lambda-coalescents and population genetic inference...neutral mutation models computing likelihoods...

IntroductionCSBPes and Fleming-Viot processes

Time change relationNeutral mutation models

Computing likelihoodsIllustration and outlook

Lambda-coalescents and population geneticinference

Matthias {Birkner1, Steinrucken2}1University of Munich and 2TU Berlin

Joint project with

Jochen Blath2

Eindhoven, 25th/27th March 2009




Outline

Introduction

Beta(2 − α,α)-coalescents

Dawson-Watanabe and Fleming-Viot processes, time-changerelation

Mutation models: Infinitely-many-alleles, infinitely-many-sites

Computing likelihoods under Λ-coalescents

Importance sampling methods

Summary & Outlook




The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class

Genetic variability at the mitochondrial cyt b locus in

Atlantic cod

468 481 487 488 490 496 508 523 562 601 631 643 649 685 691

66 t a a c a a t g a t g a c c g

17 - - - - - - c - - - - - - - -

14 - - - - - - - a - - - - - t -

8 - - - - - - - - - - - - - - t

1 - - - - - - - - - - - - - t -

2 - - - t - - - - - - - - - - -

1 - - - - - - - a - - - g - t -

1 - - - - - - - - - - - - t - -

1 - - - - g - c - - - - - - - -

1 - - - - - g - - - - - - - - -

1 - - g - - - - - - - a - - - t

1 - - - - - - c - g - - - - - -

1 g - - - - - - - - - - - - - -

1 - - - - - - - - - c - - - - -

1 - c - - - - c - - - - - - - -

(a random subsample of the sample described in Arnason, Genetics 2004)





The Great Obsession of population geneticists (J. Gillespie)

What evolutionary forces could have lead to such

divergence between individuals of the same species?





The Great Obsession of population geneticists (J. Gillespie)

What evolutionary forces could have lead to such

divergence between individuals of the same species?

In this talk, we will focus on neutral genetic variation, and thus theinterplay of mutation and genetic drift.





Wright-Fisher model: The fundamental model for ‘genetic drift’

A (haploid) population of N individuals per generation,each individual in the present generation picks a ‘parent’ atrandom from the previous generation,genetic types are inherited (possibly with a small probability ofmutation).

past





Genealogical point of view

Sample n (≪ N) individuals from the ‘present generation’

past

present





Kingman’s coalescent

Theorem (Kingman (& Hudson, Griffiths, ...), 1982)

In the limit N → ∞, the genealogy of an n-sample, measured in units of N generations, isdescribed by a continuous-time Markov chainwhere each pair of lineages merges at rate 1.





Kingman’s coalescent

Theorem (Kingman (& Hudson, Griffiths, ...), 1982)

In the limit N → ∞, the genealogy of an n-sample, measured in units of N generations, isdescribed by a continuous-time Markov chainwhere each pair of lineages merges at rate 1.

Robustness. The same limit appears for any exchangeable offspringvectors

(ν1, . . . , νN), (independent over generations),

if time is measured in units ofN

σ2generations, where

σ2 = limN→∞

Var(ν1)

(under a third moment condition on ν1).





Modeling neutral variation: Superimposing types on the

coalescent

Assume that the considered genetic typesdo not affect their bearer’s reproductive succes.

If as population size N → ∞,N

σ2× mutation prob. per ind. per generation → r ,

the type configuration in the sample can be described by puttingmutations with rate r along the genealogy.





Modeling neutral variation: Superimposing types on the

coalescent

Assume that the considered genetic typesdo not affect their bearer’s reproductive succes.

If as population size N → ∞,N

σ2× mutation prob. per ind. per generation → r ,

the type configuration in the sample can be described by puttingmutations with rate r along the genealogy.

Kingman’s coalescent is the standard model of mathematicalpopulation genetics.





Question

What if the variability of offspring numbers across individuals is solarge that reasonably

σ2 ≈ ∞ ?

This might happen e.g. in marine species (so-called reproduction

sweepstakes).





