名古屋大集中講義 iwatawiki lec03 s

(1)

2016/11/29 13:00-14:30

• 

QTL

7 8

• 

• vs.

• –  DNA

• – 

•  ²

– 

(2)

DNA

12 24

1

46

h8p://wkp.fresheye.com/wikipedia/ DNA

Kurata and Omura (1982) Jpn J Breed 32:253

A B C

B C

8

0 ₂ ₄ 6 8

0.000.050.100.150.200.250.30

k

P(X=k)

3 7

8

4 P(X = 4) = ⁸

4 "

# $ ^%

&

' × 0.3

⁴

^{× 0.7}

⁴

p 1-p

n k

n

k

"

# $ ^%

&

' = ^n!

k!(n − k)!

P(X = k) = ⁿ

k

"

# $ ^%

&

' p

^k

⁽¹ ^{− p)}

ⁿ^−k

p=0.3, n=8

(3)

p

8

4 P(X = 4 | p) = ⁸

4 "

# $ ^%

&

' p

⁴

⁽¹ ^{− p)}

⁴

p

L( p | X = 4) ∝ P(X = 4 | p) = ⁸

4 #

$ % ^&

' ( p

⁴

⁽¹ ^{− p)}

⁴

^→

•  ^L(p) ^ln L(p)

ln L(p)

ln L( p) = 4 ln p + 4 ln(1 − p) + const

0.0 0.2 0.4 0.6 0.8 1.0

-18-14-10-6

p

4 ln(p) + 4 ln(1-p)

p=0.5

d ln L( p)

dp ^{= 4}

(1− 2 p)

p(1− p) ^{= 0}

0.0 0.2 0.4 0.6 0.8 1.0

-4-3-2-10

p

log(p)

3 5 2

9

3 P(X

1

= 3, X

2

= 3, X

3

^{= 3) =}

9!

3!3!3! ^{× 0.3}

3

× 0.5

³

× 0.2

³

i i=1,2,…,k p

_i

n i x

_i

P(X

₁

= x

₁

,..., X

_k

= x

_k

) = ^n!

x

₁

! x

₂

! ^p

¹

x1

_

p

_k^x^k

= ^n!

x

₁

! x

₂

! ^p

ⁱ

x_i

i k

∏

1 1 1 2 2

2 2 2 3 3

ln L( p

₁

, p

₂

) = 3ln p

₁

+ 3ln p

₂

+ 3ln(1 − p

₁

− p

₂

) + const

p

₁

=1/3, p

₂

=1/3, p

₃

=1/3

∂ln L( p₁, p₂)

∂p₁ ^{= 3} 1 p₁⁻

1 1− p₁− p₂

$

% & ^'

( ) = ^{(1− 2 p}¹^{− p}²⁾ p₁(1− p₁− p₂)^{= 0}

∂ln L( p₁, p₂)

∂p2

= 3 ¹ p2

− ¹

1− p1− p2

$

% & ^'

( ) = ⁽¹^{− p}¹^{− 2 p}²⁾ p2(1− p1− p2⁾

= 0

p

₁

,p

₂

,p

₃

9

3 p1,p2,p3

(4)

F ₂

1 F₁

1 P1

_×

2 P2

1

A B

a b

a b a b

A B 0.5(1-r)

a b

0.5(1-r) ^r:

2

F₂

a B A

b

0.5r 0.5r

F ₂

AB 0.5(1-r)

Ab 0.5r

aB 0.5r

ab 0.5(1-r) AB

0.5(1-r) ÂABB(1-r)² ÂABbr(1-r) ÂaBBr(1-r) ÂaBb(1-r)² Ab

0.5r ÂABbr(1-r) ÂAbbr² ÂaBbr² Âabbr(1-r) aB

0.5r ÂaBBr(1-r) ÂaBbr² âaBBr² âaBbr(1-r) ab

0.5(1-r) ÂaBb(1-r)² Âabbr(1-r) âaBbr(1-r) âabb(1-r)²

1/4

AA

^Aa ^aa

BB (1-r)² 2r(1-r)

^r

2

Bb 2r(1-r) 2{r²+(1-r)²} 2r(1-r)

bb r² 2r(1-r)

^(1-r)

2

1/4

30

7

2

5

57

6

1

8

29

9 145

ln ^L (r) = 30ln(1− r)

