Approximating genealogies under genetic hitchhiking with recurrent mutation

Transcription

1 Approximaing genealogies under geneic hichhiking wih recurren muaion Peer Pfaffelhuber (join wih Joachim Hermisson) La Londe, Sepember 2008

2 Goal Goal: deec selecion in a genome Use sample variaion daa o find candidae genes Needed: predicion of sequence diversiy under various forms of selecion (Classical) selecive Sweep: Variaion around a sronlgy beneficial allele is srongly reduced Here: selecion sars acing a = 0 beneficial allele arises recurrenly during fixaion Sof sweep: beneficial allele has muliple origins

3 Sof Sweep Paerns Recurren muaion in a Wrigh-Fisher model (Pennings, Hermisson, 2006)

4 Sof Sweep Paerns Classical selecive sweep: neural variaion dragged o high frequency ogeher wih beneficial allele Sof sweps: Muliple muans inroduce differen paerns of neural variaion Consequence: Differen haploype blocks around he seleced sie

5 Lacose gene (from ishkoff e al (2007)) No all aduls can diges milk ( lacase persisence LP) Probably connecion o cale domesicaion Europe: Swedes 90% LP, Spanish 50% LP; SNP C/ associaed wih LP Asia: Chinese 1% LP Africa: Wes-African agriculuraliss 5-20% LP; G/C mos significanly associaed SNP wih LP : Differen origins of LP

6 he Wrigh-Fisher diffusion Frequency pah of beneficial allele is dx = ( θ 2 (1 X )+αx (1 X )) d+ X (1 X )dw, X 0 = 0 s selecive advanage u muaion rae N populaion size α := sn 1 θ := 2uN d Nd generaions fixaion ime 0 θ=0.1 frequency of he beneficial allele 1

7 he Wrigh-Fisher diffusion Frequency pah of beneficial allele is dx = ( θ 2 (1 X )+αx (1 X )) d+ X (1 X )dw, X 0 = 0 s selecive advanage u muaion rae N populaion size α := sn 1 θ := 2uN d Nd generaions fixaion ime θ=0.05 frequency of he beneficial allele

8 he Wrigh-Fisher diffusion Frequency pah of beneficial allele is dx = ( θ 2 (1 X )+αx (1 X )) d+ X (1 X )dw, X 0 = 0 s selecive advanage u muaion rae N populaion size α := sn 1 θ := 2uN d Nd generaions fixaion ime θ=0.5

9 Fixaion imes Le 0 := sup{ 0 : X = 0}, := 0. Fixaion imes For θ > 0, E[ ] = 1 αθ + 2 log α ( 1 + O + α α) 1 θ O( αe α), E[ ] = 2 log α ( 1 + O, α α) ( 1 ) V[ ] = O α 2. For θ 1, almos surely, =.

10 he srucured coalescen Sample n individuals a ime Genealogy a seleced/linked neural sie given by srucured coalescen Kaplan, Hudson, Langley (1989); exension by Baron, Eheridge, Surm (2004) ime 0 : random pariion ξ of {1,..., n}. Goal: describe/approximae ξ

11 he srucured coalescen Discree model: given X = x, birh evens of beneficial alleles: rae Nx 2 common ancesry of a given pair probabiliy 1 ) ( Nx 2 unscaled coalescence rae 1 Nx coalescence rae: 1 X 0 0 frequency of he beneficial allele 1

12 he srucured coalescen Discree model: given X = x, probabiliy of following a muan is u(1 x). Probabiliy of picking a beneficial allele is x. unscaled muaion rae u(1 x) x 0 0 frequency of he beneficial allele 1 muaion escape rae: θ 1 X 2 X

13 he srucured coalescen Discree model: given X = x, Frequency of recombinans of beneficial allele wih wild-ype is rx(1 x) Probabiliy of picking a beneficial allele is x. unscaled recombinaion rae r(1 x) recomb escape rae: ρ(1 X) 0 0 frequency of he beneficial allele 1

14 he srucured coalescen Discree model: given X = x, birh evens of wild-ype alleles: rae N(1 x) 2 common ancesry of a given pair probabiliy 1 ( N(1 x) 2 unscaled coalescence rae 1 N(1 x) ) 0 0 frequency of he beneficial allele 1 1 coal. in wildype: 1 X

15 he srucured coalescen Discree model: given X = x, Frequency of recombinans of beneficial allele wih wild-ype is rx(1 x) Probabiliy of picking a wild-ype allele is 1 x. unscaled recombinaion rae rx back recombinaion: ρx 0 frequency of he beneficial allele

16 he srucured coalescen rae: 1 X 0 rae: θ 1 X 2 X 0 rae: ρ(1 X) 0 1 rae: 1 X 0 rae: ρx 0

17 he srucured coalescen rae: 1 X X 0 rae: θ 1 X 2 X 0 rae: ρ(1 X) 0 1 rae: 1 X 0 rae: ρx 0 Probabiliy) O( 1 (log α) 2 ρ = O( α log α ) for

18 he srucured coalescen ime rescaling dτ = (1 X )d: dy = ( θ 2 + αy ) dτ + Y dw, Y 0 = 0. Supercriical Feller branching process wih immigraion Sop when hiing Y = 1

19 he srucured coalescen rae: 1 Y 0 rae: θ 1 2 Y 0 rae: ρ 0 Coalescen generaes a marked (rae ρ) genealogy of a supercriical Feller branching process wih immigraion (rae θ/2)

20 he Yule process approximaion spliing rae α per line, immigraion rae: θ recombinaions: rae ρ along Yule ree 2α lines

21 he Yule process approximaion spliing rae α per line, immigraion rae: θ recombinaions: rae ρ along Yule ree 2α lines muaion

22 he Yule process approximaion spliing rae α per line, immigraion rae: θ recombinaions: rae ρ along Yule ree 2α lines recombinaion muaion recomb

23 he Yule process approximaion Given: sample of size n Yule process approximaion: random pariion Υ of {1,..., n} Le ρ = γ α log α. heorem ( sup P[ξ A] P[Υ A] 1 ) = O A (log α) 2 where he error is uniform on compaca in γ, θ.

24 Relaed work θ = 0: Durre, Schweinsberg (2004,...), Eheridge, P, Wakolbinger (2006): Yule approximaion for classical sweeps ρ = 0: Pennings, Hermisson (2006): family sizes of origins of beneficial allele follow he Ewens sampling formula P, Sudeny (2007): Yule approximaion for genealogies of wo neural loci Leocard (2008): Yule approximaion for several neural loci

25 Applicaion: heerozygosiy Heerozygosiy H : probabiliy ha wo randomly picked individuals carry differen alleles Consider neural locus linked o he seleced one Assuming no muaions a neural locus during he sweep, H = P[no coalescence by 0 ] H 0.

26 Applicaion: expeced heerozygosiy Using Yule process approximaion for ρ = γ α log α : H = 1 p2 1 H 0 θ + 1 2γ 2α log α i=2 2i + θ ( (i + θ) 2 (i θ) p2 i +O 1 ) (log α) 2 wih ( p i := exp ρ α 2α j=i+1 1 ). j

27 Applicaion: expeced heerozygosiy H H θ=1 θ=0.4 θ=0 Wrigh Fisher Yule ρ

28 Summary Sof sweeps from recurren muaion generalize classical sweeps Ewens sampling formula gives family decomposiion a seleced sie Yule process wih immigraion and marks approximaes genealogy a linked neural locus

29 Oulook Lacase Persisence: parial sweep, srucured populaion Wha is a good approximaion o he genealogy under sweeps in srucured populaions?