Ancestral selection graph meets lookdown construction Ellen Baake Biomathematics and Theoretical Bioinformatics Faculty of Technology, Bielefeld University joint work with Sandra Kluth, Ute Lenz, and Anton Wakolbinger Ellen Baake ASG and LD
Moran model with mutation and selection 1 0 0 0 0 1 1 0 0 0 t N individuals, types 0 (‘good’), 1 (‘bad’) neutral reproduction, rate 1 (for all individuals) selective reproduction, rate s (for 0 individuals) mutation to 0, rate u ν 0 mutation to 1, rate u ν 1 ( ν 0 + ν 1 = 1) Ellen Baake ASG and LD
Moran model with mutation and selection 1 0 0 0 0 1 1 0 0 0 t N individuals, types 0 (‘good’), 1 (‘bad’) neutral reproduction, rate 1 (for all individuals) selective reproduction, rate s (for 0 individuals) mutation to 0, rate u ν 0 mutation to 1, rate u ν 1 ( ν 0 + ν 1 = 1) Ellen Baake ASG and LD
Moran model with mutation and selection 1 0 0 0 0 1 1 0 0 0 t N individuals, types 0 (‘good’), 1 (‘bad’) neutral reproduction, rate 1 (for all individuals) selective reproduction, rate s (for 0 individuals) mutation to 0, rate u ν 0 mutation to 1, rate u ν 1 ( ν 0 + ν 1 = 1) X t frequency of type-0 individuals at time t diffusion limit: t → t / N , N → ∞ s.t. Ns → σ , Nu → θ Ellen Baake ASG and LD
Looking back ancestors? genealogy? MRCA? 1981 neutral case ( σ = 0): Kingman’s coalescent; genealogy independent of types 1997 coalescent with selection ( σ > 0): Neuhauser and Krone’s ancestral selection graph (ASG) 1999 Donnelly and Kurtz, lookdown construction (LD) Ellen Baake ASG and LD
Ancestral selection graph: Basic idea follow back all potential ancestors..... .... and keep in mind pecking order 1 1 0 0 1 1 1 0 0 0 0 0 Ellen Baake ASG and LD
Ancestral selection graph: Basic idea step 1: backward, w/o types r (branching: rate σ per line; coalescence: rate 1 per ordered pair) Ellen Baake ASG and LD
Ancestral selection graph: Basic idea step 1: backward, w/o types r (mutation: rates θν 1 , θν 0 per line) Ellen Baake ASG and LD
Ancestral selection graph: Basic idea step 1: backward, w/o types r (mutation: rates θν 1 , θν 0 per line) step 2: forward, with types (assigned at t = 0 according to X 0 ) 0 0 r (resolve branching events) Ellen Baake ASG and LD
Ancestral selection graph: Basic idea step 1: backward, w/o types r (mutation: rates θν 1 , θν 0 per line) step 2: forward, with types (assigned at t = 0 according to X 0 ) 0 0 t (identify ancestral line) Ellen Baake ASG and LD
Ancestral selection graph: Basic idea step 1: backward, w/o types r (mutation: rates θν 1 , θν 0 per line) step 2: forward, with types (assigned at t = 0 according to X 0 ) 1 1 r (resolve branching events) Ellen Baake ASG and LD
Ancestral selection graph: Basic idea step 1: backward, w/o types r (mutation: rates θν 1 , θν 0 per line) step 2: forward, with types (assigned at t = 0 according to X 0 ) 1 1 t (identify ancestral line) Ellen Baake ASG and LD
Immortal line 1 0 0 t Ellen Baake ASG and LD
Immortal line 1 0 0 t Ellen Baake ASG and LD
Immortal line 1 0 0 t Ellen Baake ASG and LD
Immortal line 1 0 0 t h ( x ) := P (’winner’ at t = 0 is of type 0 | X 0 = x ) = x + ? σ = 0 : ? = 0 σ > 0 : ? > 0 bias towards type 0 ! Previous (analytical) results: Fearnhead (2002), Taylor (2007) Goal: probabilistic derivation, graphical construction. Ellen Baake ASG and LD
Immortal and ancestral line forward: immortal backward: ancestral 1 1 0 0 0 0 t r Ellen Baake ASG and LD
Number of lines in ASG ( θ = 0) r ( K r ) r ≥ 0 number of lines in ASG at time r = − t birth-death process with rates q K ( n , n + 1) = n σ, q K ( n , n − 1) = n ( n − 1) , n = 1 , 2 . . . distribution becomes stationary for r → ∞ ! σ n P ( K = n ) = n !(exp( σ ) − 1) , n = 1 , 2 . . . Ellen Baake ASG and LD
Ancestral line 0 t 0 ( K r ) r ≥ 0 has bottlenecks identify true ancestor of first bottleneck individual assign types to K 0 lines (stationary!) at t = 0 (by drawing iid according to X 0 ) propagate types, apply pecking order (confusing!) � bring some order into the picture! Ellen Baake ASG and LD
Ordering the ASG r ASG Ellen Baake ASG and LD
