名古屋大集中講義 iwatawiki lec04 qtlmapping s

4

QTL

1

2016/11/30 8:45-10:15

• 

•  quan4ta4ve trait

quan4ta4ve trait loci: QTL

QTL analysis; QTL mapping

Watanabe et al. (2005) Ann Bot. 95:1131

2

• 

•  ^{locus QTL}

• 

• 

• 

• 

• 

•  ^LOD

•  (interval mapping)

•  EM

•  Composite interval mapping

3

vs.

1 R r

RR Rr rr

4

vs.

QTL QTL

QTL 1 QTL Q

q

52 87 48 72 32 68

QQ QQ Qq Qq qq qq

5

QTL

•  ^gene

•  ^{locus loci}

– 

•  quan4ta4ve trait locus

– 

6

RR

rr Rr



qq Qq QQ



1.0

rr 100%

QTL

7

Frankham et al. (2002) Introduc4on to Conserva4on Gene4cs. Cambridge Univ. Press

8

East (1916) Gene4cs 1: 164

Frankham et al. (2002) Introduc4on to Conserva4on Gene4cs. Cambridge Univ. Press

corolla 9

A R

…

AARR ^aarr

AaRr F1

F2

a A

R r

F2

AA Aa aa

30 7 2

5 57 6

(RR)

(rr) (Rr)

1 8 29

(1-r)²/4

(1-r)²/4 r²/4

r²/4

r(1-r)/2

r(1-r)/2 r(1-r)/2

r(1-r)/2 {r²+(1-r)²}/2

A r=0.117 (13.33cM)

…

10

0 200 400 600 800 1000 1200

0.00.10.20.30.40.5

x

p(x)

0 200 400 600 800 1000 1200

0.00.10.20.30.40.5

0 200 400 600 800 1000 1200

0.00.10.20.30.40.5

0 200 400 600 800 1000 1200

0.00.10.20.30.40.5

AA Aa aa

? ? ?

? ? ?

QTL

QQ

qq Qq

? ? ?

(1-r)²/4

(1-r)²/4 r²/4

r²/4

r(1-r)/2

r(1-r)/2 r(1-r)/2

r(1-r)/2 {r²+(1-r)²}/2

QTL

¹¹

F

2 16.237 9.372^**

12 1.733

A a

A A _{a a}

12.3cm 15.1cm 11.8cm 14.7cm

11.3cm 10.2cm 10.8cm 12.7cm

10.1cm 10.2cm 9.8cm 12.0cm 16.2cm 11.0cm

11.5cm

QTL

A

AA, Aa, aa 3

→

^QTL

…

QTL

QTL

QTL

QTL QTL

13

2 QTL

• 

g _i

e _i

y _i

y_i: i e_i: i

y

= g

+ e

e

_{~ N(0, σ}

₎

0 σ²

y

~ N(g

, σ

)

σ^{2 14}

•  phenotypic value

– 

•  genotypic value

– 

•  environmental effect

– 

•  ^x

μ σ²

(1)

(2)

(3)

f (x) = ¹

2πσ

^{exp −}

(x − µ)

2σ

%

&

' ⁽

) *

0.00.10.20.30.4

x

f(x)

μ μ+2σ

μ-2σ

μ-4σ μ+4σ

16

•  ^QTL

qq Qq QQ

μ

μ-a μ+d μ+a

a

d

QQ < qq a < 0

Qq QQ qq d < 0

a QTL d QTL

17

qq Qq QQ

μ-a μ μ+d μ+a

y

~ N( µ + d, σ

)