Coalescents with multiple collisions, aka ‘Λ-coalescents’

While n lineages, any k coalesce at rate

λn,k =

∫

[0,1]xk−2(1 − x)n−k Λ(dx), where Λ is a finite measure on

[0, 1]. (Sagitov, 1999; Pitman, 1999).







λn,k =

∫


[0, 1]. (Sagitov, 1999; Pitman, 1999).

Interpretation:re-write λn,k =

∫[0,1] x

k(1 − x)n−k 1x2 Λ(dx) to see:

at rate 1x2 Λ([x , x + dx ]), an ‘x-resampling event’ occurs.

Thinking forwards in time, this corresponds to an event in whichthe fraction x of the total population is replaced by the offspring ofa single individual.







λn,k =

∫


[0, 1]. (Sagitov, 1999; Pitman, 1999).


∫[0,1] x




Form of rates stems from λn,k = λn+1,k + λn+1,k+1 (consistencycondition).







λn,k =

∫


[0, 1]. (Sagitov, 1999; Pitman, 1999).


∫[0,1] x




Form of rates stems from λn,k = λn+1,k + λn+1,k+1 (consistencycondition).Note: Λ = δ0 corresponds to Kingman’s coalescent.





Cannings’ models in the

‘domain of attraction of a Λ-coalescent’

Fixed population size N, exchangeable offspring numbers in onegeneration (

ν1, ν2, . . . , νN

).

Sagitov (1999), Mohle & Sagitov (2001) clarify under whichconditions the genealogies of a sequence of exchangeable finitepopulation models are described by a Λ-coalescent:

cN := pair coalescence probability over one generation → 0

( cN = 1N−1E[ν1(ν1 − 1)] )

two double mergers asymptotically negligible compared to onetriple merger

NcN Pr(a given family has size ≥ Nx

)∼

∫ 1x

y−2Λ(dy)

Time is measured in 1/cN generations (in general 6= 1/pop. size)





Note: There are many Λ-coalescents.

Maybe a natural “first candidate”:

Λ = wδ0 + (1 − w)δψ with w , ψ ∈ (0, 1)

(as considered by Eldon & Wakeley, Genetics 2006)





A ‘heavy-tailed’ Cannings model and Beta-coalescents

Haploid population of size N. Individual i has Xi potential

offspring, X1,X2, . . . ,XN are i.i.d. with mean m := E[X1

]> 1,

Pr(X1 ≥ k

)∼ Const. × k−α with α ∈ (1, 2).

Note: infinite variance.

Sample N without replacement from all potential offspring to formthe next generation.





A ‘heavy-tailed’ Cannings model and Beta-coalescents

Haploid population of size N. Individual i has Xi potential

offspring, X1,X2, . . . ,XN are i.i.d. with mean m := E[X1

]> 1,

Pr(X1 ≥ k

)∼ Const. × k−α with α ∈ (1, 2).

Note: infinite variance.

Sample N without replacement from all potential offspring to formthe next generation.

Theorem (Schweinsberg, 2003)Let cN = prob. of pair coalescence one generation back in N-thmodel.cN ∼ const. N1−α, measured in units of 1/cN generations, thegenealogy of a sample from the N-th model is approximatelydescribed by a Λ-coalescent with Λ = Beta(2 − α,α).(

Beta(2 − α,α)(dx) = 1[0,1](x) 1Γ(2−α)Γ(α) x

1−α(1 − x)α−1 dx)





Why Λ = Beta(2 − α, α)?

Heuristic argument:

Probability that first individual’s offspring provides

more than fraction y of the next generation,

given that the family is substantial (i.e. given X1 ≥ εN, for y > ε)





Why Λ = Beta(2 − α, α)?