Ordering the ASG r ordering convention: ASG continuing incoming coalescence branching Ellen Baake ASG and LD
Ordering the ASG r ordering convention: ASG continuing incoming coalescence branching r ordered ASG Ellen Baake ASG and LD
The Lookdown ASG r ordered ASG Ellen Baake ASG and LD
The Lookdown ASG r ordered ASG 5 4 3 2 1 r LD-ASG Ellen Baake ASG and LD
The Lookdown ASG 5 4 3 2 1 r LD-ASG coalescence branching Ellen Baake ASG and LD
The immune line Definition At any given time, the immune line is the line that will be immortal if all lines at that time are of type 1. 5 4 3 2 1 r t 0 0 for θ = 0: starts at bottleneck moves up at branching events � follows continuing branch! follows coalescence events downwards Ellen Baake ASG and LD
LD-ASG with types ( θ = 0) assign types (at t = 0 iid according to X 0 ) � type and level of immortal line? 1 0 1 1 1 0 1 0 0 0 0 0 Ellen Baake ASG and LD
LD-ASG with types ( θ = 0) assign types (at t = 0 iid according to X 0 ) � type and level of immortal line? 0 / 1 0 / 1 0 1 0 1 1 1 0 1 0 0 0 0 0 Ellen Baake ASG and LD
LD-ASG with types ( θ = 0) assign types (at t = 0 iid according to X 0 ) � type and level of immortal line? 0 / 1 0 / 1 0 0 / 1 0 1 0 1 1 1 1 0 1 0 0 0 0 0 Ellen Baake ASG and LD
LD-ASG with types ( θ = 0) assign types (at t = 0 iid according to X 0 ) � type and level of immortal line? 0 / 1 0 / 1 0 0 / 1 0 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 Ellen Baake ASG and LD
LD-ASG with types ( θ = 0) assign types (at t = 0 iid according to X 0 ) � type and level of immortal line? 0 / 1 0 / 1 0 0 / 1 0 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 1 1 1 Ellen Baake ASG and LD
LD-ASG with mutations LD-ASG: 5 4 3 2 1 r Ellen Baake ASG and LD
LD-ASG with mutations LD-ASG: 5 4 3 2 1 r Ellen Baake ASG and LD
LD-ASG with mutations LD-ASG: 5 4 3 2 1 r pruned LD-ASG: 5 4 3 2 1 r Ellen Baake ASG and LD
The pruned LD-ASG 5 4 3 2 1 r immune line: jumps to levels of circles Ellen Baake ASG and LD
The pruned LD-ASG 5 4 3 2 1 r relocation to top on deleterious mutation Ellen Baake ASG and LD
The pruned LD-ASG L r = highest occupied level (= number of lines) at time r ( L r ) r ∈ R Markov chain in continuous time with rates q ↑ q × q ∗ L ( n , n − 1) L ( n , n − 1) L ( n , n + 1) q ◦ L ( n , n − ℓ ) = ( n − 1) θν 1 = n ( n − 1) = n σ = θν 0 stationary distribution ( r → ∞ ): ρ n := P ( L = n ) , a n := P ( L > n ) , n = 0 , 1 , 2 , . . . given via recursion (Fearnhead/Taylor) ( n + 1 + θ + σ ) a n = ( n + 1 + θν 1 ) a n +1 + σ a n − 1 , a 0 = 1 , n →∞ a n +1 / a n = 0 . lim Ellen Baake ASG and LD
The pruned LD-ASG with types 5 4 3 2 1 r assign types: type and level of immortal line? all lines untyped (except immune line), all lines and arranged according to pecking order � results for θ = 0 carry over! Ellen Baake ASG and LD
Level and type of immortal line ( θ > 0) 5 Coin 4 3 2 x 1-x 1 r ↔ h ( x ) = P (immortal line has type 0 at t = 0 | X 0 = x ) � P ( L ≥ n )(1 − x ) n − 1 x = n ≥ 1 P ( L ≥ n ) = P (level n is occupied) , 1 = P ( L ≥ 1) ≥ P ( L ≥ 2) ≥ P ( L ≥ 3) . . . � bias towards type 0. Ellen Baake ASG and LD
Level and type of immortal line ( θ > 0) Theorem 1 The level of the immortal line in the LD-ASG with types assigned at t = 0 is either the lowest type-0 level or, if all lines at t = 0 are of type 1, it is the level of the immune line. 2 h ( x ) = P (immortal line has type 0 | x ) is the probability of at least one success when tossing L times a coin with success probability x , � P ( L ≥ n )(1 − x ) n − 1 x . h ( x ) = n ≥ 1 Ellen Baake ASG and LD
Some pictures: a n := P ( L > n ) and h ( x ) σ = 0 , 1 , 5 , 10, θ = 1, ν 1 = 0 . 5 Ellen Baake ASG and LD
Conclusion Pruned LD-ASG to identify ancestral individual and obtain its type distribution Key ingredients: equilibrium ASG (without types) ordering of lines LD-ASG pruned LD-ASG (still without types) assign types TPB 2015 Ellen Baake ASG and LD
Open problems multiple types types as sequences ! Ellen Baake ASG and LD
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