# 名古屋大集中講義 iwatawiki lec03 s

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(1) • • • • • • • • • • • 2016/11/29 13:00-14:30 vs. • – DNA • – • • • QTL 7 8 2 • –

(2) DNA 12 8 24 Kurata and Omura (1982) Jpn J Breed 32:253 46 DNA h8p://wkp.fresheye.com/wikipedia/ 1 1 3 7 8 4 " 8% P(X = 4) = \$ ' × 0.34 × 0.7 4 # 4& 1-p p B C " n% n! \$ '= # k & k!(n − k)! 0.30 0.20 0.15 P(X=k) 0.10 C 0.05 B p=0.3, n=8 0.25 " n% P(X = k) = \$ ' p k (1 − p) n −k # k& A 0 k 0.00 n 2 4 k 6 8

(3) p 8 4 1 1 1 2 2 2 2 2 3 3 " 8% P(X = 4 | p) = \$ ' p 4 (1 − p) 4 # 4& 3 i n # 8& L( p | X = 4) ∝ P(X = 4 | p) = % ( p 4 (1 − p) 4 \$ 4' 3 P(X1 = 3, X 2 = 3, X 3 = 3) = p 2 9 p 5 9! × 0.33 × 0.5 3 × 0.2 3 3!3!3! i=1,2,…,k i xi pi k n! n! x1 xk P(X1 = x1,..., X k = x k ) = p1  pk = pix i ∏ x1! x 2! x1! x 2! i -2 -3 -4 log(p) -1 0 → 0.0 0.2 0.4 0.6 0.8 1.0 9 p 3 ln L(p) L(p) • p1,p2,p3 p1,p2,p3 ln L(p) ln L( p) = 4 ln p + 4 ln(1 − p) + const ln L( p1, p2 ) = 3ln p1 + 3ln p2 + 3ln(1 − p1 − p2 ) + const -10 -14 p=0.5 -18 4 ln(p) + 4 ln(1-p) -6 d ln L( p) (1− 2 p) =4 =0 dp p(1− p) 0.0 0.2 0.4 0.6 p 0.8 1.0 \$1 ' (1 − 2 p1 − p2 ) ∂ ln L( p1, p2 ) 1 = 3& − =0 )= ∂p1 % p1 1 − p1 − p2 ( p1 (1 − p1 − p2 ) \$1 ' (1 − p1 − 2 p2 ) ∂ ln L( p1, p2 ) 1 = 3& − =0 )= ∂ p2 % p2 1 − p1 − p2 ( p2 (1 − p1 − p2 ) p1=1/3, p2=1/3, p3=1/3

(4) F2 AB 0.5(1-r) Ab 0.5r aB 0.5r ab 0.5(1-r) AB 0.5(1-r) AABB (1-r)2 AABb r(1-r) AaBB r(1-r) AaBb (1-r)2 Ab 0.5r AABb r(1-r) AAbb r2 AaBb r2 Aabb r(1-r) aB 0.5r AaBB r(1-r) AaBb r2 aaBB r2 aaBb r(1-r) ab 0.5(1-r) AaBb (1-r)2 Aabb r(1-r) aaBb r(1-r) aabb (1-r)2 1/4 F2 A A 1 P1 B × B a a b b A a B b 2 P2 1 AA Aa aa BB (1-r) 2 30 2r(1-r) 7 r 2 2 Bb 2r(1-r) 5 2{r 2+(1-r)2} 57 2r(1-r) 6 bb r 2 1 2r(1-r) 8 (1-r) 2 29 F1 A A 1 B b 0.5(1-r) 0.5r a 1/4 a 2 B b 0.5r 0.5(1-r) F2 r: 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 -130 LOD – AC rˆ = 0.117 600 -150 0.2 0.3 0.4 0.5 2 – 0.0 0.1 0.2 0.3 0.4 M (cM) Kosambi 0.5 r odds L(0.5) L( rˆ ) /L(0.5) 2 x x r LOD BC ) ( -200 0 0.1 AB • 200 -170 0.0 • 1000 ln L(r) - const -110 d ln L(r) 144 32 57(4r − 2) =− + + dr 1− r r 1− 2r + 2r 2 – Haldane " L( rˆ ) % LOD = log10 \$ ' = log10 L( rˆ ) − log10 L(0.5) = 26.8 # L(0.5) & Haldane Kosambi 1 x = − ln(1− 2r) 2 1 #1+ 2r & x = ln% ( 4 \$ 1− 2r ' 0.1 0.2 0.3 0.4 0.5 r 1 2 3 4 m a m a a m b 0.4 r = x a m b m b 0.0 0.2 0.4 0.6 0.8 1.0 x (M) r m f(x)=0 b a b Haldane 0.3 a 0.2 1000 600 -200 0 200 0.0 Kosambi 0.1 r 0.0 0 f(x)=0 x 1. f(a) f(b) b 2. a b m 3. f(m) f(a) f(b) 4. 2-3 r ∂ ln L(r) 144 32 57(4r − 2) =− + + ∂r 1− r r 1− 2r + 2r 2 0.5 (bisec]on method) Newton-Raphson . % 05 2 1.2

