)−1 by

Φ((A,B),u ⊕ v) = Au‖u‖‖Au‖α⊕Bv

‖v‖‖Bv‖β

where

α = α(log‖Au‖‖u‖ , log

‖Bv‖‖v‖ , ‖u‖, ‖v‖),

β = β(log‖Au‖‖u‖ , log

‖Bv‖‖v‖ , ‖u‖, ‖v‖)

for u ⊕ v ∈ S4(m+n)−1 such that ‖u‖, ‖v‖ �= 0 and

Φ((A,B),u ⊕ 0) =Au

‖Au‖ · α⊕ 0,

Φ((A,B), 0 ⊕ v) = 0 ⊕ Bv

‖Bv‖ · β

where

α = α(log ‖Au‖, log ‖Bem+1‖, 1, 0), β = β(log ‖Ae1‖, log ‖Bv‖, 0, 1).

We see following by the routine work.

Proposition 5.1 Defined Φ is an abstract G-action on S4(m+n)−1.

This action may not be a smooth action (cf.[5],Prop4.2).

Proposition 5.2 If α = β = 1, then Φ is not smooth.

6 Certain smooth action of SL(m,H) on Hm

In this section, we prepare for the section 7.We obtain a smooth action ϕm of SL(m,H) on H

m by

ϕm(A,u) = Au2

1 − ‖u‖2 +√

4‖Au‖2 + (1 − ‖u‖2)2.

We construct a one parameter group φm on H × S3 from the value ofϕm(D1(a + h0), e1h). That is

φm(a, (h, exp h0)) = (2 exp a · h

1 − |h|2 +√

4 exp 2a · |h|2 + (1 − |h|2)2, exp h0)

56


where a ∈ R, h ∈ H and h0 is a pure quaternion.We see that the R-action φm on H×S3 corresponds to the following vector

field on H × S3;

4∑p=1

1 − |h|21 + |h|2hp

∂

∂hp

+ O

where h = h1 + h2i+ h3j+ h4k, hp ∈ R. {( ∂∂hp

)h}4p=1 is the basis of Th(H), O

is the zero vector field on S3.

7 Construction of smooth G-actions on S4(m+n)−1

First we shall construct smooth one parameter group on F .Define diffeomorphisms k, l of H × S3 onto open sets of F by

k(h, ε) = (h√

γ2 + |h|2 ,γε√

γ2 + |h|2 ),

l(h, ε) = (γε√

γ2 + |h|2 ,h√

γ2 + |h|2 )

for h ∈ H, ε ∈ S3, where γ is a positive real number.Let ρ(t) be a smooth real valued function such that

ρ(t) = 1for|t| ≤ 1,ρ(t) = 0for|t| ≥ 2,

0 < ρ(t) < 1for1 < |t| < 2.

Put u = u1 + u2i + u3j + u4k, v = v1 + v2i + v3j + v4k, where up, vp ∈ R

and h0, h′0 are the pure quaternions. Define the tangent vector field ξ on

F = {(u, v) ∈ H2; |u|2 + |v|2 = 1} by

ξ = k∗(4∑

p=1

ρ(|h|2)1 − |h|21 + |h|2hp

∂

∂hp

) + l∗(4∑

p=1

ρ(|h|2)1 − |h|21 + |h|2hp

∂

∂hp

)

+ρ(162

7(|u|2− 53

162))·(

4∑p=1

up)·(4∑

p=1

vp)·(4∑

p=1

vp∂

∂xp−

8∑p=5

up−4∂

∂xp)

+ρ(162

7(|u|2−109

162))·(

4∑p=1

up)·(4∑

p=1

vp)·(4∑

p=1

vp∂

∂xp

−8∑

p=5

up−4∂

∂xp

)

57

Shintaro KUROKI

where {( ∂∂xp

)(h,h′)}8p=1 is the basis of T(h,h′)(H

2).

Let φ be the one-parameter group on F corresponding to the vector fieldξ. Let γ = 4. Now we define a smooth R2-action ϕ on F by

ϕ((a1, a2), (u, v)) =

φ(a1, (u, v)) for|u|2 < 19

φ(a · a1 + b · a2, (u, v)) for 19≤ |u|2 ≤ 1

2

φ(c · a1 + d · a2, (u, v)) for 12≤ |u|2 ≤ 8

9

φ(a2, (u, v)) for|u|2 > 89.

Here a, b, c, d are given any real numbers. The smoothness of φ is assuredby ξ = 0 for 1

9< |u|2 < 13

54, 67

162< |u|2 < 95

162, 41

54< |u|2 < 8

9.

Then the vector field ξ is inveriant under the involution J1 : (u, v) −→(−u, v), J2 : (u, v) −→ (u,−v). Consequentry φ(θ, Js(u, v)) = Js(φ(θ, (u, v)).So we can put ϕ((a1, a2), (u, v)) = (uα(a1, a2, r1, r2), vβ(a1, a2, r1, r2)) = (uα, vβ).Then the action Φ : G× S4(m+n)−1 −→ S4(m+n)−1 is defined by

Φ((A,B),u ⊕ v) =

Au

‖Au‖‖u‖α⊕ Bv

‖Bv‖‖v‖β (u,v �= 0)

0 ⊕ Bv

‖Bv‖ (u = 0, ||v|| = 1)

Au

‖Au‖ ⊕ 0 (v = 0, ||u|| = 1).

Remark: When |u| = 0 or |v| = 0, the vector field ξ = 0. So if |u| = 0 thenβ = 1 and if |v| = 0 then α = 1.

Now we show that Φ is smooth on neighborhoods of u = 0 and v = 0.Define k : Hm × S4n−1 −→ S4(m+n)−1 − {u ⊕ 0;u ∈ S4m−1} which is a

diffeomorphism by

k(u,v) = (u ⊕ 4v)1√

16 + |u|2 .

Then we see

Φ((A,B),u ⊕ v) = k(ϕm(A,u4√

1 − ‖u‖2), Bv

1

‖Bv‖)

for ‖u‖2 < 117

. This shows that Φ is smooth on a neighborhood of u = 0.Similary, we may show the smoothness of Φ on a neighborhood of v = 0.Therefore, we see that the G-action Φ on S4(m+n)−1 is smooth.

Theorem 7.1 There exists a smooth SL(m,H)×SL(n,H)-action on S4(m+n)−1

such that it has six open orbits and the isotropy types of them are the fol-lowing: L(m) × (D2(R) · L(n)) (two orbits), N(b : −a), N(d : −c), and

58


(D1(R) · L(m)) × L(n) (two orbits) for all real numbers a, b, c and d, suchthat (a, b) �= (0, 0), (c, d) �= (0, 0).

Where N(x : y) denotes the subgroup of N(m) × N(n) consisting of thepairs of matrices in the form:

exp tx ∗ · · · ∗0... ∗0

,

exp ty ∗ · · · ∗0... ∗0

(t ∈ R),

and D1(R) = {D1(θ); θ ∈ R}, D2(R) = {D2(θ); θ ∈ R}.Remark: By an exchange of the third term and forth term of the vector

field ξ, we can obtain another smooth G-action on S4(m+n)−1.

References

[1] T.Tomuro, Smooth SL(n,H), Sp(n,C)-actions on (4n − 1)-manifolds,Tohoku Math. J. 44(1992)243-250.

[2] F.Uchida, Real analytic SL(n,R) actions on sheres, Tohoku Math. J.33(1981)145-175.

[3] F.Uchida, Actions of Special linear groups on a product manifold, Bull.of Yamagata Univ. Nat. Sci. 10-3(1982) 227-233.

[4] F.Uchida, On a method to construct analytic actions of non-compact Liegroups on a sphere, Tohoku Math. J. 39(1987)61-69.

[5] F.Uchida, On a smooth SL(m,R) × SL(n,R) actions on Sm+n−1, Inter-discip. Inform. Sci. 18-1 (2002), 123-128.

59


On certain SL(2,R)×SL(2,R)-actions on S3∗

Fuichi UCHIDA†‡


0 Introduction

There are two types of smooth actions of K = SO(2) × SO(2) on S3,each of which has principal orbits of codimension one. In this paper, weshall study actions of G = SL(2,R) × SL(2,R) on S3 which is regardedas an extension of the K-action. Notice that K is the maximal compactsubgroup of G. As a result, we can state : for each positive integer r, thereare uncountably many topologically distinct continuous G-actions on S3, thenumber of open orbits of which is r + 2. There is a little difference in theresults between two types.

1 Standard actions of SL(2,R)× SL(2,R) on S3

Define a smooth action ψ of SL(2,R)× SL(2,R) on S3 by

ψ((A,B), X) = ‖AXB−1‖−1AXB−1

for A,B ∈ SL(2,R) and X ∈ S3. Denote by ψ0 the restricted action of ψto the compact group SO(2)×SO(2). We call ψ, ψ0 the standard actions ofthe first type. Here we identify S3 with the unit sphere of M2(R) which isthe vector space of all real 2× 2 matrices with the norm

‖X‖ = (tracetXX)1/2 for X ∈M2(R).

Denote by E11, E12, E21 and E22, the matrix units of M2(R). Put I2 =E11 + E22 and J = E11 − E22.

∗2000 AMS Mathematics Subject Classification. Primary 57S20.†Partly supported by the Grants-in-Aid for Scientific Research, The Ministry of Edu-

cation, Science and Culture, Japan No.12640056‡Department of Mathematical Sciences, Faculty of Science, Yamagata University,

Yamagata 990-8560, Japan (e-mail address : [email protected])

61

Fuichi UCHIDA

Lemma 1.1. Let X ∈ S3.(1) There exist A,B ∈ SO(2) : AXB−1 = pE11 + qE22 for certain real

numbers p, q with p ≥ |q| and p2 + q2 = 1.(2) If rankX = 2, then there exist A,B ∈ SL(2,R) and c > 0 : AXB−1 =

c(E11 + εE22)/√2, where ε = 1 for detX > 0 and ε = −1 for detX < 0. If

rankX = 1, then there exist A,B ∈ SL(2,R) : AXB−1 = E11.

Proof. Denote by X1, X2 the first and the second columns of X ∈ M2(R)respectively, that is, X = [X1, X2]. Put

R(τ) =

[cos τ sin τ

− sin τ cos τ

].

Then we see

XR(τ)−1 = [X ′1, X

′2] = [cos τX1 + sin τX2,− sin τX1 + cos τX2].

Therefore, there exists τ such thatX ′1⊥X ′

2 and ‖X ′1‖ ≥ ‖X ′

2‖. Put B = R(τ).Then, we obtain AXB−1 = pE11 + qE22 for certain A ∈ SO(2).

To show the second part, we may assume X = pE11+qE22, where p ≥ |q|.Suppose rankX = 2. Then there exists a diagonal matrix D of SL(2,R) suchthat DX = c(E11 + εE22)/

√2. Suppose rankX = 1. Then q = 0, and hence

X = E11. q.e.d.

