GWASとGS iwatawiki lec022 s

2017/11/24

2

1

(associa.on analysis)

(associa.on study)

……T……C……A……G……T………A…………A………

• 

DNA

……T……C……A……A……T………A…………A………

……T……A……G……G……T………A…………A………

……G……C……A……A……T………C…………T………

……T……A……A……G……T………A…………A………

……T……C……A……A……T………A…………A………

……T……C……A……G……T………A…………T………

……T……A……G……G……T………C…………A………

……G……C……A……G……C………A…………T………

……T……A……A……G……C………A…………A………

……T……C……A……G……C………C…………T………

……T……A……A……A……C………C…………A………

……T……C……G……G……C………A…………T………

……G……A……A……G……C………A…………A………

……T……C……A……A……C………C…………A………

A

B

C

:

: :

DNA

200

70/80

23/120

200

2

0 7 1 A

T

T

T

T

T

C

• 

•  ^QTL

• 

•  ^GWAS

•  χ

²

•  Fisher

• 

• 

•  ^{1 2}

• 

3

•  ³

AA, Aa, aa 3

^30,000

• 

• 

2005

4

Pictured by Dr. Satoshi Niikura

5

QTL

• DNA

QTL

• 

• 

• 

• 

×

6

• DNA

SNP

• 

• 

• 

ATCGAG TAGACT TATACG

ATCGAG TAGACT TATACG

ATCGAG TAGACA TATACG

ATCGAG TAGACT TATACG

ATCGAG TAGACT TATACG

ATCGAG TAGACT TATACG

ATCGAG TAGACA TATACG

ATCGAG TAGACA TATACG

ATCGAG TAGACT TATACG

ATCGAG TAGACA TATACG

7

(single nucleo.de polymorphisms: SNPs)

GeneChip Rice 44K SNP Genotyping Array

•  44,100 SNPs (10kb 1 SNP)

Tung et al. (2010) Rice 3:205-217

8

QTL

DNA

GWAS

DNA

Morrell et al. (2012) Nature Review Gene.cs 13:85

QTL vs.

linkage disequilibrium: LD

10

linkage disequilibrium: LD

QTL

(Modified from Balding 2006)

SNP

( )

SNP

( )

SNP

12

(Rafalski 2002

●^B ●^b

●^{A p}AB ^pAb ^pA

●^{a p}aB ^pab ^pa

p_B p_b

r²= ^D

2

p_Ap_ap_Bp_b

(●● 1,●● 0

)

D = p

_AB

− p

_A

p

_B

= p

_AB

p

_ab

− p

_Ab

p

_aB

r^{2= 0.25}

2

0.5× 0.5 × 0.5 × 0.5

= 1

AB

AB

AB

r^{2= 0.1024} r^{2= 0}

A B

13

• 

• 

(Rafalski 2002 )

14

LD

Gupta et al. (2005)

• 

• 

15

AA

aa

AA aa

χ² Fisher Fisher’s exact test

16

χ

²

(R) (S)

AA f11 (11) f12 (4) f1. (15) aa f21 (3) f22 (7) f2. (10) f.1 (14) f.2 (11) n (25)

p(R) = f.1 / n = 14 / 25 = 0.56 p(S) = f.2 / n = 11 / 25 = 0.44 AA p(AA) = f1. / n = 15 / 25 = 0.60 aa p(aa) = f2. / n = 10 / 25 = 0.40

↓ R AA n × p(R) × p(AA) = 25 × 0.56 × 0.60 = 8.4 R aa n × p(R) × p(aa) = 25 × 0.56 × 0.40 = 5.6 S AA n × p(S) × p(AA) = 25 × 0.44 × 0.60 = 6.6 S aa n × p(S) × p(aa) = 25 × 0.44 × 0.40 = 4.4

χ²= ^{(obs− exp)}

2

∑

exp ^{=(11− 8.4)2}

8.4 ⁺

(3− 5.6)² 5.6 ⁺

(4− 6.6)² 6.6 ⁺

(7− 4.4)² 4.4 ^{= 4.57}

(r-1)(c-1) χ² r c

5%

χ0.01

2 (1) = 6.63 >χ²= 4.57 >χ0.05 2 (1) = 3.84

5

17

:

-4 -2 0 2 4 6 8

mQQ mqq

QTL 㑇ఏᏊᆺ

QQ qq

࣐ 勖

࢝ 勖 㑇 ఏ Ꮚ ᆺ

AA

aa

40 10

10 40

-4 -2 0 2 4 6 8

₯ᅾⓗ࡞ΰྜศᕸ

-4 -2 0 2 4 6 8

࣐࣮࣮࢝

⾲⌧ᆺศᕸ

mAA

-4 -2 0 2 4 6 8

maa

y _i = u + β _j x _ij + e _i

x y

-0.2 0.2 0.6 1.0

-20246

Marker genotype

Phenotype

i AA x_i=2

aa x_i=0

x_i yi

2 D C N

83

0 2 18

-3 _{-2 -1} 0 _{1 2} 3

455055

x

y

y

_i

= α + β x

_i

+ ε

_i

= y _ˆ

_i

+ ε

_i

y

dependent (response) variable

independent (explanatory) variable

SNP

β

regression coefficient

ε

residuals y_i

ˆ

y _i=α+βx_i

x_i εi

α

intercept, constant term y

19

The method of least squares

ε

_i

= y

_i

− (α + βx

_i

)

( 2

SS_E = ε_i²

i n

∑

^{= (y}i^{−α − βx}i⁾ 2 i

n

∑

:

∂SS_E

∂β ⁼⁻² _i ^{(yi−α − βxi)xi}

n

∑

^{= 0}

∂SS_E

∂α ⁼⁻² _i ^{(yi−α − βxi)}

n

∑

^{= 0}

a = y_i

i

∑

n ^{n− b} _i^xi

∑

n n = y − bx b =

x_iy_i−

i

∑

n _i^xi

∑

n _i^yi

∑

n ⁿ

x_i²− x_i

i

∑

n

( )

^{2 n}

i

∑

n

= ^{i(xi− x )(yi− y )}

∑

n

(x_i− x )²

i

∑

n

n x_i

i

∑

n

x_i

i

∑

n ^xi 2 i

∑

n

#

$

%

%

&

' ( ( a b

#

$ % ^& ' ( = ^iyi

∑

n

x_iy_i

i

∑

n

#

$

%

%

&

' (

-3 _{-2 -1} 0 _{1 2} 3 (

455055

x

y

y_i ˆ

y _i=α+βx_i

x_i εi

20

(a b α β

y _i = u + β _j x _ij + e _i

1311

SNPs

All materials can be downloaded from hfp://ricediversity.org/

SNPs …

0e+00 1e+08 2e+08 3e+08

05102030

position (bp)

−log10(p)

LD

LD

Rafalski and Morgante 2004 Oraguzie et al. 2007) 23

:

p 

… suppose that a would-be gene7cist set out to

study the “trait” of ability to eat with chops7cks in the

San Francisco popula7on by performing an

associa7on study with the HLA complex. The allele

HLA-A1 would turn out to be posi7vely associated

with ability to use chops7cks … because the allele

HLA-A1 is more common among Asians than

Caucasians.”

Lander and Schork (1994)

HLA Human Leukocyte Antigen

24

?

1

2

(Modified from Balding 2006)²⁵

•  Indica

Japonica

y _i = u + β _j x _ij + v _k q _ik

k=1

K

∑ ^{+ e i}

1.0

0.0 0.5

Bayesian

Structure Pritchard et al. (2000) Gene.cs 155:945– 959

4

6 K=6

2

q_i1 = 0.78, q_i3 = 0.32, 0

0e+00 1e+08 2e+08 3e+08

0510152025

position (bp)

−log10(p)

…

→ ²⁹

A

•  Yu et al. (2006) Nat. Genet. 38: 203

y _i = u + β _j x _ij + v _k q _ik

k=1

K

∑ ^{+ α i + e i}

a ~ N (0, Aσ _α ² )

var(α

i

) = a(i, i)σ

_α²

cov(α

_i

,α

_i'

