Ancestral selection graph meets lookdown construction Ellen Baake - - PowerPoint PPT Presentation

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Ancestral selection graph meets lookdown construction Ellen Baake - - PowerPoint PPT Presentation

Oct 22, 2022 200 likes •646 views

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

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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

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Moran model with mutation and selection

t

1 1 1

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 uν0 mutation to 1, rate uν1 (ν0 + ν1 = 1)

Ellen Baake ASG and LD

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Moran model with mutation and selection

t

1 1 1

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 uν0 mutation to 1, rate uν1 (ν0 + ν1 = 1)

Ellen Baake ASG and LD

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Moran model with mutation and selection

t

1 1 1

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 uν0 mutation to 1, rate uν1 (ν0 + ν1 = 1) Xt 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

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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

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Ancestral selection graph: Basic idea

follow back all potential ancestors..... .... and keep in mind pecking order 1 1 1 1 1

Ellen Baake ASG and LD

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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

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Ancestral selection graph: Basic idea

step 1: backward, w/o types

r

(mutation: rates θν1, θν0 per line)

Ellen Baake ASG and LD

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Ancestral selection graph: Basic idea

step 1: backward, w/o types

r

(mutation: rates θν1, θν0 per line) step 2: forward, with types (assigned at t = 0 according to X0)

r

(resolve branching events)

Ellen Baake ASG and LD

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Ancestral selection graph: Basic idea

step 1: backward, w/o types

r

(mutation: rates θν1, θν0 per line) step 2: forward, with types (assigned at t = 0 according to X0)

t

(identify ancestral line)

Ellen Baake ASG and LD

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Ancestral selection graph: Basic idea

step 1: backward, w/o types

r

(mutation: rates θν1, θν0 per line) step 2: forward, with types (assigned at t = 0 according to X0)

r

1 1

(resolve branching events)

Ellen Baake ASG and LD

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Ancestral selection graph: Basic idea

step 1: backward, w/o types

r

(mutation: rates θν1, θν0 per line) step 2: forward, with types (assigned at t = 0 according to X0)

t

1 1

(identify ancestral line)

Ellen Baake ASG and LD

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Immortal line

t 1

Ellen Baake ASG and LD

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Immortal line

t 1

Ellen Baake ASG and LD

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Immortal line

t 1

Ellen Baake ASG and LD

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Immortal line

t 1

h(x) := P (’winner’ at t = 0 is of type 0 | X0 = 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

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Immortal and ancestral line

forward: immortal

t 1

backward: ancestral

r 1

Ellen Baake ASG and LD

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Number of lines in ASG (θ = 0)

r

(Kr)r≥0 number of lines in ASG at time r = −t birth-death process with rates qK(n, n + 1) = nσ, qK(n, n − 1) = n(n − 1), n = 1, 2 . . . distribution becomes stationary for r → ∞ ! P(K = n) = σn n!(exp(σ) − 1), n = 1, 2 . . .

Ellen Baake ASG and LD

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Ancestral line

t0 (Kr)r≥0 has bottlenecks identify true ancestor of first bottleneck individual assign types to K0 lines (stationary!) at t = 0 (by drawing iid according to X0) propagate types, apply pecking order (confusing!) bring some order into the picture!

Ellen Baake ASG and LD

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Ordering the ASG

r ASG

Ellen Baake ASG and LD

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Ordering the ASG

r ASG

  • rdering convention:

continuing incoming

coalescence branching

Ellen Baake ASG and LD

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Ordering the ASG

r ASG r

  • rdered ASG
  • rdering convention:

continuing incoming

coalescence branching

Ellen Baake ASG and LD

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The Lookdown ASG

r

  • rdered ASG

Ellen Baake ASG and LD

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The Lookdown ASG

r

  • rdered ASG

1 2 3 4 5

r LD-ASG

Ellen Baake ASG and LD

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The Lookdown ASG

1 2 3 4 5

r LD-ASG coalescence branching

Ellen Baake ASG and LD

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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.

r t0

1 2 3 4 5

for θ = 0: starts at bottleneck moves up at branching events follows continuing branch! follows coalescence events downwards

Ellen Baake ASG and LD

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LD-ASG with types (θ = 0)

assign types (at t = 0 iid according to X0) type and level of immortal line?

1 1 1 1 1 Ellen Baake ASG and LD

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LD-ASG with types (θ = 0)

assign types (at t = 0 iid according to X0) type and level of immortal line?

1 1 1 1 1 0/1 0/1 Ellen Baake ASG and LD

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LD-ASG with types (θ = 0)

assign types (at t = 0 iid according to X0) type and level of immortal line?

1 1 1 1 1 0/1 0/1 1 0/1 Ellen Baake ASG and LD

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LD-ASG with types (θ = 0)

assign types (at t = 0 iid according to X0) type and level of immortal line?

1 1 1 1 1 0/1 0/1 1 0/1 1 1 Ellen Baake ASG and LD

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LD-ASG with types (θ = 0)

assign types (at t = 0 iid according to X0) type and level of immortal line?

1 1 1 1 1 0/1 0/1 1 0/1 1 1 1 1 1 Ellen Baake ASG and LD

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LD-ASG with mutations

LD-ASG:

r

1 2 3 4 5 Ellen Baake ASG and LD

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LD-ASG with mutations

LD-ASG:

r

1 2 3 4 5 Ellen Baake ASG and LD

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LD-ASG with mutations

LD-ASG:

r

1 2 3 4 5

pruned LD-ASG:

r

1 2 3 4 5 Ellen Baake ASG and LD

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The pruned LD-ASG

r

1 2 3 4 5

immune line: jumps to levels of circles

Ellen Baake ASG and LD

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The pruned LD-ASG

r

1 2 3 4 5

relocation to top on deleterious mutation

Ellen Baake ASG and LD

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The pruned LD-ASG

Lr = highest occupied level (= number of lines) at time r

(Lr)r∈R Markov chain in continuous time with rates q↑

L(n, n − 1)

= n(n − 1) q∗

L(n, n + 1)

= nσ q◦

L(n, n − ℓ)

= θν0 q×

L (n, n − 1)

= (n − 1)θν1 stationary distribution (r → ∞): ρn := P(L = n), an := P(L > n), n = 0, 1, 2, . . . given via recursion (Fearnhead/Taylor) (n + 1 + θ + σ)an = (n + 1 + θν1)an+1 + σan−1, a0 = 1, lim

n→∞ an+1/an = 0.

Ellen Baake ASG and LD

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The pruned LD-ASG with types

r

1 2 3 4 5

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

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Level and type of immortal line (θ > 0)

r

1 2 3 4 5

Coin x ↔ 1-x h(x) = P (immortal line has type 0 at t = 0 | X0 = x) =

  • n≥1

P(L ≥ n)(1 − x)n−1x 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

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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, h(x) =

  • n≥1

P(L ≥ n)(1 − x)n−1x .

Ellen Baake ASG and LD

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Some pictures: an := P(L > n) and h(x)

σ = 0, 1, 5, 10, θ = 1, ν1 = 0.5

Ellen Baake ASG and LD

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Conclusion

Pruned LD-ASG to identify ancestral individual and obtain its type distribution Key ingredients: equilibrium ASG (without types)

  • rdering of lines

LD-ASG pruned LD-ASG (still without types) assign types TPB 2015

Ellen Baake ASG and LD

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Open problems

multiple types types as sequences !

Ellen Baake ASG and LD