Heuristic argument:

Probability that first individual’s offspring provides

more than fraction y of the next generation,

given that the family is substantial (i.e. given X1 ≥ εN, for y > ε)

≈ P

( X1

X1 + (N − 1)m≥ y

∣∣∣X1 ≥ εN)

= P

(X1 ≥ (N − 1)m

y

1 − y

∣∣∣X1 ≥ εN)

∼ const.(1 − y)α

yα= const.’ Beta(2 − α,α)([y , 1]).





The family Beta(2 − α, α), α ∈ (1, 2]

0.0 0.2 0.4 0.6 0.8 1.0

01

23

45

6

1.91.51.1

Kingman’s coalescent includedas boundary case:Beta(2 − α,α) → δ0 weakly as α→ 2.

Smaller α means tendency towardsmore extreme resampling events.

For α ≤ 1, corresponding coalescentsdo not come down from infinity.

Beta(2 − α,α)-coalescents appear as genealogies of α-stablecontinuous mass branching process (via a time-change).

Scaling relation of mutation rate per generation relative topopulation size depends on α!




Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality

Free branching: Galton-Watson processes

individuals have a random number of offspring, independently,according to a fixed probability distribution

the totality of offspring forms the next generation

da capo ...





Galton-Watson forests (with ordered individuals)tim

e





Galton-Watson forests (with ordered individuals)tim

e

ξt(n) := # descendants of founders 1, . . . , n alive in generation t .For fixed n, ξt(n), t ∈ Z+ is a Galton-Watson process, ξ0(n) = n.





Free branching & neutral typestim

e

Example: there are 2 neutral types, red and blue.





Fixed population size: Cannings models (reprise)

fixed population size N

individuals have a random number of offspring,the offspring numbers (ν1, . . . , νN) are exchangeable with∑N

i=1 νi = N

the totality of offspring forms the next generation

da capo ...





Genealogies in a Cannings modeltim

e





Genealogies in a Cannings model: Adding typestim

e

We can again think of neutral types (for the moment, withoutmutation), e.g. red and blue.





Question

Relation between

and ?





Relations?

A first answer

Condition a Galton-Watson process to have constant

population size N, obtain a model from Cannings’ class.

Easy, but is there more?





Relations?

In a Cannings model, we think ofrelative frequencies of types.

We can do the same in a free branching model by normalising withthe current total population size (at least before extinction), i.e. byconsidering

ξt(n) :=ξt(n)

ξt(N).

/ ≈ ?





Relations?

Yes, in a suitable limit of population size N → ∞.

20 40 60 80 100 120

20

40

60

80

100





Continuous state branching processes (Jirina, Silverstein, Lamperti, ... )

Z (N) = (Z(N)k )k=0,1,2,..., N ∈ N a sequence of Galton-Watson

processes (possibly with offspring distributions depending on N),

mN → 0 mass rescaling,

Z(N)0 = [m−1

N ] ( + conditions ... ).

(mNZ

(N)[Nt]

)t≥0

=⇒ X ,

where X is a continuous state branching process, i.e. an R+-valuedMarkov process that enjoys the branching property:

X ,X ′,X ′′ independent copies, X0 = x ,X ′0 = x ′,X ′′

0 = x ′′, x = x ′ + x ′′

=⇒ Xlaw= X ′ + X ′′.





Continuous state branching processes: Favourite example

Assume the N-th offspring distribution has

mean = 1 + µ/N + o(1/N),

variance = σ2 + o(1)

(and, say, uniformly bounded third moment).

Z(N)0 = N, then

(N−1Z

(N)[Nt]

)t≥0

⇒ Feller’s branching diffusion,

generator

L(2)f (z) =1

2σ2zf ′′(z) + µzf ′(z).

(i.e. Var[Zt+∆t −Zt|Zt ] ≈ σ2Zt∆t, E[Zt+∆t −Zt|Zt ] ≈ µZt∆t.)





Continuous state branching processes: stable case, α < 2

If the approximating offspring distributions are in the domain ofattraction of a stable law of index α ∈ (0, 2), i.e.