) = a(i, i ') σ

_α²

→

0e+00 1e+08 2e+08 3e+08

01234

position (bp)

−log10(p)

1SNP

SNP

SNP

A A

B B

B A

32

B A

SNPs

SNP

A

A B

B ^A

33

B

Bayesian

QTL

y _i = u + β _j γ _j x _ij

j=1

J

∑ ^{+ v k q ik}

k=1

K

∑ ^{+ α i + e i}

•  Iwata et al. (2007) Theor Appl Genet 114:1437–1449

γ

_j

= ¹

0

! "

#

j SNP

j SNP

γ

_j

~ B(1, p

_j

)

0e+00 1e+08 2e+08 3e+08

0.00.20.40.60.81.0

position (bp)

Posterior prob. of QTL

Bayesian

→

γ j

…

Chr. 3

GS3

63 kb

id3008333

546 kb

id3008127

Chr. 5

qSW5

id5002699

1.0 0.95

66 kb 0.88

0e+00 1e+08 2e+08 3e+08

0.00.20.40.60.81.0

position (bp)

Posterior prob. of QTL

SNPs

Q, K

•  Q

–  Structure (Pritchard et al. 2000)

–  Price et al. 2006)

–  Zhu and Yu 2009)

•  K:

–  Loiselle et al. (1995),

Ritland et al. (1996))

–  Zhao et al. 2007

– 

37

GWAS

Atwell et al. (2010) Nature 465: 627

→ a

³⁹

A B (

C ( D

false posi.ve rate: FPR = B / (A + B)

false nega.ve rate: FNR = C / (C + D)

false discovery rate: FDR = B / (B + D)

⁴⁰

•  FDR: false discovery rate

1.  K ^P

2.  P

3.  ^{FDR q* 5%}

i

4.  1 ⁱ

Benjamini and Hochberg (1995)

) ( )

2 ( ) 1

(

^{P P}

K

P ≤ ≤ ^! ≤

K

P

_i

iq

*

)

(

≤

•  ^p

•  ^{K P}

p K × P

• 

P

_(i)

^{≤ iq}

*

K ^{⇔ q}

*

_≥ ^KP

(i)

i

•  q^{* i}

K × P

•  ^{q* 5%}

GWAS

43

•  GWAS ^Hd6

Yano et al. (2016) Nature Gene.cs 48: 927

Yano et al. (2016) Nature Gene.cs 48: 927 44

Yano et al. (2016) Nature Gene.cs 48: 927 45

46

47

Yang et al. (2003) Am. J. Hum. Genet 73: 627

•  ACTN3 ^α-c.nin-3

•  X α-c.nin-3

•  a-ac.nin-3 ^{a-ac.nin-2 ACTN3}

126,559 GWAS

•  Fig. 1 3 ^SNPs

Rietveld et al. 2014, PNAS 13790, Ward et al. 2014 PLoS ONE e100248)

•  ^{R2 0.02%}

•  ^{Fig. 2}

48

Rietveld et al. (2013) Science 340: 1467

• 

• 

• 

• 

• 

•  ^DNA

QTL

49

50

•  2050

1.7

• 

2050 90 …

Tester M, Langridge P(2010) Science 327: 818

• 

1975

/ 292

1982

1984

Meuwissen et al. (2001) Gene.cs 157:1819

DNA 5

…

Garcia-Ruiz et al.

Proc Natl Acad Sci 113(28): E3995-4004

“The most drama7c response to genomic selec7on was observed for the lowly heritable traits DPR, PL, and SCS. Gene7c trends changed from close to zero to large and favorable, resul7ng in rapid gene7c improvement in fer7lity, lifespan, and health in a breed where these traits eroded over 7me.”

(

, ,

)

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

…

DNA

_DNA

DNA

LD

GS

DNA

y_i: i

x_ij: i j

y

x

₃

x

₄

x

₁

x

₂

x

_K

^{y = f (x}

1

^{, x}

2

^{,, x}

^K

⁾

(y) DNA (x

_i

)

f (x)

f (.)

y DNA x

₁

,...,x

_K

vs.

