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2 Topics covered.. Coalescent.
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9 突然変異の固定 集団中の頻度 3 個 1 個 突然変異の誕生 世代目 1 世代目 2 現在
10 集団中の頻度 突然変異の誕生 現在
11 集団中の頻度 突然変異の誕生 突然変異の消失 現在
12 頻度 時間
13 頻度 時間
14 頻度 時間
15 .... 集団中の頻度 現在
16 頻度 1 0 時間 現在
17 頻度 1 0 時間 現在
18 頻度 1 0 時間 現在
19 頻度 頻度 種 A 固定した変異 1 0 種 B 1 0 時間
20 頻度 頻度 固定した変異 1 0 時間 現在 共通祖先 種間変異 1 多型状態にある変異 種内変異 0 種 A 種 B 時間 現在
21 .....
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24 Wright-Fisher (2n) (n) (2n) (n) Gene pool Gene pool t t+1
25 . (N diploid individuals, 2N chromosomes).. (s = 0) Wright-Fisher
26 Gene pool 1/2N 1/2N 1/2N 2N 2N : 1 (n) 2N (1 u)2 : ( 1 1 2N ) ft 1 (1 u) 2
27 Fixation index f t = [ 1 2N + ( 1 1 2N ] )f t 1 (1 u) 2
28 Fixation index f t = [ 1 2N + ( 1 1 2N )f t 1 ] (1 u) 2 f t 1 = f t = F 1 F = 1+4Nu H = 1 F = 4Nu 1+4Nu
29 Lines of descent
30 Lines of descent 第 0 世代 第 1 世代第 2 世代第 3 世代第 4 世代第 5 世代第 6 世代
31 Lines of descent 第 0 世代 第 1 世代第 2 世代第 3 世代第 4 世代第 5 世代第 6 世代. Drift. lines of descent
32 Lines of descent 第 0 世代第 1 世代第 2 世代第 3 世代第 4 世代第 5 世代 Pr(IBD) = 1 e (1 4Nu)t 1+4Nu
33 Coalescent
34 Coalescent coalescent (gene genealogy)
35 Coalescent..
36 Coalescent: n = 2 Pr( ) 1/2N
37 Coalescent: n = 2 Pr( ) 1/2N W-F 集団 ( 親世代 ) Gene pool W-F 集団 ( 子世代 )
38 Coalescent: n = 2 Pr( ) (2N 1)/2N = 1 1/2N
39 Coalescent: n = 2 Prob(t ). t 1. t f(t) = 1 2N 1 2N e ( 1 1 2N 1 2N t ) t 1
40 Coalescent W-F W-F 集団の親世代 Gene pool 子世代
41 2N E 2 = 2N E(T 2 ) = 0 = 2N t 1 2N e 1 2N t dt
42 π θ = 4Nu E 2 = 2N E(K) = k [ k 0 = u 2 2N = 4Nu ] Pr(K = k t) f(t)dt
43 Coalescent (n) (n 1) t n coalescent n n 1 f n (t n ) = ( n 2) 2N ( n 2) ( 1 2N e (n 2) 2N t n ( n 2) 2N ) tn 1 coalescent time 2 1: E(T 2 ) = 2N 3 2: E(T 3 ) = 2N/3 n n 1: E(T n ) = 2N/ n(n 1) 2
44 Coalescent Coalescent.. {1}!2 TMRCA {2}!3!4 T5 T6 {3} {4} {5} {6} T MRCA = T total = n i=2 T i n it i i=2
45 S S u P(S = s) = 0 P(S = s t) f Ttotal (t)dt P(S = s t) = (ut)s s! e ut. u / /
46 u T total T2 TMRCA T3 T4 T5 T6 S = u n it i = 4Nu i=2 n 1 i=1 1/i = θ n 1 i=1 1 i
47 E(S) and V(S) E(S) = θ n 1 i=1 1 i V(S) = θ n 1 i=1 1 i +θ2 n 1 i=1 1 i 2
48 S s P(S = s) = 0 P(S = s t) f Ttotal (t)dt
49 S s Pr(S = s) = 0 Pr(S = s t) f(t) dt Pr(coalescent at t population model)
50 Polymorphism data as a function of genealogy
51 1... att gtat ctgacgat t atc gtaactgacgac t atc gtaactgacgac t atc gtat ctgacgac t...
52 1... att gtat ctgacgat t atc gtaactgacgac t atc gtaactgacgac t atc gtat ctgacgac t...