²

+ 7ln2r(1− r) +  + 29ln(1− r)

²

+ const

= 144 ln(1− r) + 32ln r + 57ln(1− 2r + 2r

²

) + const

(5)

ln L(r) = 144 ln(1− r) + 32ln r + 57ln(1− 2r + 2r

²

) + const

0.0 0.1 0.2 0.3 0.4 0.5

-170-150-130-110

r

ln L(r) - const

d _{ln L(r)}

dr ^{= −}

144 1− r ⁺

32 r ⁺

57(4r − 2)

1− 2r + 2r

²

0.0 0.1 0.2 0.3 0.4 0.5

-20002006001000

r

LOD odds

L(0.5)

L( ˆ r ) /L(0.5)

r = 0.117 ˆ

LOD

LOD = log₁₀ ^{L( ˆ}^{r )} L(0.5)

"

# $ ^%

&

' = log10L(ˆ r ) − log10L(0.5) = 26.8

(bisec]on method)

∂ ln L(r)

∂r ^{= −} 144 1− r⁺

32 r ⁺

57(4r − 2) 1− 2r + 2r²

0.0 0.1 0.2 0.3 0.4 0.5

-20002006001000

r

0 r

r Newton-Raphson

f(x)=0 x

1.  f(a) f(b) a

b

2.  a b m

3._f(m) f(a) a m

f(b) b m

4.  2-3 m f(x)=0

a b

1

2

b b b

3

4

a a

a

m m m

• –  AC ^AB ^BC

( )

• –  ²

x 2

x M

(cM)

–  Haldane ^Kosambi

Haldane

Kosambi

x_{= −}¹

2^{ln(1− 2r)} x₌¹

4^ln 1+ 2r 1− 2r

#

$ % ^& ' (

0.0 0.2 0.4 0.6 0.8 1.0 _1.2

0.00.10.20.30.40.5

x (M)

r

Kosambi

Haldane

r = x

. 05

% 2

(6)

1.

2. 3.  LOD

4.  LOD

(linkage groups)

5. 

6.

7. 

8.

• 1.  ^LOD

2.  LOD

3.  1 2

• •  L1-L2-L3-L4-L5-L6

•  ¹

• 

L1

L2

L3

L4

L5

L6

14.5 0.0

5.2 8.3

17.1

20.3

• –  Maximum Likelihood

–  Minimum sum of adjacent recombina]on

frac]on SARF

∑

−

=

+ +

+

+ ⁻ ⁻

=

1

1 , 1

, 1

, 1 , 1

,

^[ ^log ⁽ ¹ ⁾ ^log( ¹ ^)]

)

(

l

i

i i i

i i

i i i i

n

i

R

L _θ _θ _θ _θ

n_i,i+1

θ_i,i+1 .

∑

−

=

+ 1

1 1 , l

i i

SARF θ

i

(7)

L1

L2

L3

L4

…

L1

L2

L3

L4

L1-L3-L2-L4

…

L1

L2 L3

L4

¡ 

l 

10 … 1,804,400

l 

100 … 4.7 x 10

¹⁵⁸

→

l 

1000 … 2.0 x 10

²⁵⁶⁷

MAPMAKER

c

t k

q p

5

…

5

5!/2

= 60

a c

t k

q p

r v

(global op]mal solu]on, local op]mal solu]on)

…

Traveling salesman problem (TSP)

•  combinatorial op]miza]on problem

13,509 500

h8p://www.crpc.rice.edu/newsArchive/tsp.html

1

(8)

(ant colony op]miza]on: ACO)

• 

ACO

1.  2. 

3.  ²

4. 

5.  ^{1 4}

A

B

C ^E

F

•  ^t ^k ⁱ ^j

] .

