バイオメトリクス授業資料 iwatawiki lec08 s

(1)

060310391 0560565

8 ×

2016/11/21 13:00-14:45

@1 - 4

1

1.0 1.5 2.0 2.5 3.0

30405060

Environment

Yield

2

2 3 2

3

GxE

• 

home-site advantage

•  GxE

Frankham et al. (2002) IntroducIon to conservaIon geneIcs. Cambridge University Press

3

• 

• •  ^× ^{GxE GEI}

• 

• •  AMMI

•  QxE, QEI

• •  ^GxE -

(2)

(mulI-environment trial: MET)

A B _A B

C D C D

X Y

1 1

mulI-environment trial: MET 1

2

5

vs.

1

2

vs.

i j

6

×

Genotype x environment interacIon (GxE, GEI)

• GxE

• 

1

1.0 1.5 2.0 2.5 3.0

30405060

Environment

Yield

2

2 ²

3

7

1

1 2

2

1 2 1

2

1

1 2

2 1

1

2

G e no ty pic v al ue

GxE

1 2 1 2

1 2

3 4

1 2

noncrossover interac0on

crossover interac0on

8

(3)

GxE

•  ^GxE

– 

GxE

–  ^GxE

9

1 2 3

1 2 1 2 1 2

1 ⁹³ ⁹⁷ ⁹¹ ^{101 92} ⁹⁶

2 ⁹² ⁹⁶ ⁹⁶ 100 107 103

3 ⁹¹ ⁸⁹ ⁹⁸ 102 112 108

10

1 2 ³

9095100105110

Environment

y

y _ijk = µ + g _i + t _j + (gt) _ij + e _ijk

i j k y

_ijk

i

j

i j

k

2

11

1 2 3 g_i

1 y11. y12. y13. y1.. g1^=y1..^−y…

2 y_21. y_22. y_23. y_2.. g₂=y_2..−y_… 3 y_31. y_32. y_33. y_3.. g₃=y_3..−y_…

y_.1. y_.2. y_.3. μ = y…

tj t1^=y.1.^−y… t2^=y.2.^−y… t3^=y.3.^−y…

1 2 3

1 (gt)₁₁=y_11.−g₁−t₁−μ (gt)₁₂=y_11.−g₁−t₂−μ (gt)₁₃=y_13.−g₁−t₃−μ 2 (gt)₂₁=y_21.−g₂−t₁−μ (gt)₂₂=y_21.−g₂−t₂−μ (gt)₂₃=y_23.−g₂−t₃−μ 3 (gt)₃₁=y_31.−g₃−t₁−μ (gt)₃₂=y_31.−g₃−t₂−μ (gt)₃₃=y_33.−g₃−t₃−μ

(4)

( y

_ijk

− y

_...

)

²

k=1 K

∑

j=1 J

∑

i=1 I

∑ ⁼ ⁽ ^y

^i..

^{− y}

^...

⁾

²

k=1 K

∑

j=1 J

∑

i=1 I

∑ ⁺ ⁽ ^y

^.^j.

^{− y}

^...

⁾

²

k=1 K

∑

j=1 J

∑

i=1 I

∑

+ (y

_ij.

− y

_i..

− y

_._j.

− y

_...

)

²

k=1 K

∑

j=1 J

∑

i=1 I

∑ ⁺ ⁽ ^y

^ijk

^{− y}

^ij.

⁾

²

k=1 K

∑

j=1 J

∑

i=1 I

∑

y

_ijk

= y

_...

+ (y

_i..

− y

_...

) + ( y

_._j.

− y

_...

) + (y

_ij.

− y

_i..

− y

_._j.

+ y

_...

) + (y

_ijk

− y

_ij.

)

g_i t_j (gt)_ij e_ijk

μ

2(y_i..− y_...)(y_._j.− y_...)

k=1 K

∑

j=1 J

∑

i=1 I

∑

⁼ ²⁽^y^i..^{− y}^...⁾ ^(y^.^j.^{− y}^...⁾

j=1 J

∑

k=1 K

∑

i=1 I

∑

^{= 0}

 y...= yij.^{/ IJ} j=1

J

∑

i=1 I

∑

⁼ ^y^i..^{/ I}

i=1 I

∑

⁼ ^y^{. j.}^{/ J}

j=1 J

∑

13

i j

1 2 3

1 2 1 2 1 2

1 93 97 91 101 92 96

2 92 96 96 100 107 103

3 91 89 98 102 112 108

1 2 3

1 95 96 94 95

2 94 98 105 99

3 90 100 110 100

₉₃ ₉₈ ₁₀₃ ₉₈

1 2 3

1 5 1 -6 -3

2 0 -1 1 1

3 _-5 ₀ ₅ ₂

-5 0 5

g_i= y_i..− y...