53 1... att gtat ctgacgat t atc gtaactgacgac t atc gtaactgacgac t atc gtat ctgacgac t.....
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56 1... att gtat ctgacgat t atc gtaactgacgac t atc gtaactgacgac t atc gtat ctgacgac t...
57 ...
58 Genealogy as a function of population history
59 genealogy {1} {2} {3} f n (t) = ( n 2) 2N e (n 2) 2N t
60 genealogy {1} {2} {3} f n (t) = e (n 2) 2N 1 t 1 ( n 2) 2) e (n (t t 2N 1 ) 2 2N 2
61 genealogy {1} {2} {3} ( n 2) ( ( n 2) )( ( n 2) ) ( ( n 2) ) f n (t) = = 2N(t) ( n 2) 2N(t) e 1 2N(0) ( n 2) 2N(i) 1 2N(1) 1 2N(t 1)
62 genealogy {0, 1} {0, 2} {1, 1} {2, 1} {1, 2}. 1 coalescent:. 2 coalescent: ( ) n1 2 ( 2N 1 ) n2 2 2N migration: n 1 m migration: n 2 m 21
63 genealogy {0, 1} {0, 2} {1, 1} {2, 1} {1, 2} f n (t) = ( ( n i p i,n i (t 1) 2) + 2N 1 i=0 ( n i 2 2N 2 )) p i,n i (t) t i n i lineage
64 Structured coalescent. i coalescent: (k i. i j 2 ) c i migration: k i M ij 2 λ = ( ( k i ) 2 + M ) ij k i, c i i 2 i j M ij = 2Nm ij t/2n
65 Structured coalescent i coalescent ( ki 2) /ci. λ i j k i M ij /2. λ
66 Structured coalescent coalescent Past MRCA MRCA {1, 0} {0, 1} m12 m21 {2, 0} {1, 1} {0, 2} k1 = 3 k2 = 1 {3, 0} {2, 1} {1, 2} {0, 3} {4, 0} {3, 1} {2, 2} {1, 3} {0, 4} Present 2N x c1 2N x c2 {5, 0} {4, 1} {3, 2} {2, 3} {1, 4} {0, 5} Initial state
67 過去 現在
68 過去 現在
69
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71 positive selection Hudson (1990)
72 過去 現在
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76 {1} {1} {1} {2} {2} {3} {2} {3} {4} {3} {4} {4}
77 genealogy( )
78 positive selection
79 balancing selection
80 ...
81 ....
82
83 θ w π θ w = S n 1 i=1 1 i π = i<j d ij ( n 2) S d ij i j
84 1 aatc gtat ctgacgagt 2 aatc gtat ctgacgagt 3 aatt gtat ctgacgagt 4 aatc gtaactgacgac t 5 aatc gtaactgacgac t 6 aatc gtaactgacgagt θ w = 3 5 i=1 1/i = 1.3 π = ! 4! 2! = 1.5
85 !"!$!#
86 !"!$!# θ w = (1+1+1)/ 1/i = 1.31 π = ( 6 2) = 1.47
87 D Tajima = Tajima s D π θ w Var[π θw ] E(π) = E(θ w ) D Tajima = 0 E(π) E(θ w ) D Tajima 0
88 過去 現在 Distibution of π and θ w Distibution of D Tajima π (blue) and θ w (red) D Tajima
89 過去 現在 Distibution of π and θ w Distibution of D Tajima π (blue) and θ w (red) D Tajima
90 Distibution of π and θ w Distibution of D Tajima π (blue) and θ w (red) D Tajima
91 ..
92 Simulation
93 ms: the de facto standard simulator
94 ms: the de facto standard simulator
95 ms: the de facto standard simulator
96 ms: [msdir]$ gcc -o ms ms.c streec.c rand1t.c -lm
97 ms: [msdir]$./ms sample size replication -t 4N 0 u -r : -r 4N 0 r n sites -I : -I npop n1 n
98 ms: [msdir]$./ms 5 2 -t // segsites: 3 positions: // segsites: 4 positions:
99 sample stats: [msdir]$ gcc -o sample stats sample stats.c tajd.c -lm
100 sample stats: [msdir]$./ms 5 2 -t 2./sample stats pi: ss: 3 D: thetah: H: pi: ss: 4 D: thetah: H:
101 mbs
102 ... Coalescent
103 If you have any comments or questions, please me at kyudai.jp
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