P(more than n children) ∼ Const. × n−α

(note: in particular, no variance),the limit process Z will have discontinuous paths. Generator

L(α)f (z) = cαzf ′(z)+z

∫

(0,∞)

{f (z+h)−f (z)−h1(0,1](h)f ′(z)

}h−1−αdh.

Interpretation:if the present mass is z , a new litter of mass h is produced at ratez × h−1−α.





Stable CSBPes, some properties

Why the name?

Lamperti’s construction connects them with stable Levy

processes: Xt = Y(∫ t

0 Xs ds), where Y is a stable process

without negative jumps, stopped upon hitting 0.

Scaling properties

Equation for Laplace transforms

Existence of mean:α > 1 ⇒ Ex Xt <∞,α < 1 ⇒ Ex Xt = ∞ for all t > 0.

Extinction/Explosion:α < 1 ⇒ X has growing paths, explodes in finite time,α > 1 ⇒ X becomes extinct in finite time.





Dawson-Watanabe superprocesses

For our purposes here: the continuous mass & timeanalogue of the “ragged boundary” picture:

Z a given CSBP. The corresponding Dawson-Watanabe

superprocess∗ is a process X with values in the measures on [0, 1]such that

for B ⊂ [0, 1]: (Xt(B))t≥0law= Z , started from Z0 = |B |,

X·(B1),X·(B2), . . . ,X·(Bn) are independent if B1, . . .Bn aredisjoint.

Interpretation: Xt(B) = mass of descendants alive at time t

whose ancestors where ∈ B∗Note: this is a “toy version”.





Λ-Fleming-Viot processes (Donnelly & Kurtz, Bertoin & Le Gall)

For our purposes here: the continuous mass & timeanalogue of the “straight boundary” picture:

(ρt)t≥0 a Markov process with values in the probability measureson [0, 1]. Interpretation (if ρ0 = uniform measure):

ρt(B) =fraction of mass alive at time t whose ancestors attime 0 where in B ⊂ [0, 1].

On test functions of the formF (ρ) =

∫. . .

∫ρ(dx1) . . . ρ(dxp)f (x1, . . . , xp), the generator acts as

LFV ,ΛF (ρ) =∑

J⊂{1,...,p},

|J|≥2

λp,|J|

∫. . .

∫ρ(dx1) . . . ρ(dxp)

(f (aJ

1 , . . . , aJp) − f (a1, . . . , ap)

).

↑aJi = amin J if i ∈ J





Λ-FV processes: Interpretation of “non-Kingman” part

N a Poisson point process on [0,∞) × (0, 1] with intensitymeasure dt ⊗ y−2Λ(dy).

