The tree-valued Fleming-Viot process with mutation and selection Peter Pfaffelhuber University of Freiburg Joint work with Andrej Depperschmidt and Andreas Greven
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
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
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
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
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
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
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.
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
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
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
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
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.
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 )
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
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
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
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
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,
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,
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
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
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
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
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
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)
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
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
Long-time behavior Theorem: The tree-valued Fleming-Viot process is ergodic iff the measure-valued Fleming-Viot process is ergodic
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 ] = 1 1 + 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
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