By this lemma, we see that the standard action ψ of SL(2,R)×SL(2,R)on S3 has just three orbits. Two of them are open orbits through (E11 +εE22)/

√2 for ε = ±1, respectively, and the other is a compact orbit through

E11. In particular, two singular orbits of restricted action ψ0 of SO(2) ×SO(2) on S3 are contained in open orbits of the action ψ, respectively.

By direct calculation, we obtain the condition that an element (A,B) ∈SO(2)×SO(2) belongs to the isotropy subgroup at pE11+qE22, where p ≥ |q|and p2 + q2 = 1. In fact, if p > |q| > 0, then A = B = εI2, where ε = ±1. Ifp = q = 1/

√2, then A = B, and if p = −q = 1/

√2, then B = A−1.

Moreover, we obtain the condition that an element (A,B) ∈ SL(2,R)×SL(2,R) belongs to the isotropy subgroup at pE11+ qE22, where p ≥ |q| andp2 + q2 = 1. In particular, if p = q = 1/

√2, then A = B. If p = −q = 1/

√2,

then B = JAJ−1. If p = 1, q = 0, then

A =

[a ∗0 ∗

], B =

[b 0∗ ∗

], ab > 0.

Define a smooth action ϕ of SL(2,R)× SL(2,R) on S3 by

ϕ((A,B),u ⊕ v) = ‖Au ⊕Bv‖−1(Au ⊕Bv)

62

On certain SL(2,R) × SL(2,R)-actions on S3

for A,B ∈ SL(2,R) and u,v ∈ R2 such that ‖u⊕v‖ = 1. Denote by ϕ0 therestricted action of ϕ to the compact group SO(2) × SO(2). We call ϕ, ϕ0

the standard actions of the second type.Denote by e1, e2 the canonical basis of R

2. We obtain the following resultimmediately.

Lemma 1.2. Let u⊕ v ∈ S3.(1) There exist A,B ∈ SO(2) : Au ⊕ Bv = ‖u‖e1 ⊕ ‖v‖e1.(2) If ‖u‖ · ‖v‖ �= 0, then there exist A,B ∈ SL(2,R) : Au ⊕ Bv =

(e1 ⊕ e1)/√2.

By this lemma, we see that the standard action ϕ of SL(2,R)×SL(2,R)on S3 has just three orbits. One of them is an open orbit through (e1⊕e1)/

√2

and the others are compact orbits through e1 ⊕ 0 and 0⊕ e1, respectively.By direct calculation, we obtain the condition that an element (A,B) ∈

SL(2,R) × SL(2,R) belongs to the isotropy subgroup at pe1 ⊕ qe1, wherep ≥ 0, q ≥ 0 and p2 + q2 = 1. In particular, if pq �= 0, then

A =

[a ∗0 ∗

], B =

[b ∗0 ∗

]; a > 0, b > 0.

2 Certain closed subgroups of SL(2,R) × SL(2,R)

Put G = SL(2,R) × SL(2,R), K = SO(2) × SO(2). Here we searchclosed subgroups of G which may be an isotropy subgroup at some point ofS3, with respect to an extended G-action of the standard K-action ψ0 or ϕ0.

Lemma 2.1. Let S be a closed subgroup of G. Suppose S satisfies thefollowing conditions.

(1) S ∩K = {(εI2, εI2)|ε = ±1}, {(A,A)|A ∈ SO(2)} or {(A,A−1)|A ∈SO(2)},

(2) dimS ≥ 3.Then, S is conjugate with one of the followings :

(a) N2,2 =

{([a ∗0 ∗

],

[b ∗0 ∗

]) ∣∣ ab > 0

},

(b) {(A,A)|A ∈ SL(2,R)}, {(A, JAJ−1)|A ∈ SL(2,R)},

(c)

{([a ∗0 ∗

],

[b 00 ∗

]) ∣∣ ab > 0

},

{([a 00 ∗

],

[b ∗0 ∗

]) ∣∣ ab > 0

},

(d) S(α, β, γ) =

{([εeαt−βγk ∗0 ∗

],

[εeβt+αγk ∗0 ∗

]) ∣∣ t ∈ R, k ∈ Z, ε = ±1}

63

Fuichi UCHIDA

for (α, β) �= (0, 0) and γ ≥ 0.Proof. Let Si be the image of S by the natural projection of G =

SL(2,R) × SL(2,R) to the i-th factor, for i = 1, 2. Let S(i) be the closedsubgroup of SL(2,R) such that

S(1) × {I2} = S ∩ (SL(2,R)× {I2}), {I2} × S(2) = S ∩ ({I2} × SL(2,R)).

Then, S(i) is a normal subgroup of Si, for i = 1, 2. Moreover, S(1) × S(2) isa normal subgroup of S, S1 × S2 contains S as a subgroup, and there arenatural isomorphisms

S1/S(1)∼= S2/S(2)

∼= S/(S(1) × S(2)).

By (1), we see Si �= S(i) for i = 1, 2. By (2), we may assume dimS1 ≥ 2,without loss of generality. Since −I2 ∈ S1 by (1), we see S1 is conjugate withone of the followings :

SL(2,R) or N(2) =

{[ ∗ ∗0 ∗

]}.

Denote by T 0 and N(T ) the identity component and the normalizer of aclosed subgroup T of SL(2,R), respectively.

First, we assume S1 = SL(2,R). Then, S(1) = {I2} or C2. Here, C2 ={±I2} is the center of SL(2,R). Moreover, we see S2 = S1 and S(2) = S(1).If S(1) = S(2) = C2, then the corresponding S does not satisfy the condition(1). If S(1) = S(2) = {I2}, then we see

S = {(A,PAP−1)|A ∈ SL(2,R)},

for some P ∈ GL(2,R). This is the case (b).Next, we assume S1 = N(2). Then, dimS(1) ≤ 1 or S(1) = N(2)0.

Suppose dimS(1) = 0. Then we see dimS2 = 2 and dimS(2) = 0, and hencedimS = 2, which contradicts to (2). Suppose dimS(1) = 1. Then we seedimS2 = 2 and dimS(2) = 1. Moreover, we can assume S1 = S2 = N(2).Since S(i) is a normal subgroup of Si, we see the identity component of S(i)

coincides with L(2). Here

L(2) =

{[1 ∗0 1

]}.

Then N(2) is the normalizer of L(2). Consider the natural projection

π : N(2)×N(2) → N(2)/L(2)×N(2)/L(2) ∼= R× × R×.

64


Then we see S = π−1(T ) for some 1-dimensional closed subgroup T of R× ×R×. Thus we obtain S = S(α, β, γ) for some (α, β) �= (0, 0) and γ ≥ 0. Thisis the case (d).

Finally, we assume S1 = N(2) and S(1) = N(2)0. Then we see

dimS2 = dimS(2) = 1 or 2.

If dimS2 = 2, then S is the group listed in (a). Suppose dimS2 = 1. Thenthe identity component S0

2 is conjugate with L(2),SO(2) or D(2)0, up toconjugation, where

D(2) =

{[ ∗ 00 ∗

]}.

Since NSO(2) = SO(2), the case S02 = SO(2) is omitted. Suppose S0

2 =D(2)0. Then we see S2 = D(2) and S(2) = D(2)0, by (1). This is the case(c). Suppose S0

2 = L(2). Then we see

S =

{([a ∗0 ∗

],

[εbk ∗0 ∗

]) ∣∣ aε > 0, ε = ±1, k ∈ Z

},

where b ≥ 1 is a constant. This is a special type of (d). q.e.d.

Lemma 2.2. Let S(α, β, γ) be the closed subgroup of G listed in Lemma2.1 (d). If γ �= 0, then S(α, β, γ) is not realized as an isotropy subgroupat some point of S3 with respect to any extended G-action of the standardK-action ψ0 of the first type.

Proof. First we shall show that the restricted K-action on the quotientG/S(α, β, γ) has only one isotropy type ∆C2 = {(εI2, εI2)|ε = ±1}. SinceS(α, β, γ) is a subgroup of N(2) × N(2), we obtain a natural projectionπ : G/S(α, β, γ) → G/(N(2)×N(2)). The restricted K-action on G/(N(2)×N(2)) has only one isotropy type C2 × C2. Since π is K-equivariant, we seethat the restricted K-action on the quotient space G/S(α, β, γ) has only oneisotropy type ∆C2.

By this result, we see that the quotient space G/S(α, β, γ) is open 3-manifold, if S(α, β, γ) is realized as an isotropy subgroup at some point ofS3 with respect to an extended G-action of the standard K-action ψ0.

Now we shall show that G/S(α, β, γ) is a compact 3-manifold, if γ �= 0.Notice that L(2) × L(2) is a normal subgroup of S(α, β, γ) and there is anatural G-equivariant diffeomorphism

G/(L(2)× L(2)) ∼= R20 × R2

0.

Here R20 = R2 − {0} and the action of SL(2,R) on R2

0 is canonical. Theright action of S(α, β, γ)/(L(2) × L(2)) on G/(L(2) × L(2)) corresponds to

65

Fuichi UCHIDA

certain scalar multiplication on R20×R2

0. In fact, the right action of the cosetrepresented by

([εeαt−βγk ∗0 ∗

],

[εeβt+αγk ∗0 ∗

]); t ∈ R, k ∈ Z

corresponds to the scalar multiplication

(u,v) → (εeαt−βγku, εeβt+αγkv).

Consider the natural diffeomorphism

S1 × S1 × R+ × R+ → R20 × R2

0

defined by (u,v, λ, µ) → (λu, µv). Here, R+ is the set of all positive realnumbers. Then the above scalar multiplication corresponds to the following :

(u,v, λ, µ) → (εu, εv, eαt−βγkλ, eβt+αγkµ).

Hence we see that for each u,v ∈ S1, the curve f(s) = (u,v, e−βs, eαs) in-tersects transversely each S(α, β, γ)/(L(2)×L(2))-orbit through (u,v, λ, µ).Therefore, we obtain a diffeomorphism

G/S(α, β, γ)0 ∼= S1 × S1 × R.

If γ = 0, then G/S(α, β, 0) is K-equivariantly diffeomorphic to

(SO(2)×∆C2 SO(2))× R.

But if γ �= 0, then G/S(α, β, γ) is K-equivariantly diffeomorphic to

(SO(2)×∆C2 SO(2))× S1,

and hence G/S(α, β, γ) is a compact 3-manifold. q.e.d.

Lemma 2.3. Let S be a closed subgroup of G. Suppose S satisfies thefollowing conditions.

(1) S ∩K = {(I2, I2)}, SO(2)× {I2} or {I2} × SO(2),(2) dimS ≥ 3.

66


Then, S is conjugate with one of the followings :

(a) SL(2,R)× {I2}, {I2} × SL(2,R),

(b) SL(2,R)×N(2)0, N(2)0 × SL(2,R),

(c) SL(2,R)×{[

eγk ∗0 ∗

] ∣∣ k ∈ Z

},

{[eγk ∗0 ∗

] ∣∣ k ∈ Z

}× SL(2,R),

(d) SO(2)×N(2)0, N(2)0 × SO(2),

(e) N(2)0 ×N(2)0,

(f) T (α, β, γ) =

{([eαt−βγk ∗0 ∗

],

[eβt+αγk ∗0 ∗

]) ∣∣ t ∈ R, k ∈ Z

}

for (α, β) �= (0, 0) and γ ≥ 0.Proof. Consider the case S ∩ K = {(I2, I2)} and the remainig case,

respectively. The proof is quite similar to the one of Lemma 2.1. So we omitthe details. q.e.d.