0 20 40 60 80 100

0.00.2

0.40.6

0.81.0

· · ·· ··

·

·· ··· ·· ···· · ·

··

··

·· ·· ·· ·

·

·

·

··

··

·· · · ·····

··

··· ···

·

·

··

·· ···

·

· · ·

·

· ·· ·· ··

··· ··

· ·· ·· ·· · ··· ·· ·· ···

· ·

·

·· · ·· ···· ··· ·· ·

·

···· ·· ··· ··· ··· ···

··

· ··

· ·

· · ·

·

· · ·

·

·

· ·· · ··

·· ·

·

··

·

··

· · ·· ·

·· ·· ·

·

· · ···

·

·

· · ···

··

· · · ······ · ··

·

·

·

··· · ·

··

· ··· ·· ···

·

·· ·· · · ·

·

·· · ··· ··· ··

·

· ·· ·· ·

·

··

·

· · ·

··

· ··

·

··· · ··

·

·

· ·

· ···

· ·· · ·····

· ···· · ··· ··

·

· ··

·

·

·· · ·

·

· ·

·

·

·

···· ·· ·

·

·

·

·

· ·· ··· · ··· ··

· ·· ·

·· ·

· ··· · ·· ··

·

·· ··· ··

· ···

·· ··

·

· · ·· ··

··· · ·· · ·

·

·· ·· ··· ··

·

··· · ····· ···

·· ··

·

· ·

·

· ···· ··

···

·

·

··· · ··· ··

··

· ···

·

· · ···· ·· · ·

·

·· ·· ·

·

·

·

··

·

···

·

·

···

·

· ·· ·

·

···

·

··

·

··· ··

·

· ···· ·· · ···

·

· ·· ··

··

·

·· ·

·

···

·

·

·

·

· ···

·· · ·· ·· ·

·

·

·

··· ·· ···

··

·

··

· · ··

·

···· · ··

·

· ·· ·

·· ·

·

·

·

·

·

··

·

··· ··

·

···

·

·

·

· ·· ·· · ·

·

··

·

······

·

· ··

·

·· ·· · · ·· ····

··· · ·

·

·

·

··

···

·

·

··

···

····

·· · ·· ·

··

·

·

·

·

·

·· ·

··

· · ··· ··· · ··

·

··· ·

·

···

· ···· ·

·

···

·

· ··· ·· · ··

·

·

·

· ··· · ·· ·· ·

·

·· · ·

·

· ·

·

· ·

·

· ····

· ··

· ·

·

·

·

·

· ···· ·· ·· ·

·

·

·

·· ·

·

·· ·· ·· ·· · · · ·· · ·

·

··

··· ·

·

·

·

·

·

··

·· ··

·

··· · ·· ··

·

· ·

·

·· ·

·

·· · ··· ··

·· ···

·

···

· ···· ·

···

··

·· ·· ·· ·

·

·

·

··

·

·· ·· · ·

·

·· ·

·

·

· ·

· ·

· ··

·

· ··

···

·

·

·

···

·

·· · ···

·

· · ··

·

· ·· · ·· · ··

·

··· ·· ·

·

·· · ·· ·· ·

·

· · ····

· ···

·

· ··· ··

·

···

·

·· ·· · ··· ·· ··

·

··· ··· ·· ·

··

· · ·· ···

·

· ·· ··· ·· · · · ·

·

·

·

···

·

·· ·

y

time

If (t, y) is a point of N , an individual X is chosen according to thecurrent ρ(t−), and ρ→ yδX + (1 − y)ρ.

Note: alternative form of generator

LFV ,λG(ρ) =∫

y−2Λ(dy)∫ρ(dx)

(G(yδx + (1 − y)ρ) − G(ρ)

)





Duality between coalescent and FV

For a partition γ of {1, . . . , p} with |γ| classes and a probabilitymeasure ρ on [0, 1] consider test functions of the type

Φf (γ, ρ) :=

∫ρ(dx1) . . . ρ(dx|γ|)f

(~y(γ; x1, . . . , x|γ|)

)

↑yj = xi if j ∈ i -th class of γ

(“assign an independent ρ-sample to each class of γ”), wheref : [0, 1]p → R.

Then we have with(Γ

(p)t ) . . . . . . . . restriction of Λ-coalescent to {1, . . . , p},

(ρt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Λ-FV process

E

[Φf (Γ

(p)0 , ρt)

]= E

[Φf (Γ

(p)t , ρ0)

]for all t ≥ 0.





Sample interpretation of the duality

E

[Φf (Γ

(p)0 , ρt)

]= E

[Φf (Γ

(p)t , ρ0)

]ge

nerat

ions

20 40 60 80 100

20

40

60

80

100





Pathwise duality via Donnelly & Kurtz’ (modified)

lookdown construction

new particleat level 3

new particleat level 6

post−birth types

pre−birth types

a

b

a

g

b

c

e

f

d

g

b

c

d

b

e

f

7

6

5

4

3

2

1

pre−birth labels

post−birth labels

9

8

7

6

5

4

3

2

1




TheoremHeuristics (discont. case)

Let (Xt) be a Dawson-Watanabe process (for simplicity, type space[0, 1], no mutation), Zt := Xt([0, 1]) its total mass process,Rt := Xt/Zt .