1 2 3

1 2 1 2 1 2

1 _-2 ₂ _-5 ₅ _-2 ₂

2 -2 2 -2 2 2 -2

3 1 -1 -2 2 2 -2

×

(gt)_ij= yij.− yi..− yj..+ y...

t_j= y_{. j.}− y...

e_ijk= y_ijk− y_ij.

3×2×{(-5)²+0²+5²} = 300

3×2×{(-3)²+1²+2²} = 84

2×(5²+(-5)²+1²+(-1)²+6²+1²+5²)= 228

(-2)²+2²+(-2)²+2²+1²+(-1)²+(-5)²+5²+(-2)²+2²+(-2)²+2²+(-2)²+2²+2²+(-2)²+2²+(-2)² =108 14

• 

– 

• 

– 

• 

•  ⁼ ^– = IJK – 1

•  ⁼ ^– ^= I – 1

•  ⁼ ^– ^= J – 1

•  ⁼ ^×

= (I-1) × (J-1) =IJ – I – J + 1

•  ⁼ ^– ^– ^–

= (IJK – 1) – (I – 1) – (J – 1) – (IJ – I – J + 1) = IJK - IJ (yi..^{− y}...⁾

2 k=1

K

∑

j=1 J

∑

i=1 I

∑ ^/(I⁻¹⁾

15

F

• –  2 m, n

s

₁²

, s

₂²

F = s

₁²

/ s

₂²

m-1, n-1 F

→

0 5 10 15

0.00.20.40.60.81.0

F

f(F)

(2, 9) (4, 9) F

F

F p (

) X

16

(5)

• •  ^1%

•  ^× ^5%

> summary(aov(y ~ Var*Env, data = gxe))

Df Sum Sq Mean Sq F value Pr(>F) Var 2 84 42 3.50 0.075085 . Env 2 300 150 12.50 0.002526 ** Var:Env 4 228 57 4.75 0.024531 * Residuals 9 108 12 ---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

42 / 12

Var MS / Residuals MS

17

GxE

• 

Bernardo (2002) Breeding for quanItaIve traits in plants. Stemma Press, MN, USA

•  GxE

•  ^(y

i..

⁾

• 

19

• 

(6)

• 

– 

D_jj'= 2(1−¹

n^{)(1− r}^jj'⁾ ^r^jj'⁼

(y_ij.− y_{. j.})(y_ij'.− y_{. j'.})

i=1 n

∑

(y_ij.− y_{. j.})²

i=1 n

∑

^(yij'.^{− y}. j'.⁾

2 i=1

n

∑

(Pearson product-moment correlaIon)

1 2 3

1 95 94 90 93

2 96 98 100 98

3 ₉₄ ₁₀₅ ₁₁₀ ₁₀₃

₉₅ ₉₉ ₁₀₀ ₉₈

1 2 3

1 ₂ ₁ _-3 ₉₃

2 -2 0 2 98

3 _-9 ₂ ₇ ₁₀₃

y_ij.− y_{. j.}

Bernardo (2002) Breeding for quanItaIve traits in plants. Stemma Press, MN, USA 21

• 

1 2 … N

1 (gt)₁₁ (gt)₁₂ … (gt)₁₃

2 (gt)₂₁ (gt)₂₂ … (gt)₂₃

: : … :

M (gt)₃₁ (gt)₃₂ … (gt)₃₃

GxE

→ GxE

1 2 3

1 5 0 -5 -3

2 ₁ _-1 ₀ ₁

3 -6 1 5 2

-5 0 5

×

→

Bernardo (2002) Breeding for quanItaIve traits in plants. Stemma Press, MN, USA 22

•  GxE ^GxE

–  (stability analysis)

–  AMMI: addiIve

main-eﬀects and mulIplicaIve interacIon

23

•  Joint-regression analysis

(Yates and Cochran 1938)

• t

_j

•  ^t

j

•  ¹

-10 -5 0 5 10

8090100110120

Environmental index

y

t

_j

ij.

y

_ijk

= µ + g

_i

+ b

_i

t

_j

+ δ

_ij

+ e

_ijk

y

_ijk

= µ + g

_i

+ t

_j

+ (gt)