Lemma 2.4. Let S be the closed subgroup of G listed in Lemma 2.3.The subgroups (a),(c),(d) and (f) for γ �= 0 are not realized as an isotropysubgroup at some point of S3 with respect to any extended G-action of thestandard K-action ϕ0 of the second type.

Proof. Considering the isotropy types of the natural K-action on thecoset space G/S, we can delete (a), (c), (d). Moreover, we see

G/T (α, β, 0) ∼= K × R, G/T (α, β, γ) ∼= K × S1 (γ �= 0)

as K-manifolds. So we can delete (f) for γ �= 0. q.e.d.

Lemma 2.5. For any extended continuous G-action of the standard K-action ϕ0 of the second type, the two singular orbits of the K-action ϕ0 areG-invariant.

Proof. This is the direct consequence of Lemma 2.4. q.e.d.

3 Certain smooth actions of SL(2,R)×SL(2,R) on 3-manifolds, I

3.1. Let ψ be the standard action of G = SL(2,R) × SL(2,R) of thefirst type on S3. Then the isotropy subgroup at E12 coincides with N2,2. Theorbit through E12 is the only one compact orbit.

The isotropy subgroups at I2/√2 and J/

√2 coincide with

{(A,A)|A ∈ SL(2,R)} and {(A, JAJ−1)|A ∈ SL(2,R)},

67

Fuichi UCHIDA

respectively. The orbits through these points are just two open orbits. Put

S3+ = {X ∈ S3 | detX ≥ 0}, S3

− = {X ∈ S3 | detX ≤ 0}.

These are closed invariant subsets. Put

D2+ =

{c(a, b)

[1 + a bb 1− a

] ∣∣ a2 + b2 ≤ 1

}

= {c(λ)(I2 + λJR(θ)) | θ ∈ R, 0 ≤ λ ≤ 1},

where c(a, b) = 1/√2(1 + a2 + b2) and c(λ) = 1/

√2(1 + λ2). This is a closed

subset of S3+ and there are K-equivariant diffeomorphisms :

K ×∆SO(2) D2+

f−→ SO(2)× D2+

g−→ S3+.

Here, K ×∆SO(2) D2+ is the quotient manifold of K ×D2

+ by the equivalencerelation

((A,B), X) ∼ ((AC−1, BC−1), CXC−1) for C ∈ SO(2)

and the mappings f, g are defined by

f([(A,B), X]) = (AB−1, BXB−1), g(A,X) = AX.

Moreover the K-action on SO(2)× D2+ is defined by

(A,B) · (C,X) = (ACB−1, BXB−1).

Put

D2− =

{c(a, b)

[a+ 1 −bb a− 1

] ∣∣ a2 + b2 ≤ 1

}

= {c(λ)(J + λR(θ)) | θ ∈ R, 0 ≤ λ ≤ 1},

where c(a, b) = 1/√2(1 + a2 + b2) and c(λ) = 1/

√2(1 + λ2). This is a closed

subset of S3− and there are K-equivariant diffeomorphisms :

K ×∆JSO(2) D2−

f ′−→ SO(2)× D2−

g′−→ S3−.

Here, K ×∆JSO(2) D2− is the quotient manifold of K ×D2

− by the equivalencerelation

((A,B), X) ∼ ((AC−1, BC), CXC−1) for C ∈ SO(2)

68


and the mappings f ′, g′ are defined by

f ′([(A,B), X]) = (AB,B−1XB−1), g′(A,X) = AX.

Moreover the K-action on SO(2)× D2− is defined by

(A,B) · (C,X) = (ABC,B−1XB−1).

3.2. Let ψ(α,β) be the G-action on (S1 ×∆C2 S1)× S1 defined by

ψ(α,β)((A,B), (u,v, (λ, µ)))

= (‖Au‖−1Au, ‖Bv‖−1Bv, (cλ‖Au‖α‖Bv‖β, cµ‖Au‖−α‖Bv‖−β)),

where c is a positive real number. Here α, β are fixed real numbers suchthat (α, β) �= (0, 0) and S1 ×∆C2 S

1 is the quotient manifold of S1 × S1 bythe equivalence relaion (u,v) ∼ (−u,−v). Then the isotropy subgroup at(e1, e1, (λ, µ)) coincides with

N2,2 for λµ = 0 and S(−β, α, 0) for λµ �= 0,

respectively. Put

M(α,β) = (S1 ×∆C2 S1)× S1

+,

where S1+ = {(λ, µ) ∈ S1 | λ ≥ 0, µ ≥ 0}. This is an invariant closed subset.

3.3. Let ψ1D be the G-action on T 3/∼ defined by

ψ1D((A,B), (u,v,w)) = (‖Au‖−1Au, ‖Bv‖−1Bv, ‖Bw‖−1Bw),

where T 3/∼ is the quotient manifold of T 3 = S1×S1×S1 by the equivalencerelation (u,v,w) ∼ (−u,−v,−w). Then the isotropy subgroup at (e1, e1, e2)coincides with {([

a ∗0 ∗

],

[b 00 ∗

])| ab > 0

}.

The isotropy subgroup at (e1, e1, e1) coincides with N2,2. Put

M1D = {(u,v,w) | det[v,w] ≥ 0}.This is an invariant closed set.

Let ψ2D be the G-action on T 3/∼ defined by

ψ2D((A,B), (u,v,w)) = (‖Au‖−1Au, ‖Av‖−1Av, ‖Bw‖−1Bw).

69

Fuichi UCHIDA

Then the isotropy subgroup at (e1, e2, e1) coincides with{([a 00 ∗

],

[b ∗0 ∗

])| ab > 0

}.

The isotropy subgroup at (e1, e1, e1) coincides with N2,2. Put

M2D = {(u,v,w) | det[u,v] ≥ 0}.This is an invariant closed set.

Remark. S3+, S

3−,M(α,β),M1D and M2D are compact 3-manifolds with

smooth G-action, each of which has a boundary. S3+, S

3− have a connected

boundary, and the others have just two boundary components. Each bound-ary component of the above 3-manifolds is diffeomorphic to G/N2,2 as asmooth G-manifold.

4 Continuous actions of SL(2,R)× SL(2,R) on S3 of the first type

In the previous section, we prepare the compact 3-manifolds S3+, S

3−,

M(α,β),M1D and M2D with smooth action of G = SL(2,R)×SL(2,R). Con-sider the disjoint union

M0 ∪M1 ∪ · · · ∪Mr ∪Mr+1.

Here, M0 = S3+, Mr+1 = S3

− and each Mi(i = 1, 2, · · · , r) is one of M(α,β),M1D and M2D, respectively. Put M the space obtained from the abovedisjoint union pasting the boundaries one after another by equivariant G-diffeomorphisms. Then, we see M is naturally diffeomorphic to S3 with thestandard K-action of the first type. Moreover,M has the natural continuousG-action. In such a way, we obtain an extended continuous G-action of thestandard K-action of the first type on S3. In consequence, we obtain thefollowing result.

Theorem 4.1. For each positive integer r, there exist uncountably manytopologically distinct continuous SL(2,R) × SL(2,R) actions on S3, whichare extensions of the standard SO(2)×SO(2) action of the first type and thenumber of open orbits of which is r + 2.

Considering the differentiability of the above G-actions, we obtain thefollowing result.

Theorem 4.2. Each action of SL(2,R) × SL(2,R) on S3 is not C2-differentiable for any decomposition :

S3 ∼= M0 ∪M1 ∪ · · · ∪Mr ∪Mr+1 (r ≥ 1).

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Proof. We shall show that G-action on M0 ∪M1 is not C2- differentiable.

First, we introduce local coordinates (τ, θ, λ) on S3+,M(α,β),M1D and M2D,

as follows. Put

X = X(τ, θ, λ) =1√

2(1 + λ2)R(τ)(I2 + λJ)R(θ)−1,

where 0 < λ ≤ 1. Then X ∈ S3+ and X(0, 0, 1) = E11. This is the local

coordinates on S3+. Suppose[

a bc d

]= X(τ, θ, λ).

Then we obtain the following relations :

λ =

√1− 4(ad− bc)2

1 + 2(ad− bc), tan 2τ =

−2(ac+ bd)

a2 + b2 − c2 − d2, tan 2θ =

−2(ab+ cd)

a2 − b2 + c2 − d2.

Moreover, we obtain the following relations :

tan(τ − θ) =b− c

a+ d, tan(τ + θ) =

b+ c

d− a.

Put

Y0(τ, θ, λ) = (R(τ)e1, R(θ)e2, (λ,√1− λ2)), 0 ≤ λ ≤ 1.

This is the local coordinates on M(α,β). Put

Y1(τ, θ, λ) = (R(τ)e1, R(θ)(λe1 + e2)/√1 + λ2, R(θ)e2),

Y2(τ, θ, λ) = (R(τ)e1, R(τ)(e1 + λe2)/√1 + λ2, R(θ)e2).

These are the local coordinates on M1D and M2D, respectively.Next, we consider the tangent vector fields on these manifolds correspond-

ing to the one-parameter group {g(t) | t ∈ R}, where

g(t) =

([1 t0 1

],

[1 t0 1

]).

Put

X(τ(t), θ(t), λ(t)) = ψ(g(t), X(τ, θ, λ)).

Then we see

τ(0) = (1 + λ−1 sin(τ + θ) sin(τ − θ)− λ cos(τ + θ) cos(τ − θ))/2,

θ(0) = (1 + λ−1 sin(τ + θ) sin(τ − θ) + λ cos(τ + θ) cos(τ − θ))/2,

λ(0) = (1− λ2)(sin 2θ − sin 2τ)/2.

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Fuichi UCHIDA

The tangent vector field on S3+ can be described by these data. Put

Y0(τ(t), θ(t), λ(t)) = ψ(α,β)(g(t), Y0(τ, θ, λ)).

Then we see

τ(0) = sin2 τ, θ(0) = cos2 θ, λ(0) = λ(λ2 − 1)(α sin 2τ − β sin 2θ).

The tangent vector field on M(α,β) can be described by these data. Put

Y1(τ(t), θ(t), λ(t)) = ψ1D(g(t), Y1(τ, θ, λ)).

Then we see

τ(0) = sin2 τ, θ(0) = cos2 θ, λ(0) = −λ2(1 + sin 2θ)/(1 + λ sin 2θ).

The tangent vector field on M1D can be described by these data. Put

Y2(τ(t), θ(t), λ(t)) = ψ2D(g(t), Y2(τ, θ, λ)).

Then we see

τ(0) = sin2 τ, θ(0) = cos2 θ, λ(0) = λ(sin 2τ − λ cos 2τ).