QQ

Qq

qq

y

~ N( µ + a, σ

)

y

~ N( µ − a, ^σ

⁾

18

•  ^{i QTL QQ}

y

…

φ(y | µ,σ²) = ¹ 2πσ^{2exp −}

(y− µ)² 2σ²

& ' (

) * +

p(y_i)=

φ

µ

σ

QQ y_i

QQ

y

19

Qq QQ

qq

y

y

_y

3

^y

^y

_y

^y

_y

^y

y ,…, y₉ L₁

L

^{= φ(y}

^{| µ − a,σ}

^)φ(y

^{| µ − a,σ}

^)φ(y

^{| µ + d,σ}

^)φ(y

^{| µ + a,σ}

⁾

QTL

→ QTL

→ QTL 20

2

H

: QTL a ≠ 0 d ≠ 0

Qq QQ

qq

y₁ y₂ y₃y₄ y₅ y₆y₇ y₈ y₉

H

: QTL a = 0 d = 0

y₁ y₂ y₃y₄ y₅ ^y6^y7^y8 ^y9

L

^{= φ(y}

^{| µ − a,σ}

^)φ(y

^{| µ − a,σ}

^)φ(y

^{| µ + d,σ}

^)φ(y

^{| µ + a,σ}

⁾

L

= φ(y

| µ,σ

)

i=1 9

∏ max L

max L

maxL₁/maxL₀

N(μ, σ²)

QTL

21

LOD =log

(maxL

/maxL

)

maxL

/maxL

QTL

H

QTL

QTL

maxL

/maxL

H

QTL

QTL

*

22

A B C D E

QTL

QTL

LOD

LO D

H0 H1

100

QTL …

23

QTL

QTL

QTL

QTL

QTL

24

QTL

A Q B

r

r

P1

P2

F1 ^F2

A _A

r

B B Q Q

a _a

b q q

b

A a

B b

Q q

A _A

B b

QTL

QQ (1- r_A )(1- r_B )/(1- r_AB ) × (1- r_A ) r_B/ r_AB Qq (1- r_A )(1- r_B )/(1- r_AB ) × r_A(1- r_B )/r_AB

r_A r_B/(1- r_AB ) × (1- r_A ) r_B/r_AB

qq r_A r_B /(1- r_AB ) × r_A(1- r_B )/r_AB A, B QTL

AABb QTL

…

? ?

25

A a

B b

Q q

A Q B

r_AQ r_QB

r_AB

A, B QTL

F1

A-?-B :

1 2^{(1− rAB)}

A-Q-B : ¹

2^{(1− rA)(1− rB)}

A-q-B : ¹

2^rArB A-?-B Q (1− rA)(1− rB) /(1− rAB⁾

A-?-B q _{rArB/(1− rAB})

A-?-b :

1 2^rAB

A-Q-b : ¹

2^{(1− rA)rB}

A-q-b : ¹

2^{rA(1− rB)} A-?-b Q (1− rA)rB/ rAB

A-?-b q rA(1− rB) / rAB

P(X | Z) =^{P(X ∩ Z)} P(Z)

:

Q q

AB q₁ q2

Ab q₃ q₄

aB q₄ q₃

ab q2 q1

q₁= (1− rA)(1− rB) /(1− rAB⁾

q₂= r_Ar_B/(1− r_AB)

QTL

q₃= (1− r_A)r_B/ r_AB q₄= r_A(1− rB) / rAB

QQ (p_QQ) Qq (p_Qq) qq (p_qq)

AABB q1^{2 2q}1q2 q2²

AABb q₁q₃ q₁q₄+q₂q₃ q₂q₄

AAbb q₃² 2q₃q₄ q₄²

AaBB q₁q₄ q₁q₃+q₂q₄ q₂q₃ AaBb z₁q₁q₂+z₂q₃q₄ z₁(q₁²+q₂²)

+z₂(q₃²+q₄²)

z₁q₁q₂+z₂q₃q₄

Aabb q₂q₃ q₁q₃+q₂q₄ q₁q₄

aaBB q₄² 2q₃q₄ q₃²

aaBb q₂q₄ q₁q₄+q₂q₃ q₁q₃

aabb q₂² 2q₁q₂ q₁²

z₁= (1− r_AB)²/{(1− r_AB)²+ r_AB²} z₂= r_AB²/{(1− r_AB)²+ r_AB²}

27

…

y _i

Qq QQ

qq

Qq

QQ

qq

i QTL QQ, Qq, qq yi

…

ϕ_iqq= p(y_i| qq) =φ(y_i|µ− a,σ²) = ¹ 2πσ^2exp−

{y_i− (µ− a)}² 2σ² ' ( )