[

)]

(

[

]

[

)]

(

) [

(

_i^k

N

l ^il ^il

ij k ij

ij

^j ^N

d

t

d

t t

p

k i

∈

= ∀

∑

∈

−

− β α

β α

τ

d ij

)

ij

(t

τ

⎩ ⎨

⎧

∉

= ∈

Δ

)

(

)

,

(

if

0 )

(

)

,

(

if

)

) (

( i j T t

t

T

j

i

t

L

t Q

k k k

k

τ

ij

∑

=

^Δ

+

=

+

ij ^m_k ij^k

ij

⁽ ^t ¹ ⁾ ^ρτ ⁽ ^t ⁾

₁

^τ ⁽ ^t ⁾

τ

k

ρ 1

(9)

…

AntMap

•  AntMap

• 

•  AntMap

•  OS Windows, Mac, Linux, Solaris

Java h8p://lbm.ab.a.u-tokyo.ac.jp/~iwata/antmap/

bootstrap

•  RGP ^Web h8p://rgp.dna.aﬀrc.go.jp/

AntMap

① / DHLs 169

② /Kasalath// BILs 245

③ /IR24 RILs 375

(10)

• 

–  ^{① 2.5} –  ② 3 –  ③ 7

CPU Intel Mobile Pen]um 1.6GH

•  100

–  ^{③ 38}

•  ① ②

③ 5, 2, 5

①

②

③

•  1,200

•  10

2.5 (Swarm intelligence)

• 

• (par]cle swarm op]mizer: PSO)

• 

(11)

PSO

A

B

C

A c1r1(xp− x)

c₂r₂_(x_g_{− x)}

xg

xp

x wvt₋₁

v_t

x

t

^{← x}

t−1

^{+ v}

t

v

t

^{← wv}

t−1

^{+ c}

1

^r

1

^(x

p

^{− x) + c}

2

^r

2

^(x

g

^{− x)}

$ %

&

f (x)

w, c1, c2 1 (w < 1)

r₁, r₂ [0, 1]

PSO

f (x) = 10n + ^x

i

2

−10 cos2 π x

_i

( )

i=1 n

∑

Global minimum (

x = 0 f (x) = 0

Rastrigin

n = 2

R code

# load required packages require(rgl)

require(pso)

# set x and y arrays x <- (-50:50)/10 y <- x

# set a objective function (surface) z <- matrix(NA, length(x), length(y)) for(i in 1:length(x)) {

for(j in 1:length(y)) {

# Rastrigin function

z[i,j] <- 20 + (x[i]^2 - 10 * cos(2 * pi * x[i]))

+ (y[j]^2 - 10 * cos(2 * pi * y[j])) }

}

# show the objective surface open3d()

persp3d(x, y, z, col = "green")

# optimization

o1 <- psoptim(rep(NA,2),function(x) 20+sum(x^2-10*cos(2*pi*x)), lower=-5,upper=5,control=list(abstol=1e-8)) show(o1)

•  QTL

• 

名古屋大集中講義 iwatawiki lec03 s

2016/11/29 13:00-14:30

•

•

•

•

•

•

•

•

•

•

•

vs.

•

– DNA

•

–

• 2

–

DNA

1

1

8

3 7

8

4

P(X = 4) = 8

4

"

# $ %

&

' × 0.3

× 0.7

p 1-p

n k

n

k

"

# $ %

&

' = n!

k!(n − k)!

P(X = k) = n

k

"

# $ %

&

' p

(1 − p)

p

8

4

P(X = 4 | p) = 8

4

"

# $ %

&

' p

(1 − p)

p

p

L( p | X = 4) ∝ P(X = 4 | p) = 8

4

#

$ % &

' ( p

(1 − p)

→

• L(p) ln L(p)

ln L(p)

ln L( p) = 4 ln p + 4 ln(1 − p) + const

p=0.5

d ln L( p)

dp = 4

(1− 2 p)

p(1− p) = 0

3 5 2

9

3

• 

• 

• 

• 

• 

• 

• 

• 

• 

• 

• 

• 

–  DNA

• 

– 

•  ²

– 

P(X = 4) = ⁸

# $ ^%

^{× 0.7}

# $ ^%

' = ^n!

P(X = k) = ⁿ

# $ ^%

⁽¹ ^{− p)}

P(X = 4 | p) = ⁸

# $ ^%

⁽¹ ^{− p)}

L( p | X = 4) ∝ P(X = 4 | p) = ⁸

$ % ^&

⁽¹ ^{− p)}

^→

•  ^L(p) ^ln L(p)

dp ^{= 4}

p(1− p) ^{= 0}

^{= 3) =}

3!3!3! ^{× 0.3}

) = ^n!

! ^p

_

= ^n!

! ^p

F ₂

_×

F ₂

ln ^L (r) = 30ln(1− r)

d _{ln L(r)}

dr ^{= −}

1− r ⁺