The tree-valued Fleming-Viot process with mutation and selection

Transcription

1 The tree-valued Fleming-Viot process with mutation and selection Peter Pfaffelhuber University of Freiburg Joint work with Andrej Depperschmidt and Andreas Greven

2 Population genetic models Populations of constant size have been modelled by Markov Chains (Wright-Fisher-model, Moran model) Diffusion approximations (Fisher-Wright diffusion) dx = αx (1 X )dt + X (1 X )dw or Measure-valued diffusions (Fleming-Viot superprocess) New: Extend Fleming-Viot process by genealogical information Tree-valued Fleming-Viot process

3 The Moran model with mutation and selection time resampling, rate 1 per pair mutation, rate ϑ/2 selection, rate α/n per pair only can use selective arrow Goal: construct a tree-valued stochastic process U = (U t ) t 0 describes genealogical relationships dynamically makes forward and backward picture implicit

4 The Moran model with mutation and selection time resampling, rate 1 per pair mutation, rate ϑ/2 selection, rate α/n per pair only can use selective arrow Goal: construct a tree-valued stochastic process U = (U t ) t 0 describes genealogical relationships dynamically makes forward and backward picture implicit

5 The Moran model with mutation and selection time Goal: construct a tree-valued stochastic process U = (U t ) t 0 describe genealogical relationships dynamically make forward and backward picture implicit

6 The Moran model with mutation and selection time t s Goal: construct a tree-valued stochastic process U = (U t ) t 0 describe genealogical relationships dynamically make forward and backward picture implicit

7 The Moran model with mutation and selection time t s Goal: construct a tree-valued stochastic process U = (U t ) t 0 describe genealogical relationships dynamically make forward and backward picture implicit

8 Summary: The tree-valued Fleming-Viot process Theorem: The (Ω, Π)-martingale problem is well-posed. Its solution the tree-valued Fleming-Viot process arises as weak limit of tree-valued Moran models. Theorem: Tree-valued processes for different α are absolutely continuous with respect to each other. Theorem: The measure-valued Fleming-Viot process is ergodic iff the tree-valued Fleming-Viot process is ergodic. Theorem: The distribution of R12 α, the distance of two randomly sampled points in equilibrium, can be computed.

9 Formalizing genealogical trees Leaves in genealogical trees form a metric space; leaves are marked by elements of I (compact) A tree is given by:state space of U: U :={isometry class of (X, r, µ) : (X, r) complete and separable metric space, µ P(X I )} r(x 1, x 2 ) defines the genealogical distance of individuals x 1 and x 2 µ marks currently living individuals

10 Formalizing genealogical trees Leaves in genealogical trees form a metric space; leaves are marked by elements of I (compact) A tree is given by:state space of U: X :={isometry class of (X, r, µ) : (X, r) complete and separable metric space, µ P(X I )} r(x 1, x 2 ) defines the genealogical distance of individuals x 1 and x 2 µ marks currently living individuals

11 Formalizing genealogical trees Leaves in genealogical trees form a metric space; leaves are marked by elements of I (compact) A tree is given by:state space of U: X :={isometry class of (X, r, µ) : (X, r) complete and separable metric space, µ P(X I )} r(x 1, x 2 ) defines the genealogical distance of individuals x 1 and x 2 µ marks currently living individuals

12 Formalizing genealogical trees Leaves in genealogical trees form a metric space; leaves are marked by elements of I (compact) State space of U: X :={isometry class of (X, r, µ) : (X, r) complete and separable metric space, µ P(X I )} r(x 1, x 2 ) defines the genealogical distance of individuals x 1 and x 2 µ marks currently living individuals

13 Martingale Problem Given: Markov process X = (X t ) t 0. The generator is 1 ΩΦ(x) := lim h 0 h E x[φ(x h ) Φ(x)]. Given: Operator Ω on Π. A solution of the (Ω, Π)-martingale problem is a process X = (X t ) t 0 if for all Φ Π, ( t ) Φ(X t ) ΩΦ(X s )ds 0 t 0 is a martingale. The MP is well-posed if there is exactly one such process.

14 Polynomials on P(I ) Π: functions on U of the form (polynomials) Φ ( X, r,µ ) := µ N, φ := φ ( r(x, x),u ) µ N (d(x,u)) for (x,u) = ((x 1,u 1 ), (x 2,u 2 ),...), φ C b (R (N 2) I N ) depending on finitely many coordinates Π separates points in P(I )

15 Polynomials on U Π: functions on U of the form (polynomials) Φ ( X, r,µ ) := µ N, φ := φ ( r(x, x),u ) µ N (d(x,u)) for (x,u) = ((x 1,u 1 ), (x 2,u 2 ),...), φ C b (R (N 2) I N ) depending on finitely many coordinates Π separates points in X

16 Generator for the Fleming-Viot process: measure-valued Ω := Ω grow +Ω res + Ω mut + Ω sel Ω grow : tree growth Ω res : resampling Ω mut : mutation Ω sel : selection