(6) 1. L1 0.0 • 2. 3. LOD 4. LOD (linkage groups) L2 5.2 L3 8.3 L1-L2-L3-L4-L5-L6 1 • 5. 6. 7. 8. • L4 14.5 L5 17.1 L6 20.3 • • • 1. – LOD l −1 L( R)=∑ ni ,i +1[θ i ,i +1 log θ i ,i +1 + (1 − θ i ,i +1 ) log(1 − θ i ,i +1 )] 2. i =1 LOD – 3. 1 2 Maximum Likelihood frac]on SARF Minimum sum of adjacent recombina]on l −1 SARF = ∑ θ i ,i +1 i =1 θi,i+1 ni,i+1 .

(7) (global op]mal solu]on, local op]mal solu]on) L1 L1 L1 L2 L3 L3 … L2 L3 L2 L4 L4 … L1-L3-L2-L4 … L4 Traveling salesman problem (TSP) ¡ l l l 10 … 1,804,400 100 … 4.7 x 10158 1000 … 2.0 x 102567 MAPMAKER • → combinatorial op]miza]on problem c c a t k q p … k 5 5!/2 = 60 t 5 q p 1 v r 13,509 500 h8p://www.crpc.rice.edu/newsArchive/tsp.html

(8) (ant colony op]miza]on: ACO) t • k ij k p (t ) = • i j [τ ij (t )]α [dij ]− β ∑l∈N k [τ il (t )]α [dil ]− β ∀j ∈ Nik . i • d ij τ ij (t ) • ACO 1. 2. 3. 2 4. 5. 1 4 m τ ij (t + 1) = ρτ ij (t ) + ∑k =1 Δτ ijk (t ) F B ρ 1 A C E k ⎧Q Lk (t ) if (i, j ) ∈ T k (t ) Δτ (t ) = ⎨ if (i, j ) ∉ T k (t ) ⎩0 k ij

(9) … … bootstrap AntMap • AntMap • RGP Web • h8p://rgp.dna.aﬀrc.go.jp/ AntMap ① • AntMap / DHLs 169 ② /Kasalath// BILs 245 OS Windows, Mac, Linux, • Solaris Java h8p://lbm.ab.a.u-tokyo.ac.jp/~iwata/antmap/ ③ /IR24 RILs 375

(10) (Swarm intelligence) • ① 2.5 ② 3 ③ 7 – – – ① • CPU Intel Mobile Pen]um 1.6GH • 100 – ② • ③ 38 • ① ② 5, 2, 5 ③ ③ (par]cle swarm op]mizer: PSO) • 1,200 • 10 2.5 • • • •

(11) PSO wv t−1 R code B c 2 r2 (x g − x) A x vt c1r1 (x p − x) xg C f (x) A x p \$ x t ← x t−1 + v t % & v t ← wv t−1 + c1r1 (x p − x) + c 2 r2 (x g − x) w, c1, c2 r1, r2 1 [0, 1] (w < 1) # 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) PSO • Rastrigin QTL n f (x) = 10n + ∑ ( xi2 −10 cos2π xi ) • i=1 Global minimum ( x=0 f (x) = 0 • n = 2

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