The tangent vector field on M2D can be described by these data.Now we introduce new local coordinates (τ1, θ1, µ) on S3

+ by

X1(τ1, θ1, µ) = X(τ1 + a(µ), θ1 + b(µ), c(µ)).

Here, a(µ), b(µ), c(µ) are smooth functions on 0 < µ ≤ 1 satisfying a(1) =b(1) = 0, c(1) = 1 and c′(1) > 0. Put

X1(τ1(t), θ1(t), µ(t)) = ψ(g(t), X1(τ1, θ1, µ)).

Then we see

τ1(0) = τ(0)− a′(µ)µ(0), θ1(0) = θ(0)− b′(µ)µ(0), µ(0) = (c−1)′(c(µ))λ(0).

In particular, we see

µ(0) = 0 (µ = 1),∂

∂µ(µ(0)) = sin 2τ1 − sin 2θ1 (µ = 1).

Hence we obtain

∂

∂µ(τ1(0)) =

−c′(1)2

�= 0 at(τ1, θ1, µ) = (0, 0, 1).

On the other hand,

∂

∂λ(τ(0)) = 0

for M(α,β),M1D and M2D, respectively.This fact shows the G-action on S3

+ ∪ M1 is not C2-differentiable forM1 = M(α,β),M1D or M2D. q.e.d.

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5 Certain smooth actions of SL(2,R)×SL(2,R) on 3-manifolds, II

5.1. Considering the stereographic projection, we obtain the smoothaction φ of SL(2,R) on R2 as follows :

φ(A,X) =2

1− ‖X‖2 +√(1− ‖X‖2)2 + 4‖AX‖2

AX.

The origin is invariant and the other invariant sets are the followings :

{X ∈ R2 | 0 < ‖X‖ < 1}, {X ∈ R2 | ‖X‖ = 1}, {X ∈ R2 | ‖X‖ > 1}.

Put M+ = S1 × D2 and M− = D2 × S1 and define

φ+((A,B), (u,v)) = (‖Au‖−1Au, φ(B,v)),φ−((A,B), (u,v)) = (φ(A,u), ‖Bv‖−1Bv).

Then φ+ is a smooth G-action on M+ and φ− is a smooth G-action on M−.The isotropy subgroup at (e1, 0) ∈ M+ coincides with N(2)0 × SL(2,R),

the one at (0, e1) ∈ M− coincides with SL(2,R) × N(2)0, and the one at(e1, e1) coincides with N(2)0 ×N(2)0 for M+ and M−.

5.2. Let ϕ(α,β) be the G-action on T 3 = S1 × S1 × S1 defined by

ϕ(α,β)((A,B), (u,v, (λ, µ)))

= (‖Au‖−1Au, ‖Bv‖−1Bv, (cλ‖Au‖α‖Bv‖β, cµ‖Au‖−α‖Bv‖−β)),

where c is a positive real number. Here α, β are fixed real numbers such that(α, β) �= (0, 0).

The isotropy subgroup at (e1, e1, (λ, µ)) coincides with

N(2)0 ×N(2)0 for λµ = 0 and T (−β, α, 0) for λµ �= 0,

respectively. Put

M(α,β) = S1 × S1 × S1+,

where S1+ = {(λ, µ) ∈ S1 | λ ≥ 0, µ ≥ 0}. This is an invariant closed subset.

Remark. M+,M− and M(α,β) are compact 3-manifolds with smooth G-action, each of which has a boundary. M+,M− have a connected boundaryand M(α,β) has just two boundary components. Each boundary componentof the above 3-manifolds is diffeomorphic to G/(N(2)0 ×N(2)0) as a smoothG-manifold.

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Fuichi UCHIDA

5.3. Let ξ be a smooth R2-action on R. Now, we shall define a smoothG-action ξ on S1 × S1 × R by ξ. Let

ρ : SO(2)×D(2)0 × L(2) → SL(2,R)

be the smooth bijection defined by the matrix multiplication ρ(x, y, z) = xyz.The inverse mapping of ρ defines a smooth decomposition of SL(2,R) toSO(2)×D(2)0 × L(2). Put

D(α) =

[eα 00 e−α

].

We identify R2 with the following subgroup of G.

{(D(α), D(β)) | α, β ∈ R}.Define

ξ((A,B), (R(τ)e1, R(θ)e1, t)) = (R(τ ′)e1, R(θ′)e1, ξ((α, β), t)).

Here A,B ∈ SL(2,R) and

AR(τ) = R(τ ′)D(α)L1, BR(θ) = R(θ′)D(β)L2

for some L1, L2 ∈ L(2). Then we see ξ is a well-defined smooth G-action onS1 × S1 × R.

6 Continuous actions of SL(2,R)×SL(2,R) on S3 of the second type

In the previous section, we prepare the compact 3-manifoldsM+,M− andM(α,β) with smooth action of G = SL(2,R)×SL(2,R). Consider the disjointunion

M0 ∪M1 ∪ · · · ∪Mr ∪Mr+1.

Here, M0 = M+,Mr+1 = M− and Mi = M(αi,βi)(i = 1, 2, · · · , r). Put Mthe space obtained from the above disjoint union pasting the boundariesone after another by equivariant G-diffeomorphisms. Then, we see M isnaturally diffeomorphic to S3 with the standard K-action of the second type.Moreover, M has the natural continuous G-action. In such a way, we obtainan extended continuous G-action of the standardK-action of the second typeon S3. In consequence, we obtain the following result.

Theorem 6.1. For each positive integer r, there exist uncountably manytopologically distinct continuous SL(2,R) × SL(2,R) actions on S3, which

74


are extensions of the standard SO(2)× SO(2) action of the second type andthe number of open orbits of which is r + 2.

Next, we shall construct smooth G-actions on S3, which are extensionsof the standard K-action of the second type.

Let σ(t) be a smooth real valued function such that

σ(t) = 1 for |t| ≤ 1,σ(t) = 0 for |t| ≥ 2,

0 < σ(t) < 1 for 1 < |t| < 2.

Define a tangent vector field Ξr on R by

Ξr =

r+1∑i=0

εiσ(t+ 2− 5i)d

dt.

Here ε0 = −1, εr+1 = 1 and εi = ±1 for i = 1, 2, · · · , r. The support of thevector field Ξr is contained in the closed interval [−4, 5r + 5].

Let φr be the one-parameter group on R corresponding to the vector fieldΞr. Now we define a smooth R2-action ξr on the open interval (−4, 5r + 5)by

ξr((s, t), x) =

φr(t, x) for − 4 < x < 0φr(s, x) for 5r + 1 < x < 5r + 5φr(ais+ bit, x) for 5i− 4 < x < 5i (i = 1, 2, · · · , r).

Here, ai, bi are given real numbers such that (ai, bi) �= (0, 0). The smoothnessof ξr is assured by the existence of the neutral zones.

By 5.3, we obtain a smooth G-action ξr on Mr = S1 × S1 × (−4, 5r + 5)from ξr. Then the isotropy subgroup at (e1, e1,−2) coincides with N(2)0 ×L(2), the one at (e1, e1, 5r + 3) coincides with L(2)× N(2)0 and the one at(e1, e1, 5i+ 3) coincides with T (−bi, ai, 0).

Put M+ = S1 × R2 and M− = R2 × S1 and define

φ+((A,B), (u,v)) = (‖Au‖−1Au, φ(B,v)),

φ−((A,B), (u,v)) = (φ(A,u), ‖Bv‖−1Bv).

Then φ+ is a smooth G-action on M+ and φ− is a smooth G-action on M−.Consider the disjoint union

M+ ∪ Mr ∪ M−.

Put M the space obtained from the above disjoint union, identifying the openset S1 × {X ∈ R2 | ‖X‖ > 1} of M+ with the open set S1 × S1 × (−4, 0)

75

Fuichi UCHIDA

of Mr by G-diffeomorphism and the open set {X ∈ R2 | ‖X‖ > 1} × S1 ofM− with the open set S1 ×S1 × (5r+1, 5r+5) of Mr by G-diffeomorphism.Then, M has a smooth G-action. In consequence, we obtain the followingresult.

Theorem 6.2. For each positive integer r, there exist uncountably manytopologically distinct smooth SL(2,R) × SL(2,R) actions on S3, which areextensions of the standard SO(2)× SO(2) action of the second type and thenumber of open orbits of which is r + 4.

Remark. Let c be a positive real number. Define

ϕc((A,B),u ⊕ v) = eθAu ⊕ ecθBv.

Here, θ ∈ R is determined by the condition

e2θ‖Au‖2 + e2cθ‖Bv‖2 = 1.

Then, ϕc is a smooth action of SL(2,R)×SL(2,R) on S3, which is called thetwisted linear action [4]. The isotropy subgroup at (e1 ⊕ e1)/

√2 coincides

with T (1, c, 0).

7 Final remark

On smooth actions of non-compact semi-simple Lie groups on low dimen-sional manifolds, T.Asoh [1] and O.Yokoyama [9] have studied on smoothSL(2,C)-actions on 3-manifolds, and K.Mukoyama [2] has studied on smoothSp(2,R)-actions on the 4-sphere. On the present article, we study on certaincontinuous SL(2,R)× SL(2,R)-actions on the 3-sphere.

Let G be a non-compact semi-simple Lie group and K be the maximalcompact subgroup of G. On smooth G-actions on a sphere of which restrictedK-action has principal orbits of codimension one, F.Uchida [5,6,7,8] has stud-ied such actions for G = SO0(p, q),Sp(p, q) and SL(m,R)× SL(n,R), andK.Mukoyama [3] has studied such actions for G = SU(p, q). On the presentarticle, a complementary part of [8] is studied.

References

[1] T.Asoh, On smooth SL(2,C) actions on 3-manifolds, Osaka J. Math. 24(1987), 271–298.

[2] K.Mukoyama, Smooth Sp(2,R)-actions on the 4-sphere, Tohoku Math.J. 48 (1996), 543–560.

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[3] K.Mukoyama, Smooth SU(p, q)-actions on the (2p+2q−1)-sphere and onthe complex projective (p+ q− 1)-space, Kyushu J. Math. 55 (2001),213–236.

[4] F.Uchida, On a method to construct analytic actions of non-compact Liegroups on a sphere, Tohoku Math. J. 39 (1987), 61–69.

[5] F.Uchida, On smooth SO0(p, q)-actions on Sp+q−1, Osaka J. Math. 26(1989), 775–787.

[6] F.Uchida, On smooth SO0(p, q)-actions on Sp+q−1,II, Tohoku Math. J.49 (1997), 185–202.

[7] F.Uchida, On smooth Sp(p, q)-actions on S4p+4q−1, Osaka J. Math. 39(2002), 293–314.

[8] F.Uchida, On smooth SL(m,R) × SL(n,R) -actions on Sm+n−1, Inter-discip. Inform. Sci. 18 (2002), 123–128.

[9] O.Yokoyama, A note on smooth SL(2,C)-actions on 3-dimensional closedmanifolds, Interdiscip. Inform. Sci. 2-1 (1996), 89–102.