Theorem (B., Blath, Capaldo, Etheridge, Mohle, Schweinsberg,Wakolbinger (2005), Hiraba (2000), Perkins (1991))The ratio process (Rt)0≤t<τ can be time-changed with an additivefunctional of the total mass process (Zt) to obtain a Markovprocess if and only if

Z is a CSBP of some index α ∈ (0, 2].

If α = 2, Tt =∫ t

0 σ2Z−1

s ds and T−1(t) = inf{s : Ts > t}. Theprocess (RT−1(t))t≥0 is the classical (non-spatial) Fleming-Viotprocess, dual to Kingman’s coalescent.If α ∈ (0, 2), Tt = const ·

∫ t

0 Z 1−αs ds and (RT−1(t))t≥0 is the

Beta(2 − α,α)-Fleming-Viot process.





Let X and X ′ be two independent CSBP’s with the samecharacteristics (ν), St := Xt + X ′

t , Rt := Xt/St .

At rate St−ν(dh)dt, a new family of size h > 0 is created , therelative mass of the newborns is y := h/(St− + h), so

∆Rt =

{y(1 − R) with probability Rt− and−yR with probability (1 − Rt−) .

To eliminate the dependence of the relative jump size y on thecurrent total population size St−, ν must satisfy

sν({h : h/(h + s) > y}) = sν({h : h > sy/(1 − y)}) = g(s)f (y)

for all s, y > 0 for some functions f , g .





‘Meta-mathematic’ associations

Brownian motion ↔ Kingman’s coalescent∩ ∩

Stable processes ↔ Beta(2 − α,α)-coalescents∩ ∩

General Levy processes ↔ General Λ-coalescents




A computer experimentInfinitely many allelesInfinitely many sites(Λ-) Ethier-Griffiths’ urn

Playing god with simulated “full trees”

1.0

1.0

true α

αM

L,fulltree

1.2

1.2

1.4

1.4

1.6

1.6

1.8

1.8

2.0

2.0

ML estimates of α for simulted datasets with sample size n = 100,estimate based on full genealogical tree

(400 replicates for each value of α).





Consider n-Λ-coalescent with mutation rate r per line (and infinitealleles mutation model). n = (n1, . . . , nℓ), possible type

configuration

Theorem (Mohle) The probability p(n) of observing a typeconfiguration n = (n1, . . . , nℓ) satisfies the recursion given byp(1) = 1 and

p(n) =nr∑n

k=2

(nk

)λn,k + nr

ℓ∑

j=1nj =1

1

ℓp(n(j))

+1∑n

k=2

(nk

)λn,k + nr

n∑

k=2

ℓ∑

j=1nj≥k

(n

k

)λn,k

nj − k + 1

n − k + 1p(n − (k − 1)ej ).

(n(j) = (n1, . . . , nj−1, nj+1, . . . , nk) )





Watterson’s infinitely many sites model

Model genetic locus as infinite sequence of completely linked sites,mutations always hit a new site.

Mathematical abstraction:

a gene is [0, 1]

a type is a configuration of points on [0, 1]

Ethier & Griffiths (1987) parametrisation:

type space E = [0, 1]N

mutation operator

Bf((x1, x2, . . . )

)= r

∫ 1

0f((u, x1, x2, . . . )

)−f

((x1, x2, . . . )

)du





Asymptotics of the frequency spectrum

Consider an n-Beta(2 − α,α)-coalescent, mutations at rate r

according to the infinitely-many-sites model (assuming knownancestral types). Let

M(n) := #total number of mutations in the sample,

Mk(n) := #number of mutations affecting exaktly k samples,

k = 1, 2, . . . , n − 1.Theorem (Berestycki, Berestycki & Schweinsberg)

M(n)

n2−α→ r

α(α− 1)Γ(α)

2 − α,

Mk(n)

n2−α→ rα(α− 1)2

Γ(k + α− 2)

k!

in probability as n → ∞.