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

DNA

DNA

y_i=µ + β_jx_ij+ e_i

j= 0 N

∑

...

DNA

vs.

yi⁼µ + βjxij^{+ e}i j= 0

N

∑

vs.

y_i=µ + β_jx_ij+ e_i

j= 0 N

∑

hfp://www.soran.net/index_html/A0084070.htm

1 3

F1

6

GS GS

GS GS

3

Marker assisted selec.on: MAS

…

×

↓

QTL QTL

MAS

•  ^≠

–_QTL

–  ^{QTL QTL}

•  ^QTL

–  ^QTL

–  ^QTL

GS

•  QTL

→ QTL

x

y = f (x)

\\\

_x

y

273 142

304 423

173 373 234 138 203133 223

y = _{f (x)}

GWAS GS

GS model

GWAS model

“large p small n” problem

p >> n

x, y 1

↓

↓

60K …

…

y

_i

= _µ + _β

_j

x

_ij

+ e

_i

j=0 J

∑

PRESS

S_e

GS

0 2 4 6 8 10

024681012

0 2 4 6 8 10

024681012

x

y

0 2 4 6 8 10

024681012

x b b y₌ 0₊ 1

∑

=

+

=

7

1 0

k k k^x

b b y

(n-fold cross-validation)

1.  n

2.  i

3. _i 2

4. _{2, 3 n}

5. 

2

PRESS

n leave-one-out

y _ˆ

i

y

i

y _ˆ

i

y ˆ

i

y

i

0e+00 1e+08 2e+08 3e+08

01234

GWAS result

Position (bp)

−log10(p)

SNPs

x

212

x

468

x

₉₈₅

x

₂₇₆

^x

647

S: SNPs

y _i ⁼ x _ij b _j

j∈S

∑ ^{+ e i}

●

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6 7 8 9 10 11 12

78910

y.obs

y.pred

MLR

r ² = 0.36

argmin

b

(E) = argmin

b

(y

_i

− x

_i^T

^b )

²

i

∑ ^{+ λ b}

²

#

$ % ^&

' (

y _i = x _ij b _j

j

J

∑ ^{+ e i = x i T b + e i}

^b

²

= b

_j²

j J

∑

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6 7 8 9 10 11 12

7.58.08.59.09.5

y.obs

y.pred

r ² = 0.50

MLR

Δr ² = +0.14

LASSO

argmin

b

(y

_i

− x

_i^T

^b )

²

i

∑ ^{+ λ b}

1

#

$ % ^&

' (

y

_i

= x

_ij

b

_j

j J

∑ ^{+ e}

ⁱ

^{= x}

^iT

^{b + e}

ⁱ

LASSO:

b

1

^{= b}

^j

j J

∑

-2 -1 ⁰ 1 2

0.00.51.01.52.0

x

y

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6 7 8 9 10 11 12

7.07.58.08.59.09.510.0

y.obs

y.pred

LASSO

r ² = 0.54

MLR

Δr ² = +0.18

ridge

0e+00 1e+08 2e+08 3e+08

−0.040.000.02

Ridge

Position (bp)

Coefficients

0e+00 1e+08 2e+08 3e+08

−0.20.00.10.2

LASSO

Position (bp)

Coefficients

Ridge, LASSO

Ridge LASSO

0 shrink

GS3

LASSO

GS3

↓

Ridge polygene LASSO

QTL

←

Elas.c net

argmin

b

(y

_i

_{− x}

i T

_b)

² i

∑ ^{+ λ( 1 −α}

2

b

²

_{+ α b}

1

⁾

#

$ % ^&

' (

y

_i

= x

_ij

b

_j

j J

∑ ^{+ e}

ⁱ

^{= x}

^iT

^{b + e}

ⁱ

b

1

^{= b}

^j

j J

∑

b

²

_{= b}

j 2 j J

∑

α = 1: lasso

α = 0: ridge