77


Inhibitions of Growth and Lateral BranchDevelopment by Calmodulin Antagonists in

Hairy Roots of Lithospermum erythrorhizon,Atropa belladonna and Daucus carota

Ryoichi Kato†, Miyuki Arashida†, Masahiro Kodama†,Hiroshi Kamada‡ and Takashi Suzuki††

(Received March 29, 2002)

Abstract

Hairy roots of Lithospermum erythrorhizon, belladonna and carrot,which were induced by inoculation with Agrobacterium rhizogenes har-boring the Ri plasmid, were cultured on a medium containing 0.1, 1,10, 30 or 100µM W-7 or W-5, calmodulin antagonists. Growth ratesof L. erythrorhizon and belladonna hairy roots cultured on all mediacontaining W-7 were lower than that of the roots cultured without W-7. Growth of carrot hairy roots was inhibited by W-7 above 30µM.Inhibition rates of the root growth by high concentrations of W-5 werelower than those of the growth by the same concentrations of W-7. Inthe case of the development of lateral roots on hairy roots, 30 and100 µM W-7 or W-5 inhibited formation of lateral roots. The numberof lateral roots formed by culturing on a medium containing W-7 waslower than that of the roots formed on the medium containing W-5.These results strictly suggest that calmodulin acts upon the growthof hairy roots and the development of lateral roots on hairy roots.

†Biology Laboratory, Faculty of Education, Yamagata University, Yamagata, 990–8560Japan

‡Institute of Biological Sciences, University of Tsukuba, Tsukuba, Ibaragi, 305–8572Japan

††Science Laboratory, Faculty of Education, Yamagata University, Yamagata, 990–8560Japan

79

Ryoichi Kato, Miyuki Arashida, Masahiro Kodama, Hiroshi Kamada and Takashi Suzuki

Abbreviations

W-7, N-(6-aminohexyl)-5-chloro-1-naphthalenesulfonamide; W-5, N-(6-aminohexyl)-naphthalenesulfonamide.

Key words

Atropa belladonna—Calmodulin antagonist—Daucus carota—Hairyroots—Root growth and lateral branch development—Lithospermumerythrorhizon.

1 Introduction

Calmodulin plays important roles in many physiological processes in plantcells as well as animal cells (Dieter 1984, Sun 2000, Chung et al. 2000,Lenartowska et al. 2001). Muto and Hirosawa (1987) reported that growth ofadventitious roots formed in stem cuttings of Tradescantia fluminensis couldbe inhibited by incubation in a solution containing a low concentration of acalmodulin antagonist: trifluoperazine or chlorpromazine, and they discussedinvolvement of calmodulin in the growth of the roots. These antagonists bindvery tightly to isolated calmodulin (Levin and Weiss 1976, 1977). However,an affinity of the antagonists for cytoplasmic membrane is so high that theantagonists seriously damage cultured mammalian cells (Osborn and Weber1980). These reports suggest that using of trifluoperazine or chlorpromazinein in vivo experiments is unsuitable for detecting the action of calmodulin.

It has been clearly demonstrated that the Ri plasmid present in Agrobac-terium rhizogenes causes the transformation of plant cells via the introduc-tion of T-DNA from the Ri plasmid into the genomic DNA of plant cells. Ithas also been demonstrated that the transformed plant cells give rise to mas-sive roots, the so-called hairy roots (White and Nester 1980, Tepfer 1984,Shanks and Morgan 1999, Giri and Narasu 2000). The transformed plantcells can continue to produce hairy roots even after Agrobacterium was elim-inated. Hairy roots grow vigorously in a phytohormone-free medium andprovide a useful material for studies on secondary metabolite production(Kamada et al. 1986, Bourgaud et al. 2001, Facchini 2001, Kim et al. 2002).But, there are very few physiological investigations into hairy root growth(Kato et al. 1989, Bonhomme et al. 2000, Tanaka et al. 2001) and lateralbranch development on hairy roots (Morgan and Shanks 2000, Balvanyos etal. 2001).

In this report, we investigated the involvement of calmodulin in thegrowth of hairy roots and the development of lateral branches on the roots,

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Hairy roots and calmodulin

using W-7 and W-5 as calmodulin antagonists. W-7 and W-5 can penetratecytoplasmic membrane and distribute mainly in cytoplasm (Hidaka et al.1981).

2 Materials and Methods

Plant material—The procedures for the induction of hairy roots were de-scribed by Kato et al. (1989) and by Shimomura et al. (1991). Carrot seeds(Daucus carota L. cv. US-Harumakigosun) were sown on vermiculite at 25◦Cunder the continuous light (ca. 3 k lux). One-week-old seedlings were surface-sterilized with 1% sodium hypochlorite solution for 15min and rinsed threetimes with sterilized distilled water. Sterile belladonna (Atropa belladonnaL.) and Lithospermum erythrorhizon Sieb. et Zucc. plants were obtained byculturing the shoot tips of field-grown plants. They were maintained by re-peated shoot culture on a Murashige and Skoog’s (MS) medium containing3% sucrose and 1% agar, at 25◦C under a 18-h light (ca. 4 k lux) / 6-h darkcondition. Carrot hypocotyls, belladonna and L. erythrorhizon internodeswere cut into 20-mm-long segments and placed with the basal part uppermoston the MS medium. The cut ends of these segments were inoculated by a nee-dle with Agrobacterium rhizogenes, which had been grown on YEB medium(Vervliet et al. 1974) containing 1.5% agar. The segments of belladonna andL. erythrorhizon were infected with A. rhizogenes strain 15834 and those ofcarrot were infected with strain 2659. Hairy roots appeared at the inocula-tion sites three weeks later. The growing root tips of the hairy roots werecut off and cultured on an MS medium containing 3% sucrose, 1% agar andan antibiotic (1mg/ml carbenicillin) at 25◦C under a 16-h light (ca. 3 k lux)/ 8-h dark condition, to eliminate the bacteria from the roots. The roottips were subcultured three more times at two-week intervals on a fresh MSmedium supplemented with the antibiotic. The root tips of belladonna andcarrot hairy roots were re-subcultured on an MS medium containing 3% su-crose and 0.2% Gellan Gum (Kelco, Division of Merck & Co., Inc., San Diego,U.S.A) without antibiotic at monthly intervals at 25◦C. The tips of L. ery-throrhizon hairy roots were re-subcultured at two-month intervals at 25◦C onthe same medium and later on a Root Culture (RC) medium [EMBO coursemanual 1982 as follows (mg/l): Ca(NO3)2 ·4H2O (288), KNO3 (80), KCl (65),MgSO4 ·7H2O (370), Na2SO4 ·10H2O (226.7), NaH2PO4 ·4H2O (21.5), H3BO3(1.5), ZnSO4 ·7H2O (2.65), KI (0.75), Na2MoO4 ·2H2O (0.25), CuSO4 ·5H2O(0.02), MnCl2 ·4H2O (6.0), FeCl3 (3.2), EDTA-2Na (7.4), thiamine-HCl (0.1),nicotinic acid (0.5), pyridoxin-HCl (0.1) and glycine (3.0)] containing 1.5%sucrose and 0.3% Gellan Gum (pH 4.9).

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Incubation with calmodulin antagonists—Segments (length: 15mm) ofroot tips were cut from these hairy roots. Ten segments of belladonna andcarrot hairy roots were put side by side in a Petri dish (diameter: 94mm)on an MS medium containing 3% sucrose, 0.2% Gellan Gum and variousconcentrations (zero, 0.1, 1, 10, 30 and 100µM) of W-7 or W-5. Ten seg-ments of L. erythrorhizon hairy roots were put side by side in the same dishon a RC medium containing 1.5% sucrose, 0.3% Gellan Gum and variousconcentrations of W-7 or W-5. The dishes were covered with a double thick-ness of aluminum foil. They were incubated at 24◦C for 6, 8, 13 or 16 days.Growth of each hairy root was measured with a slide caliper. At the carrothairy roots, the number of lateral branches formed on the roots was visuallycounted.

W-7 and W-5 were obtained from Seikagaku Corporation, Tokyo, Japan.

3 Results

Established cultures of hairy roots exhibited rapid growth with lateralroots on the hormone-free media. In order to confirm that true transfor-mation had occurred, the presence of opines (mannopine and agropine, orcucumopine) was monitored by the method described earlier (Kamada et al.1986, Isogai et al. 1990), and both opines were detected (data not shown).

Growth rates—Thirty segments of Lithospermum erythrorhizon hairy rootswere cultured on a RC medium containing zero, 0.1, 1, 10, 30 or 100µM W-7for 13 days. Growth rates of the hairy roots cultured on the all media con-taining W-7 were lower than that of the roots cultured without W-7 (zero),and then the rates of the roots became lower as W-7 concentrations increased(Fig. 1). Segments of L. erythrorhizon hairy roots were cultured for 13 dayson the medium containing W-5 in the same concentrations of W-7. Growthrates of hairy roots cultured on the media containing W-5 below 10µM wereequal to that of the roots cultured without W-5 (zero), but then the ratesof the roots cultured on the media containing W-5 above 30µM were lowerthan that of the roots cultured without W-5 (Fig. 1).

We tried to determine whether the calmodulin antagonists inhibit thegrowth of hairy roots in other plant species. Segments of belladonna hairyroots were cultured on an MS medium containing zero, 0.1, 1, 10, 30 or100µM of W-7 or W-5 for 6 days. Growth of the roots was inhibited byW-7 above 0.1µM or by W-5 above 1µM (Fig. 2). Growth rates of the rootscultured on the medium containing W-7 were lower than those of the rootscultured on the medium containing W-5 at 10, 30 and 100µM antagonists,respectively. And then a difference between inhibition rates by W-7 and W-5

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Fig. 1. Effects of W-7 and W-5 on the growthof Lithospermum erythrorhizon hairy roots. Seg-ments (length: 15mm) of the tips of L. erythrorhi-zon hairy roots were cultured on an MS mediumcontaining zero, 0.1, 1, 10, 30 or 100µM W-7 (●)or W-5 (○), in the dark at 24◦C for 13 days. Eachpoint represents the average of measurements fromat least 30 segments of hairy roots, with the stan-dard error.

Fig. 2. Effects of W-7 and W-5 on the growth ofbelladonna hairy roots. Segments of the tips ofbelladonna hairy roots were cultured on an MSmedium containing zero, 0.1, 1, 10, 30 or 100µMW-7 (●) or W-5 (○), in the dark at 24◦C for 6days. Each point represents the average of mea-surements from at least 30 segments of hairy roots,with the standard error.

became higher as antagonists concentrations increased (Fig. 2).Segments of carrot hairy roots were cultured on an MS medium containing

zero, 0.1, 1, 10, 30 or 100µM of W-7 or W-5 for 8 days. Growth rates of theroots cultured on three media containing 0.1, 1 and 10µM antagonists wereequivalent to those of the roots cultured without antagonist (zero) (Fig. 3).Thirty micro-mole W-5 did not inhibit the root growth, but 30µM W-7 did(Fig. 3). Inhibition rate of the root growth by 100µM W-7 was higher thanthat of the growth by 100µM W-5 (Fig. 3).