_ij

+ e

_ijk

b_i i

24

(7)

-10 -5 0 5 10

9095100105110

x

y = a + bx

ε

_i

ε

_i

= y

_i

− (a + bx

_i

)

SS_E = ε_i

i n

∑

⁼ ^(yi^{− a − bx}i⁾ 2 i

n

∑

:

∂SS_E

∂b ⁼⁻² _i ⁽^yⁱ^{− a − bx}ⁱ⁾^xⁱ

n

∑

^{= 0}

∂SS_E

∂a ⁼⁻² _i ⁽^yⁱ^{− a − bx}ⁱ⁾

n

∑

^{= 0}

a = y_i n

i n

∑

^{− b} ^xi ⁿ i n

∑

^{= y − bx}

b = (x_i− x )(y_i− y )

i n

∑

^(xi^{− x )}

2 i n

∑

y

_i

← y

_ijk

x

_i

← t

_j

Joint regression analysis

a← µ + g

_i

^b ← b

_i

^ε

i

^← ^δ

ij

^{+ e}

ijk

25

b _i

•  0

–  ⁱ

•  1

–  ⁱ

•  1

– 

•  1

– 

-10 -5 0 5 10

8090100110120

Environmental index

y

t

_j

ij.

Bernardo (2002)

• 

b_i=0 Type I stability

→

• 

b_i=1 (Type 2 stability)

• 

Type 3 stability Bernardo (2002)

-10 -5 0 5 10

859095100105110115

Environmental index

y

27

AMMI

1 2 3

1 5 1 -6

2 ₀ _-1 ₁

3 -5 0 5

×

(gt)_ij= y_ij.− y_i..− y_j..+ y_...

→ (gt)ij

× gt _ij

Samonte et al. (2005) Crop Sci 45:2414

(8)

…

•  QTL

• QTL

29

QTL QxE QEI

•  QTL

•  ^QTL

Mathews et al. (2008) TAG 117: 1077

30

QTL

0 5 10 15 20

7/30 8/6 8/13 8/20 8/27 9/3 9/10 9/17

Number of lines .

a

0 5 10 15 20

7/30 8/6 8/13 8/20 8/27 9/3 9/10 9/17

Number of lines .

b

0 5 10 15 20

7/30 8/6 8/13 8/20 8/27 9/3 9/10 9/17

Number of lines .

c

0 5 10 15 20

7/30 8/6 8/13 8/20 8/27 9/3 9/10 9/17

Number of lines .

d

0 5 10 15 20

7/30 8/6 8/13 8/20 8/27 9/3 9/10 9/17

Flowering date

Number of lines .

e

Kasalath

Kasalath RILs

5

31

32

(9)

0 0.2 0.4 0.6 0.8 1 1.2

0 10 20 30 40 50

Temperature (^oC)

T)

α₌₂ α=5 α₌₈

G

g(T ; ^α )

1

0 0.2 0.4 0.6 0.8 1 1.2

10 12 14 16 18 20 22 24 Photoperiod (hour)

g(P)

β=0.01 β=0.1 β=1 β=10

h(P; _β )

Δ = ^{g(T ;} ^α ^)h(P; ^β ⁾

G

1

α T)

•  α

f (Ti) =

T_i_{− T}_b T_o_{− T}_b

"

#$ ^%

& ' ^T^c^{− T}ⁱ

T_c_{− T}_o

"

#$ ^%

& '

T_c_−T_o ( )(^To−Tb)

( )

**

+ , --

α

,

₍

T_b_{≤ T}_i_{≤ T}_c

₎

0,

₍

^Ti^{< T}b^{, T}i^{> T}c

₎

/

0 1 1 1

2 1 11

(T_b=8 , T_o=30 , T_c=42 )

0 0.2 0.4 0.6 0.8 1 1.2

0 10 20 30 40 50

Temperature (^oC)

T)

α₌₂ α=5 α=8

β P)

•  β

g(P_i) =

P_i− P_b P0− Pb

"

#$ ^%

& ' ^P^c^{− P}ⁱ

P_c− P0

"

#$ ^%

& '

P_c−P0

( )^/(P₀−Pb)

( )

**

+ , --

β

,

(

P_i≥ P0

)

1,

(

P_i< P0

)

/

0 11 1

2 11 1

(P_b=0 h, P_o=10 h, P_c=24 h)

0 0.2 0.4 0.6 0.8 1 1.2

g(P)

β=0.01 β=0.1 β=1 β=10

35

α β

Chr. 3 Chr. 6 Chr. 7

Chr. 1 Chr. 3 Chr. 6 Chr. 7

Chr. 1

Hd6 Hd8

Hd1 Hd9

Hd2

α

β

QTL

(10)

(

( )

4

395210/

8 67

QTL

37

L

1 3 2 4 0 0 0 .