* + , ϕiQq= p(yi| Qq) =φ(yi| µ + d,σ²) = ¹

2πσ^2exp−

{yi− (µ + d)}² 2σ² ' ( )

* + , ϕ_iQQ= p(y_i| QQ) =φ(y_i| µ + a,σ²) = ¹

2πσ^{2exp −}

{y_i− (µ + a)}² 2σ² ' ( )

* +

28 ,

y_i

Qq QQ qq

QQ

AABb i

QTL QQ y_i

ϕ

A _A

B b

? ?

p(QQ) = q

q

p

p

ϕ

= q

q

φ (y

| µ + a, σ

)

QTL

29

EM

QTL

→

EM (expecta4on-maximiza4on)

(Dempster et al. 1977. J Roy Stat Soc. Ser B: 39: 1-39)

(E ) i QTL

z

= ϕ

p

(ϕ

p

+ ϕ

p

+ ϕ

p

)

z

= ϕ

p

(ϕ

p

+ ϕ

p

+ ϕ

p

)

z

= ϕ

p

(ϕ

p

+ ϕ

p

+ ϕ

p

)

p_i** i QTL

(M ) ln L μ, a, b, σ²

ln L = _{ z

ln(ϕ

p

) + z

ln(ϕ

p

) + z

ln(ϕ

p

) _}

i=1 n

∑ ^{+ const}

³⁰

A B C D E

QTL

LOD

LO D

31

•  CIM (composite interval mapping)

–  ^{QTL QTL}

QTL

•  MIM (mul4ple interval mapping)

–  CIM ^QTL

QTL MIM

QTL

•  Bayesian mul4ple QTL mapping

–  ^:QTL:

QTL

GxE

32

IM

•  IM ^{1 QTL}

= QTL +

1 2

Q

Q

Qq QQ qq

QQ qq Qq

Q^b Q^a

33

IM

QTL

QTL A

QTL B

B A

34

CIM

= QTL +

= 1 QTL + QTL +

Q^b

Q^b

Σ

(

QQ qq Qq

Σ

CIM

QQ qq Qq

35

CIM

QTL

QTL B

QTL A

B ^A

36

QTL

Q^a Q^b

QTL

LOD _LOD

CIM IM

CIM > IM

…

QTL

QTL

³⁷

SIM

CIM

1 22cM a=0.27

4.7%1 66cM a=0.43 11.6%

2 56cM a=0.29

6.2 %

IM CIM

100cM

2

55cM

QTL3 (a=0.3)

100cM

QTL1 (a=0.4)

1

25cM

QTL2 (a=0.35)

75cM

CIM ₃₈

Mapmaker/QTL

ftp://ftp-genome.wi.mit.edu/distribution/software/newqtl/

QTL Cartographer (WinQTLCart)

http://statgen.ncsu.edu/qtlcart/ WinQTLCart GUI

IM, CIM, MIM 1

Qgene

http://coding.plantpath.ksu.edu/qgene/ eQTL

R/qtl

http://www.rqtl.org/

R QTL

2

Mul4mapper

hqp://www.rni.helsinki.fi/~mjs/

QTL

QTL

MAPL98

hqp://lbm.ab.a.u-tokyo.ac.jp/~ukai/mapl98.html

QTL

QTLBIM hqp://www.qtlbim.org/

R QTL

QTL

2

GxE

QTL

39

(1)

Gene pyramiding:

1

SW2443

0.0

SW256

20.4

SW240

44.9

FSHB

55.3

SW942

56.3

S0091

62.9

SW395

65.0

SW766

69.9

SW1879

94.0

SWR308

129.7

Shear_value
0
5
10
15
20
25

chr2

QTL posi4on : 71.4cM (chromosome 2) QTL effect : a=0.67, d= 0.16

P1 _P2

F1 F1

F2

) (2

42

•  QTL

•  ^-QTL-

•  QTL ^QTL

•  QTL

GxE

•  ^QTL

43