Number of lateral branches—The effects of the calmodulin antagonists onthe development of lateral roots on hairy roots were investigated in the nextexperiments. Thirty segments of carrot hairy roots were cultured on an MSmedium containing zero, 0.1, 1, 10, 30 or 100µM of W-7 or W-5 for 16 days.The number of the lateral roots formed on the media containing antagonists

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Fig. 3. Effects of W-7 and W-5 on the growth ofcarrot hairy roots. Segments of the tips of car-rot hairy roots were cultured on an MS mediumcontaining zero, 0.1, 1, 10, 30 or 100µM W-7 (●)or W-5 (○), in the dark at 24◦C for 8 days. Eachpoint represents the average of measurements fromat least 30 segments of hairy roots, with the stan-dard error.

Fig. 4. Effects of W-7 and W-5 on the developmentof lateral roots on carrot hairy roots. Segmentsof the tips of carrot hairy roots were cultured onan MS medium containing zero, 0.1, 1, 10, 30 or100µM W-7 (●) or W-5 (○), in the dark at 24◦Cfor 16 days. Each point represents the average ofmeasurements from at least 30 segments of hairyroots, with the standard error.

below 10µM was equal to that of the roots formed on an antagonist-freemedium (zero), and then the number of lateral roots formed on antagonistsabove 30µM was lower than that of the roots formed on an antagonist-freemedium (Fig. 4). At 30 and 100µM the antagonists, the number of lateralroots induced on the medium containing W-7 was lower than that of theroots induced on the medium containing W-5 (Fig. 4).

4 Discussion

Calmodulin regulates various physiological responses in animal and plantcells (Dieter 1984, Sun 2000, Chung et al. 2000, Lenartowska et al. 2001).It was shown that calmodulin involves in growth of adventitious roots in-duced in stem cuttings of Tradescantia fluminensis, by an incubation of theroots in a solution containing trifluoperazine or chlorpromazine (Muto andHirosawa 1987). These calmodulin antagonists, however, tightly combine

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with cytoplasmic membrane and damage the cells without direct inhibitionof the calmodulin action (Osborn and Weber 1980). Applications of theseantagonists to intact cells or tissues are unfit for a detection of the calmod-ulin function. Hairy roots, which are originated from transformation of plantcells by the Ri plasmid of Agrobacterium rhizogenes, have been induced fromsome plant materials to produce secondary metabolites (Kamada et al. 1986,Bourgaud et al. 2001, Facchini 2001, Kim et al. 2002). But, only a few studieshave been performed on the analyses of hairy roots growth (Kato et al. 1989,Bonhomme et al. 2000, Tanaka et al. 2001) and lateral branch developmenton hairy roots (Morgan and Shanks 2000, Balvanyos et al. 2001).

We determined involvement of calmodulin in the growth of hairy rootsusing W-7 and W-5, which are able to penetrate cytoplasmic membrane(Hidaka et al. 1981). Segments of L. erythrorhizon, belladonna and carrothairy roots were cultured on media containing various concentrations of W-7. Growth rates of L. erythrorhizon and belladonna hairy roots culturedon the media containing W-7 were lower than those of the roots culturedwithout W-7 (Figs. 1 and 2). Growth of carrot hairy roots was inhibited byW-7 above 30µM (Fig. 3). Segments of those roots were cultured on mediacontaining W-5, a chlorine-deficient analogue of W-7 that weakly interactswith calmodulin (Hidaka et al. 1981), at the same concentrations of W-7 tomake a close examination of the role of calmodulin in the growth of hairyroots. Inhibition rates of the root growth by high concentrations of W-5 werelower than those of the growth by the same concentrations of W-7 (Figs. 1, 2and 3). These results strictly suggest that calmodulin is concerned in hairyroot growth. To the best of our knowledge, this is the first report provedexactly on involvement of calmodulin in the growth of plant roots.

In order to determine whether the calmodulin antagonists inhibit devel-opment of lateral roots on hairy roots, segments of carrot hairy roots werecultured on an MS medium containing various concentrations of W-7 or W-5.At 30 and 100µM the antagonists, both W-7 and W-5 inhibited formation oflateral roots and the number of lateral roots formed on the medium contain-ing W-7 was lower than that of the roots formed on the medium containingW-5 (Fig. 4). These results show that calmodulin acts upon the developmentof lateral roots on hairy roots.

The involvement of calmodulin in cell division in plant cells is suggestedby the following facts. (1) Young dividing and growing cells contained morecalmodulin than matured non-growing cells in pea seedlings (Muto and Miya-chi 1984). (2) Distribution of calmodulin related to that of tubulin in thecell cycle of onion and pea root meristematic cells (Wick et al. 1985). (3)Calmodulin was localizing at kinetochore microtubles in the mitotic appara-tus in Haemanthus endosperm cells (Vantard et al. 1885). Growth of hairy

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roots consists of cell division and cell elongation. An important event inthe development of lateral roots on hairy roots is cell division, which causesprimordium formation of lateral roots. Inhibition rate of hairy root growthby each concentration of W-7 or W-5 shown in Figure 3 is similar to thatof formation of lateral roots of the roots by W-7 or W-5 shown in Figure 4.These results suggest that calmodulin plays an important role in cell di-vision in hairy roots. These may have some reference to the reports thatcalmodulin-binbing protein is involved in cell division in pea seedlings (Dayet al. 2000) and that expression of calmodulin mRNA is closely correlatedwith cell division in tobacco anthers (Chen et al. 1999).

References

[Balvanyos et al. 2001] Balvanyos, I., L. Kursinszki and E. Szoke, PlantGrowth Regul. 34: 339-345. 2001.

[Bonhomme et al. 2000] Bonhomme, V., D. Laurain-Mattar and M. A. Flini-aux, J. Nat. Prod. 63: 1249-1252. 2000.

[Bourgaud et al. 2001] Bourgaud, F., A. Gravot, S. Milesi and E. Gontier,Plant Sci. 161: 839-851. 2001.

[Chen et al. 1999] Chen, S. R., Y. T. Lu and H. Y. Yang, Chinese Sci. Bull.44: 142-146. 1999.

[Chung et al. 2000] Chung, W. S., S. H. Lee, J. C. Kim, W. D. Heo, M. C.Kim, C. Y. Park, H. C. Park, C. O. Lim, W. B. Kim and J. F. Harper,Plant Cell 12: 1393-1407. 2000.

[Day et al. 2000] Day, I. S., C. Miller, M. Golovkin and A. S. N. Reddy, J.Biol. Chem. 275: 13737-13745. 2000.

[Dieter 1984] Dieter, P., Plant Cell Environ. 7: 371-380. 1984.

[EMBO course manual 1982] EMBO course manual, The use of Ti plasmidas cloning vector for genetic engineering in plants, August 4-23, p. 109.1982.

[Facchini 2001] Facchini, P. J., Annu. Rev. Plant Physiol. 52: 29-66. 2001.

[Giri and Narasu 2000] Giri, A. and M. L. Narasu, Biotechnol. Adv. 18: 1-22. 2000.

[Hidaka et al. 1981] Hidaka, H., Y. Sasaki, T. Tanaka, T. Endo and S. Ohno,

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Proc. Natl. Acad. Sci. U.S.A 78: 4354-4357. 1981.

[Isogai et al. 1990] Isogai, A., N. Fukuchi, M. Hayashi, H. Kamada, H.Harada and A. Suzuki, Phytocem. 29: 3131-3134. 1990.

[Kamada et al. 1986] Kamada, H., H. Okamura, M. Satake, H. Harada andK. Shimomura, Plant Cell Reports 5: 239-242. 1986.

[Kato et al. 1989] Kato, R., H. Kamada and M. Asashima, Plant Cell Phys-iol. 30: 605-608. 1989.

[Kim et al. 2002] Kim, Y., B. E. Wyslouzil and P. J. Weathers, In Vitro CellDev-Pl. 38: 1-10. 2002.

[Lenartowska et al. 2001] Lenartowska, M., M. I. Rodriguez-Garcia and E.Bednarska, Acta Biol. Cracov. Bot. 43: 117-123. 2001.

[Levin and Weiss 1976] Levin, R. M. and B. Weiss, Mol. Pharmacol. 12: 581-589. 1976.

[Levin and Weiss 1977] Levin, R. M. and B. Weiss, Mol. Pharmacol. 13: 690-697. 1977.

[Morgan and Shanks 2000] Morgan, J. A. and J. V. Shanks, J. Biotechnol.79: 137-145. 2000.

[Muto and Hirosawa 1987] Muto, S. and T. Hirosawa, Plant Cell Physiol.28: 1569-1574. 1987.

[Muto and Miyachi 1984] Muto, S. and S. Miyachi, Z. Pflanzenphysiol. 114:421-431. 1984.

[Osborn and Weber 1980] Osborn, M. and K. Weber, Exp. Cell Res. 130:484-488. 1980.

[Shanks and Morgan 1999] Shanks, J. V. and J. Morgan, Curr. Opin.Biotech. 10: 151-155. 1999.

[Shimomura et al. 1991] Shimomura, K., H. Sudo, H. Saga and H. Kamada,Plant Cell Reports 10: 282-285. 1991.

[Sun 2000] Sun, D. Y., Acta, Bot. Sin. 42: 441-445. 2000.

[Tanaka et al. 2001] Tanaka, N., Y. Fujikawa, M. A. M. Aly, H. Saneoka, K.Fujita and I. Yamashita, Plant Cell Tiss. Org. 66: 175-182. 2001.

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[Tepfer 1984] Tepfer, D. A., Cell 37: 959-967. 1984.

[Vantard et al. 1985] Vantard, M., A.-M. Lambert, J. De Mey, P. Picquotand L. J. Van Eldik, J. Cell Biol. 101: 488-499. 1985.

[Vervliet et al. 1974] Vervliet, G., M. Holsters, H. Teuchy, M. Van Montaguand J. Schell, J. Gen. Virol. 26: 33-48. 1974.

[White and Nester 1980] White, F. F. and E. W. Nester, J. Bacteriol. 141:1134-1141. 1980.

[Wick et al. 1985] Wick, S. M., S. Muto and J. Duniec, Protoplasma 126:198-206. 1985.