Asymptotics of the frequency spectrum

Consider an n-Beta(2 − α,α)-coalescent, mutations at rate r

according to the infinitely-many-sites model (assuming knownancestral types). Let

M(n) := #total number of mutations in the sample,

Mk(n) := #number of mutations affecting exaktly k samples,

k = 1, 2, . . . , n − 1.Theorem (Berestycki, Berestycki & Schweinsberg)

M(n)

n2−α→ r

α(α− 1)Γ(α)

2 − α,

Mk(n)

n2−α→ rα(α− 1)2

Γ(k + α− 2)

k!

in probability as n → ∞.

Thus M1(n)/M(n) ≈ 2 − α for n large, which suggests

αBBS := 2 −M1(n)

M(n)as an estimator for α.





Infinitely-many-sites model

Model genetic locus as infinite sequence of completely linked sites,mutations always hit a new site

Example: segr. siteSeq. 1 2 3 41 1 0 0 02 1 1 0 03 0 0 1 14 0 0 1 15 0 0 1 0

(0=wild type, 1=mutant

assume known ancestral types)

Obs. fit IMS ⇐⇒ no sub-matrix1 01 10 1

aa bb cc(nor row permutation).





Infinitely-many-sites model, II

If the infinitely-many-sites model applies, the observationscorrespond to a unique rooted perfect phylogeny (or ‘genetree’).

Sequences, Genetree, obs. types

segr. siteSeq. 1 2 3 41 1 0 0 02 1 1 0 03 0 0 1 14 0 0 1 15 0 0 1 0

1

2

3

4

1 2 3,4 5

type multiplicity(1, 0) 1(2, 1, 0) 1(4, 3, 0) 2(3, 0) 1

Construct e.g. using Gusfield’s (1991) algorithm.

Note: purely combinatorial, does not depend on a probabilisticmodel for the observations.





Simulating samples under the IMS model

The Ethier-Griffiths urn (1987) can be used to generate arandom sample of size n under Kingman’s coalescent(with mutation rate r per line):

Start with 2 leaves.

When there are k leaves:

Add a mutation to a leaf w. prob. 2r2r+(k−1) ,

split one leaf w. prob. k−12r+(k−1)

(leaf picked uniformly among the k).

Stop when n + 1 leaves, delete last leaf.





Simulating samples under the IMS model: Λ-case

(Y(n)t )≥0 block counting process of Λ-coalescent starting from n

blocks:

Jump from i to j ∈ {1, 2, . . . , i − 1} at rateqij :=

(i

i−j+1

)λi ,i−j+1.

τ1 := inf{t ≥ 0 : Y(n)t = 1}.

Y(n)t := Y

(n)(τ1−t)− time-reversed block counting process

(Y(n)t = ∂ for t ≥ τ1).

Jump rates q(n)ji = gniqi j

gnj, q

(n)n∂ = −qnn =

∑n−1j=1 qnj ,

P(Y(n)0 = k) = P(Y

(n)τ1−

= k) = gnkqk1.

gni := E∫ ∞0 1(Y

(n)t = i) dt is the Green function (in general,

not known explicitly, but easy recursion).





Simulating samples under the IMS model: Λ-case, cont.

The n-Λ-“Ethier-Griffiths urn” (mutation rate r).

Begin with K leaves, P(K = k) = P(Y(n)0 = k).

While there are k leaves:

Add a mutation to a leaf w. prob. r

kr−eq(n)kk

,

split one leaf into ℓ w. prob.eq(n)

k,k+ℓ−1

eq(n)kk

,

if k = n goto stop w. prob. −eq(n)nn

kr−eq(n)nn

(leaf picked uniformly among the k).

Stop.




Tree probabilitiesImportance samplingPerformance

Recursion for tree probabilities

Can calculate P(r ,Λ)

(observed sequence data (t,n)

)=: p(t,n) via

1 2

34

0





Recursion for tree probabilities

Can calculate P(r ,Λ)

(observed sequence data (t,n)

)=: p(t,n) via

1 2

34

0

p(t,n) =1

rn + λn

∑

i :ni≥2

ni∑

k=2

(n

k

)λn,k

ni − k + 1

n − k + 1p(t,n− (k − 1)ei

)

+r

rn + λn

∑

i :ni =1,xi0unique,

s(xi )6=xj∀j

p(si (t),n

)

+r

rn + λn

1

d

∑

i :ni =1,

xi0unique

∑

j:s(xi )=xj

(nj + 1)p(ri (t), ri (n + ej)

).