1 3 4 . 0 0 0

1 3 2 4757 . 0 0 0

1 3 4757 0 . 0 0

1 3 2 * 5 . . . 0

1 3 5 0 . . .

1 3 2 * 5 0 0 . .

4

5 *

606Q R 7.

C

1 3

C

1 3 C

C

KN8:

9G T

1 3

38

0 0.2 0.4 0.6 0.8 1 1.2

0 10 20 30 40 50

Temperature (^oC)

T)

α=2 α₌₅ α=8

G

g(T ; ^α )

1

0 0.2 0.4 0.6 0.8 1 1.2

g(P)

β=0.01 β=0.1 β=1 β=10

h(P;β )

Δ = ^{g(T ;} ^α ^)h(P; ^β ⁾

G

1

α _{= f}

α

(x) ^β = f

β

(x) ^G = f

G

(x)

x

7.5

x

x ^{α, β, G}

T P Δ =^{g(T ;α}^)h(P;β⁾

G

0 20 40 60 80 100 120

0.00.20.40.60.81.0

0 20 40 60 80 100 120

0.00.20.40.60.81.0

0 20 40 60 80 100 120

0.00.20.40.60.81.0

0 20 40 60 80 100 120

0.00.20.40.60.81.0

0 20 40 60 80 100 120

0.00.20.40.60.81.0

0 20 40 60 80 100 120

0.00.20.40.60.81.0

0 20 40 60 80 100 120

0.00.20.40.60.81.0

0 20 40 60 80 100 120

0.00.20.40.60.81.0

0 20 40 60 80 100 120

0.00.20.40.60.81.0

0 20 40 60 80 100 120

0.00.20.40.60.81.0

0 20 40 60 80 100 120

0.00.20.40.60.81.0

α_{= f}_α_(x) _β_{= f}

β(x) ^G= fG(x)

102

1 1

(11)

•  GxE

•  GxE ^GxE

GxE

•  GxE ^QTL

GxE

•  GxE

41

• 

• : _(2002/09)

ISBN-10: 4757804008

ISBN-13: 978-4757804005

•  A5 ³⁹²

6,900

hpp://www.igaku.co.jp/2_02_bioscience/2_02_1_ryoteki.htm 42

8 1.  ^#21 ^1, 2, 3 ^r

jj’

D

_jj’

3

2 2.  1 ²

3.  ³ ¹ ²

43

バイオメトリクス授業資料 iwatawiki lec08 s

060310391 0560565

8

×

2016/11/21 13:00-14:45

@1 - 4

GxE

•

•

• × GxE GEI

•

•

•

•

• AMMI

• QxE, QEI

•

• GxE -

(mulI-environment trial: MET)

vs.

vs.

vs.

×

Genotype x environment interacIon (GxE, GEI)

•

GxE

•

G e no ty pic v al ue

GxE

• GxE

–

GxE

– GxE

y ijk = µ + g i + t j + (gt) ij + e ijk

i j k y

2

( y

− y

)

∑

∑

∑ = ( y

− y

)

∑

∑

∑ + ( y

− y

)

∑

∑

∑

+ (y

− y

− y

− y

)

∑

∑

∑ + ( y

− y

)

∑

∑

∑

y

= y

+ (y

− y

) + ( y

− y

) + (y

− y

− y

+ y

) + (y

− y

)

∑

∑

• 

• 

•  ^× ^{GxE GEI}

• 

• 

• 

• 

•  AMMI

•  QxE, QEI

• 

•  ^GxE -

• 

• 

•  ^GxE

– 

–  ^GxE

y _ijk = µ + g _i + t _j + (gt) _ij + e _ijk

∑ ⁼ ⁽ ^y

^{− y}

⁾

∑ ⁺ ⁽ ^y

^{− y}

⁾

∑ ⁺ ⁽ ^y

^{− y}

⁾

• 

• 

• 

• 

–  2 m, n

• 

•  ^1%

•  ^× ^5%

• 

• 

• 

•  GxE

•  ^(y

⁾

• 

• 

• 

• 

• 

• 

•  GxE ^GxE

–  (stability analysis)

–  AMMI: addiIve