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In Vitro Phosphorylation of Proteins inIAA-Treated Primary Roots and Coleoptiles in

Zea mays

Ryoichi Kato†, Sachiko Takatsuna†, Tsuyoshi Wada†,Yumi Narihara† and Takashi Suzuki‡

(Received March 29, 2002)

Abstract

Five-mm sections of elongation zones which were cut from primaryroots or coleoptiles of Zea mays were incubated for designated periodswith various concentrations of IAA. In vitro protein phosphorylationin the soluble fractions prepared from these sections was analyzed bySDS-PAGE. The phosphorylation of proteins in sections of primaryroots incubated for 20 or 40min in the presence of 10−7 M IAA wasgreater than that in the sections incubated in the absence of IAA. Thephosphorylation of proteins in sections of primary roots incubated for20min or 2 h in the presence of 10−8, 10−7 or 10−6 M IAA was higherthan that in the sections incubated in the absence of IAA. An incuba-tion for 20min or 2 h with 10−4 M IAA inhibited the phosphorylationof proteins in sections of primary roots. The growth of the sectionsof primary roots incubated for 2 h in the presence of 10−7 M IAA orhigher concentrations was lower than that of the sections incubated inthe absence of IAA. These results suggest that the phosphorylationof proteins which was increased by IAA treatment is independent ofan inhibition of the growth induced by IAA in maize primary roots.The phosphorylation of proteins in sections of coleoptiles incubatedfor 10, 20 or 40min in the presence of 10−7 or 10−5 M IAA was equalto that in the sections incubated in the absence of IAA. These resultsshow that IAA regulates growth of maize coleoptiles in no relationwith phosphorylation of the proteins and that IAA regulates growth

†Biology Laboratory, Faculty of Education, Yamagata University, Yamagata, 990–8560Japan

‡Science Laboratory, Faculty of Education, Yamagata University, Yamagata, 990–8560Japan

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Ryoichi Kato, Sachiko Takatsuna, Tsuyoshi Wada, Yumi Narihara and Takashi Suzuki

of maize mesocotyls via phosphorylation of the proteins, comparingwith earlier results [Kato et al. (1996) Plant Cell Phsiol. 37:667].

Abbreviations

EGTA, ethyleneglycol bis(2-aminoethyl ether)tetraacetic acid;IAA, indole-3-acetic acid; MAPK, mitogen-activated protein kinase;MAPKKK, mitogen-activated protein kinase kinase kinase;SDS-PAGE, sodium dodecyl sulfate-polyacrylamide gel electrophore-sis.

Key words

Coleoptile—IAA treatment—Primary root—Protein phosphoryla-tion—Zea mays.

1 Introduction

Auxins control many aspects of plant growth and development. However,the mechanism by which auxin regulates diverse physiological processes isnot clear. Some investigations have been reported regarding protein phos-phorylation in relation with physiological responses in higher plants (Kato etal. 1983, 1984, Ranjeva and Boudet 1987, Kato and Fujii 1988, Briggs andHuala 1999, Harmon et al. 2000, Meszaros and Pauk 2000, Rock 2000, Leonet al. 2001, Muday and DeLong 2001). Auxin action has been associated withchanges in the phosphorylation status of nuclear proteins (Murray and Key1978, Schafer and Kahl 1981), of membrane-located proteins (Morre et al.1984, Varnold and Morre 1985), of ribosomal proteins (Perez et al. 1990) andof soluble proteins (Veluthambi and Poovaiah 1986, Reddy et al. 1987). Ithas been indicated a possibility that auxin influences activation of a nuclearprotein kinase CK2 (Hidalgo et al. 2001), expression of gene of a calmod-ulin like domain protein kinase (Davletova et al. 2001) and MAPK cascades(Mizoguchi et al. 1994, Nakashima et al. 1998, Morris 2001, Wrzaczek andHirt 2001, Zwerger and Hirt 2001) which transduce a large variety of externalsignals in plants (Machida et al. 1997, Meskiene and Hirt 2000, Tena et al.2001). Further, Chono et al. (1998) showed a protein kinase gene responsiveto auxin and gibberellins in cucumber hypocotyls.

Kato et al. (1996) reported that IAA promotes cell elongation via proteinphosphorylation that depends on calmodulin-dependent protein kinase andprotein kinase C in maize mesocotyls. In this report, we determined whichauxin affects phosphorylation of proteins in primary roots and coleoptiles of

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Protein phosphorylation enhanced by IAA

Zea mays.

2 Materials and Methods

Plant materials—The procedures used were previously described by Katoet al. (1991) and Kato et al. (1995). Caryopses of Zea mays L., cv. GoldenCross Bantam 70 (Sakata Seed Co., Yokohama, Japan) were immersed inrunning tap water for 36 h in darkness. The imbibed caryopses were planted,embryo-upwards, on agar-solidified water (0.4% agar) in containers (286 ×190 × 50mm3). The containers were covered with a black vinyl sheet andkept in darkness at 25± 0.5◦C. After an incubation for 24–36 h, the primaryroots with 20–25mm long were selected. Five-mm sections (elongation zone),starting from 1mm below the root tip, were exactly cut from the roots. Onthe other hand, the imbibed caryopses were pierced on the surface of moistvermiculite in containers and covered with a vinyl sheet. The seedlings wereallowed to elongate at 25± 0.5◦C in the dark. Five-mm sections (elongationzone) were cut from 72-h-old coleoptiles starting from 5mm below the tip.Primary leaves were discarded. Handling of the caryopses and other oper-ations were performed under a very weak green light (intensity, less than100µW cm−2; transmittance, 490–580 nm; peak, 530 nm), which was ob-tained by filtering the light from a fluorescent lamp through two sheets ofgreen cellophane and two sheets of blue cellophane.

Phosphorylation of proteins—The procedures were those described byKato et al. (1996). A group consisting of 30 sections of primary roots orcoleoptiles was put into Petri dish (diameter, 51mm; height, 22mm) with1ml of 20mM potassium-phosphate buffer (K-buffer; pH 5.8) containing var-ious concentrations of IAA (Wako Pure Chemical Industries, Ltd., Osaka,Japan). They were incubated in darkness at 25 ± 0.5◦C for various periodswith shaking at 60 rpm. The sections were then rinsed twice with distilledwater. The following procedures were performed at 4◦C or in an ice-bath.The sections were homogenized by use of an ice-chilled mortar and pestlewith 100µl of 30mM Tris-HCl (pH 7.4) containing 1mM EDTA, 1mM β-mercaptoethanol, 2mM MgCl2 and 8% sucrose (Buffer A). The above oper-ations were performed under the weak green light (described above) and thefollowing procedures were performed under the light condition. The mortarand pestle were rinsed three times with 100µl of Buffer A and the rinses werecombined with the homogenate. The homogenate was centrifuged at 3,000×g for 20min. The resulting supernatant was then centrifuged at 85,000 ×gfor 90min. The supernatant obtained after the second centrifugation wasmixed with 1.6ml (in the case of primary roots) or 1.8ml (in the case of

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coleoptiles) of Buffer A. The supernatant was applied to a Centricon (Ami-con, Inc., Beverly, U.S.A., Model: SR-10), which was centrifuged at 5,000 ×gfor 90min. A further 1.7ml (in the case of primary roots) or 1.8ml (in thecase of coleoptiles) of Buffer A was added to the concentrated supernatant.The procedure was repeated twice. The final centrifugation was at 5,000 ×gfor 120min (in the case of primary roots) or 90min (in the case of coleop-tiles). Approximately 300µl of concentrated and demineralized supernatantwas finally obtained as a sample. The amount of proteins in each sample wasdetermined by use of a Bio-Rad Protein Assay Kit (Bio-Rad Laboratories,California, U.S.A., Model: Kit II), with Bovine serum albumin as a standard.

The reaction mixture (final volume, 100µl) contained 3.3 × 10−2 µM[γ-32P]ATP (3.0–6.0 × 106 cpm pmol−1; Radiochemical Centre, Amersham,Bucks., U.K.) and 80µl of the sample. In some experiments, a cyclic nu-cleotide, a calmodulin antagonist or a protein kinase inhibitor was added tothe mixture. The reactions were initiated by the addition of [γ-32P]ATP,and were allowed to proceed for 4min (in the case of primary roots) or1min (in the case of coleoptiles) at 30◦C. Each reaction mixture was thencombined with 25µl of a Stop solution [50mM Tris-HCl (pH7.0) containing5mM EDTA, 15% sodium dodecyl sulfate (SDS), 5% β-mercaptoethanol,25% glycerol and 0.05% Bromophenol Blue] and heated at 95◦C for 5min.Each volume of the reaction mixture, containing exactly 30µg (in the caseof primary roots) or 20µg (in the case of coleoptiles) of protein, was appliedto lanes of SDS-PAGE. A portion of the reaction mixture was subjected toSDS-PAGE (concentration of polyacrylamide, 11%) according to the proce-dure of Laemmli (1970). The proteins in the gel were stained with Coomassiebrilliant blue and the gel was dried and exposed to Kodak X-Omat AR Film(Eastman Kodak Co., Rochester, New York, U.S.A.) for 12–24 h. The driedgel and the exposed film were severally scanned by a scanner (Seiko EpsonCo., Suwa-shi, Nagamo, Japan, Model: GT-9000). Total amounts of the gra-dations of the blue in the gel and of the darkness on the film were calculatedwith a personal computer and a software package for image processing (NIHimage) at each lane. We confirmed that the same quantity of protein waspresent in each lane of the gel (data not shown).

Measurement of elongation of primary roots—A group consisting of 30sections of primary roots was put into the Petri dish with 1 ml of K-buffercontaining 10−12～10−4M IAA. The sections were floated on the K-bufferand incubated in darkness at 25± 0.5◦C for 2 h with shaking at 60 rpm. Thelength of each section was measured by use of a travelling microscope (PikaSeiko, Ltd., Tokyo, Japan, Model: PRM-D2).

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3 Results

Phosphorylation of proteins in primary roots—Sections of elongationzones of primary roots were incubated for 10, 20 or 40min in the presence orabsence of 10−7M IAA. At least nine phosphorylated proteins, with molecu-lar masses of 53, 42, 41, 35, 33, 31, 21.5, 18.5 and 16.5 kDa, were detected inthe soluble fraction prepared from the sections (Fig. 1). The extent of phos-phorylation of these proteins in the sections incubated for 20 or 40min withIAA was higher than that in the sections incubated without IAA (Fig. 1).But, a 10-min incubation with IAA had no effect on the phosphorylation ofproteins (Fig. 1).

The root sections were incubated for 20min in the presence of 10−8～10−4M or 10−12～10−8M IAA. The extent of phosphorylation of proteins inthe soluble fraction prepared from the sections incubated with 10−8, 10−7 or10−6M IAA was higher than that in the fraction from the sections incubatedwithout IAA (control) (Fig. 2, 3). The extent of phosphorylation of proteinsin the sections incubated with 10−12～10−9M or 10−5M IAA was nearlyequal to that of the control (Fig. 2, 3). The phosphorylation of proteins inthe sections incubated with 10−4M IAA was lower than that of the control(Fig. 2).

The root sections were incubated for 2 h in the presence of 10−8～10−4MIAA. The extent of phosphorylation of proteins in the soluble fraction pre-pared from the sections incubated with 10−8, 10−7 or 10−6M IAA was higherthan that in the fraction from the sections incubated without IAA (Fig. 4).IAA at 10−4M had the significant effect on the decrease of the phosphoryla-tion of proteins (Fig. 4).

Phosphorylation of proteins in coleoptiles—Sections of elongation zonesof coleoptiles were incubated for 10, 20 or 40min in the presence or absenceof 10−7 or 10−5M IAA. Nine phosphorylated proteins, similar to proteinsdetected in primary roots, were detected in the soluble fraction preparedfrom the sections (Fig. 5, 6). The extent of phosphorylation of these proteinsin the sections incubated with IAA was equal to that in the sections incubatedwithout IAA (Fig. 5, 6).