17 Generator for the Fleming-Viot process: tree-valued Ω := Ω grow +Ω res + Ω mut + Ω sel Ω grow : tree growth Ω res : resampling Ω mut : mutation Ω sel : selection

18 Tree Growth When no resampling occurs the tree grows ε Time ε passes.. sample. full tree Distances in the sample grow at speed 2 Ω grow Φ(X, r, µ) = 2 µ N, i<j φ. r ij

19 Resampling: measure-valued Ω res Φ(X, r,µ):= k<l µ N, φ θ k,l φ with (θ k,l (u)) i := { u i, u k, i l i = l In addition, r(x i, x j ), if i, j l, (θ k,l r(x, x)) i,j := r(x i, x k ), if j = l, r(x k, x j ), if i = l,

20 Resampling: tree-valued Ω res Φ(X, r,µ):= k<l µ N, φ θ k,l φ with (θ k,l (u)) i := { u i, u k, i l i = l In addition, r(x i, x j ), if i, j l, (θ k,l r(x, x)) i,j := r(x i, x k ), if j = l, r(x k, x j ), if i = l,

21 Mutation: measure-valued ϑ: total mutation rate ϑ β(u, dv): mutation rate from u to v Ω mut Φ(X, r,µ)= ϑ µ N, β k φ φ with β k (u, dv) acting on kth variable k

22 Mutation: tree-valued ϑ: total mutation rate ϑ β(u, dv): mutation rate from u to v Ω mut Φ(X, r,µ)= ϑ µ N, β k φ φ with β k (u, dv) acting on kth variable k

23 Selection: measure-valued α: selection coefficient χ(u) [0, 1]: fitness of type u (continuous) Ω sel Φ(X, r,µ) := α n µ N, χ k φ χ n+1 φ k=1 where φ only depends on sample of size n with χ k acting on kth variable

24 Selection: tree-valued α: selection coefficient χ(u) [0, 1]: fitness of type u (continuous) Ω sel Φ(X, r,µ) := α n µ N, χ k φ χ n+1 φ k=1 where φ only depends on sample of size n with χ k acting on kth variable

25 Selection: tree-valued Why is selection the same as for measure-valued case? Ω sel N : generator for finite model of size N φ: only depends on first n N individuals Ω sel N Φ(X, r, µ) α N α = α N µ N, χ k (φ θ k,l φ) k,l=1 n µ N, χ n+1 (φ θ n+1,l φ) l=1 n µ N, χ l φ χ n+1 φ l=1

26 The tree-valued Fleming-Viot process Theorem: The (Ω, Π)-martingale problem is well-posed. Its solution X = (X t ) 0, X t = (X t, r t, µ t ) the tree-valued Fleming-Viot process arises as weak limit of tree-valued Moran models and satisfies: P(t Xt is continuous) = 1, P((Xt, r t ) is compact for all t > 0) = 1, X is Feller (hence strong Markov)

27 Girsanov transform: measure-valued Theorem: Let α, α R, X solution of (Ω, Π)-MP for selection coefficient α, and M = Ψ(X, r,µ) := (α α) µ N, χ 1 ( t ) Ψ(X t, r t,µ t ) Ψ(X 0, r 0,µ 0 ) ΩΨ(X s, r s,µ s )ds. 0 t 0 Then, Q, defined by dq = e Mt 1 2 [M]t dp Ft solves (Ω, Π)-MP for selection coefficient α

28 Girsanov transform: tree-valued Theorem: Let α, α R, X solution of (Ω, Π)-MP for selection coefficient α, and M = Ψ(X, r,µ) := (α α) µ N, χ 1 ( t ) Ψ(X t, r t,µ t ) Ψ(X 0, r 0,µ 0 ) ΩΨ(X s, r s,µ s )ds. 0 t 0 Then, Q, defined by dq = e Mt 1 2 [M]t dp Ft solves (Ω, Π)-MP for selection coefficient α

29 Long-time behavior Theorem: The tree-valued Fleming-Viot process is ergodic iff the measure-valued Fleming-Viot process is ergodic

30 Application: distances is equilibrium Theorem: I = {, }, χ(u) = 1{u= } ( is fit, is unfit) ϑ 2 : mutation rate and R12 α : distance of two randomly sampled points in equilibrium E[e λrα 12 /2 ] = λ 4ϑ(2 + λ + 2ϑ)λ + (1 + ϑ)(1 + λ + ϑ)(6 + λ + ϑ)(1 + λ)(6 + 2λ + ϑ) α2 + O(α 3 ) Proof: Use E[Ω µ N, e λr(x 1,x 2 )/2 ] = 0

31 Summary and outlook Once the right state-space is chosen, construction of tree-valued Fleming-Viot process straight-forward Genealogical distances can be computed using generators Next step: Include recombination