Elongation of primary roots—Sections of elongation zones of primaryroots were incubated for 2 h in the presence of 10−12～10−4M IAA. Thegrowth rates of the sections incubated with 10−7M IAA or higher concentra-tions were lower than the rates of the sections incubated without IAA (Fig. 7).The inhibition of growth by IAA became stronger as the IAA concentrationincreased (Fig. 7).

All experiments were repeated three times with essentially identical re-sults.

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Fig. 1. Phosphorylation of proteins in sections of maize primary roots incu-bated for 10, 20 or 40min in the presence of 10−7 M IAA. Thirty sections ofthe roots were incubated for 10, 20 or 40min in the presence or absence of10−7 M IAA. The sections were homogenized with an ice-chilled mortar andpestle. The homogenate was centrifuged at 3,000 ×g for 20min. The obtainedsupernatant was centrifuged at 85,000 ×g for 90min. The resulting super-natant was concentrated and demineralized with a Centricon, and each samplewas obtained. The sample was incubated with [γ-32P]ATP for 4min at 30◦C.After stopping the reaction by the addition of a Stop solution, just 30µg pro-teins were separated by SDS-PAGE (concentration of polyacrylamide, 11%).The radioactivity of the proteins was determined by autoradiography. Theexposed film was scanned with a scanner and the amount of the gradationof the darkness on the film was calculated with a personal computer at eachlane. A: Autoradiography. The positions of 53 (53K), 42 (42K), 41 (41K), 35(35K), 33 (33K), 31 (31K), 21.5 (21.5K), 18.5 (18.5K) and 16.5 (16.5K) k Daproteins are indicated by arrows. B: Total amounts of the gradations of thedarkness on the film. Numbers in the bars indicate the relative extent of theamount.

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Fig. 2. Phosphorylation of proteins in sections of maize primary roots incu-bated for 20min in the presence of 10−8～10−4 M IAA. Thirty sections of theroots were incubated for 20min in the presence of 10−8, 10−7, 10−6, 10−5

or 10−4 M IAA. The procedures were the same as those described in thelegend of Fig. 1. A: Autoradiography. The symbols are the same as indicatedin Fig. 1. B: Total amounts of the gradations of the darkness on the film.Numbers in the bars indicate the relative extent of the amount.

4 Discussion

It is well known that auxin regulates many physiological processes. How-ever, these roles at the molecular level are not well understood. We previouslyreported phosphorylation of proteins relating to auxin-induced elongation inthe sections of elongation zones of maize mesocotyls (Kato et al. 1996) whosegrowth is promoted by auxin (Vanderhoef and Brigges 1978, Walton and Ray1981, Yahalom et al. 1988). This report presented evidence for the role ofauxin in protein phosphorylation in the sections of elongation zones of maizeprimary roots whose growth is inhibited by auxin (Bonner and Koepfli 1939,

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Fig. 3. Phosphorylation of proteins in sections of maize primary roots incu-bated for 20min in the presence of 10−12～10−8 M IAA. Thirty sections ofthe roots were incubated for 20min in the presence of 10−12, 10−11, 10−10,10−9 or 10−8 M IAA. The procedures were the same as those described in thelegend of Fig. 1. A: Autoradiography. The symbols are the same as indicatedin Fig. 1. B: Total amounts of the gradations of the darkness on the film.Numbers in the bars indicate the relative extent of the amount.

Pilet and Saugy 1987) and of maize coleoptiles whose growth auxin promotes(Kato and Fujii 1982, Karcz et al. 1990, Cleland 1991).

The phosphorylation of proteins in sections of primary roots incubatedfor 20min or 2 h with 10−8, 10−7 or 10−6M IAA was higher than that in thesections incubated without IAA (Fig. 2, 4). The phosphorylation of proteinsin sections of primary roots incubated for 20min with 10−12～10−9 or 10−5MIAA was almost the same as that in the sections incubated without IAA(Fig. 2, 3). The growth of sections of the roots incubated for 2 h with 10−7MIAA or higher concentrations was lower than that of the sections incubated

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Fig. 4. Phosphorylation of proteins in sections of maize primary roots incu-bated for 2 h in the presence of 10−8～10−4 M IAA. Thirty sections of theroots were incubated for 2 h in the presence of 10−8, 10−7, 10−6, 10−5 or10−4 M IAA. The procedures were the same as those described in the leg-end of Fig. 1. A: Autoradiography. The symbols are the same as indicatedin Fig. 1. B: Total amounts of the gradations of the darkness on the film.Numbers in the bars indicate the relative extent of the amount.

for 2 h without IAA (Fig. 7). These results suggest that the phosphorylationof 53, 42, 41, 35, 33, 31, 21.5, 18.5 and 16.5 kDa proteins is not related withthe inhibition of growth induced by auxin in maize primary roots, althoughauxin regulates the growth via the phosphorylation of these proteins in maizemesocotyls (Kato et. al. 1996).

A 2-h incubation of sections of primary roots with 10−8, 10−7 or 10−6MIAA had the promotive effect on the phosphorylation of proteins in the rootsections (Fig. 4). On the other hand, no changes in the phosphorylation ofproteins were observed in sections of maize mesocotyls incubated for 2 h in the

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Fig. 5. Phosphorylation of proteins in sections of maize coleoptiles incubatedfor 10, 20 or 40min in the presence of 10−7 M IAA. Thirty sections of coleop-tiles were incubated for 10, 20 or 40min in the presence or absence of 10−7 MIAA. The sections were homogenized, and the homogenate was centrifuged at3,000 ×g for 20min. The obtained supernatant was centrifuged at 85,000 ×gfor 90min. The resulting supernatant was concentrated and demineralized,and each sample was obtained. The sample was incubated with [γ-32P]ATPfor 1min at 30◦C. After stopping the reaction, just 20µg proteins were sep-arated by SDS-PAGE (concentration of polyacrylamide, 11%). The otherprocedures were the same as those described in the legend of Fig. 1. A: Au-toradiography. The symbols are the same as indicated in Fig. 1. B: Totalamounts of the gradations of the darkness on the film. Numbers in the barsindicate the relative extent of the amount.

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B

Fig. 6. Phosphorylation of proteins in sections of maize coleoptiles incubatedfor 10, 20 or 40min in the presence of 10−5 M IAA. Thirty sections of coleop-tiles were incubated for 10, 20 or 40min in the presence or absence of 10−5 MIAA. The procedures were the same as those described in the legend of Fig. 5.A: Autoradiography. The symbols are the same as indicated in Fig. 1. B: To-tal amounts of the gradations of the darkness on the film. Numbers in thebars indicate the relative extent of the amount.

presence of IAA (Kato et al. 1996). The difference of the phosphorylation inprimary roots from the phosphorylation in mesocotyls after a 2-h incubationin the presence of IAA may be caused by difference between response ofprimary roots and that of mesocotyls to IAA treatment.

The phosphorylation of proteins in sections of primary roots incubated for20min or 2 h with 10−4M IAA was lower than that in the sections incubatedfor 20min or 2 h without IAA (Fig. 2, 4). This is the first report to our

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Fig. 7. Effects of IAA concentrations on the growth of sections of maize pri-mary roots. Thirty sections of the roots were incubated for 2 h in the presenceof 10−12, 10−11, 10−10, 10−9, 10−8, 10−7, 10−6, 10−5 or 10−4 M IAA. Thelength of each section was measured with a travelling microscope. Each pointrepresents an average of values from 30 sections, with standard errors as in-dicated.

knowledge on the inhibition of protein phosphorylation induced by auxin.The proteins in the soluble fraction (85,000 ×g supernatant) prepared

from sections of primary roots which had been incubated for 20min with10−7M IAA were phosphorylated in the presence of 10µM cyclic nucleotide(cAMP or cGMP), 100µM calmodulin antagonist (W-7 or W-5) or 20µMprotein kinase inhibitor (H-7 or HA1004). However, the addition of cyclicnucleotides or calmodulin antagonists had no effect on the phosphorylation ofproteins, and there was no difference between the extent of phosphorylationof proteins in the presence of H-7 and that of phosphorylation of protein inthe presence of HA1004 (data not shown). A calmodulin-dependent proteinkinase and protein kinase C phosphorylate the proteins when auxin promotesthe growth in maize mesocotyls (Kato et. al. 1996), however, these kinasesmight not be involved in the phosphorylation of the proteins increased byauxin in maize primary roots.

Mizoguchi et al. (1994) reported that the activity of the MAPKs increasedwhen auxin-starved cultured cells of tobacco were treated with auxin for 5or 10min and that the activity did not increase after auxin treatment for15min or longer. Nakashima et al. (1998) showed that transcripts of a proteinkinase gene which is related to MAPKKK were detected as early as 3 h after

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the onset of incubation in the presence of both auxin and cytokinin in leafdiscs of tobacco, but very limited amounts of the transcripts were found byan incubation in auxin alone. Chono et al. (1998) reported that the mRNAlevels of a protein kinase began to increase 30min after treatment with auxinin cucumber hypocotyls and reached a maximal level in 2 h. On the otherhand, a 10-min incubation of the sections with 10−7M IAA had no effecton the phosphorylation of proteins in maize primary roots, but a 20-min,40-min or 2-h incubation with 10−7M IAA had the promotive effect on thephosphorylation (Fig. 1, 4). A difference in respose time to auxin suggestthat the phosphorylation of proteins which is enhanced by auxin in maizeprimary roots might be not related to the MAPK cascades in a tobacco plantand to the protein kinase in cucumber.

The phosphorylation of proteins in sections of coleoptiles incubated for10, 20 or 40min with 10−7 or 10−5M IAA was not higher than that in thesections incubated for 10, 20 or 40min without IAA (Fig. 5, 6). The growthof sections of coleoptiles incubated with 10−7M IAA or higher concentrationswas greater than that of the sections incubated without IAA and the pro-motion of growth of coleoptiles by IAA was greater at higher concentrationsof IAA (Kato and Fujii 1982). On the other hand, the phosphorylation ofproteins in sections of maize mesocotyls incubated for 10, 20 or 40min with10−5M IAA was higher than that in the sections incubated for 10, 20 or40min without IAA and an incubation of maize mesocotyls for 20min with10−7M IAA increased the phosphorylation of proteins (Kato et al. 1996).These results show that auxin regulates growth of coleoptiles in no relationwith phosphorylation of the proteins and that auxin regulates growth ofmesocotyls via phosphorylation of the proteins, in spite of auxin promotingboth growth of coleoptiles (Kato and Fujii 1982, Karcz et al. 1990, Cleland1991) and mesocotyls (Vanderhoef and Brigges 1978, Walton and Ray 1981,Yahalom et al. 1988), in Zea mays. There is a possibility that physiologicalprocess of auxin which promotes growth of coleoptiles was very different fromthe process of auxin which promotes growth of mesocotyls in Zea mays.

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