Extends Ethier & Griffiths (1987) to Λ-coalescents andMohle’s recursion (2005) to IMS model.





Compute probabilities

Use exact recursions for moderate sample complexities.Approach more complex samples by version of Griffiths &Tavare’s (1994)

Monte Carlo method

p(t,n) = E(t,n)

[ τ−1∏

i=0

f(r ,Λ)(Xi)]

For suitable Markov chain Xi on sample configurations.

Estimate expectation via empirical mean of independent runs.

extension to Λ-coalescents by B. & Blath (2008)





Artifical sample of size 12 analysed with r = 1 and α = 1.5:

5.0 5.5 6.0 6.5 7.0

−2.

5−

2.0

−1.

5−

1.0

−0.

50.

0

Griffiths & Tavaré

log10(

b sdest

imate

)

log10(#runs)





HIStory

Interpret genealogy as sequency of historical states:H =

(H−τ = ((1), (0)),Hτ−1 , . . . ,H−1,H0 = (t,n)

)

1

2

3

0

(((3, 1, 0), (1, 0), (2, 1, 0), (0)

), (1, 2, 1, 1)

)





HIStory


(H−τ = ((1), (0)),Hτ−1 , . . . ,H−1,H0 = (t,n)

)

1

2

3

0

H0

H−1

H−2

H−3

H−4

H−5

H−6

(((3, 1, 0), (1, 0), (2, 1, 0), (0)

), (1, 2, 1, 1)

)





HIStory


(H−τ = ((1), (0)),Hτ−1 , . . . ,H−1,H0 = (t,n)

)

1

2

3

0

1

2

3

0

(((3, 1, 0), (1, 0), (2, 1, 0), (0)

), (1, 2, 1, 1)

)





HIStory


(H−τ = ((1), (0)),Hτ−1 , . . . ,H−1,H0 = (t,n)

)

1

2

3

0

1

2

3

0

1

2

3

0

(((3, 1, 0), (1, 0), (2, 1, 0), (0)

), (1, 2, 1, 1)

)





HIStory


(H−τ = ((1), (0)),Hτ−1 , . . . ,H−1,H0 = (t,n)

)

1

2

3

0

1

2

3

0

1

2

3

0

different histories can lead to same sample(((3, 1, 0), (1, 0), (2, 1, 0), (0)

), (1, 2, 1, 1)

)





Importance sampling

We have

p(t,n) = P(r ,Λ)

(H0 = (t,n)

)=

∑

H:H0=(t,n)

P(r ,Λ)

(H

)

=∑

H:H0=(t,n)

P(r ,Λ)

(H

)

Q(H

)︸︷︷︸

=:w(H)importance weight

Q(H),

for any law Q on histories s.th. P(r ,Λ)

∣∣∣{H0=(t,n)}

≪ Q.

Thus,

p(t,n) ≈1

R

R∑

i=1

w(H(i)),

where H(1), . . . ,H(R) are independent samples from Q





Simply the best

p(t,n) =∑

H:H0=(t,n)

P(r ,Λ)

(H

)

Q(H

) Q(H

)≈

1

R

R∑

i=1

w(H(i))

Qopt(·) := P(r ,Λ)

(·∣∣H0 = (t,n)

)is optimal (Stephens &

Donnelly 2000):

Variance of estimator is zero since w(H(i)) ≡ p(t,n).

Finding Qopt is as hard as the original problem.

H0,H−1, . . . is Markov chain under Qopt.

Remark: Transistion probabilities qGT(Hi |Hi+1) ∝ P(r ,Λ)(Hi+1|Hi )gives (Λ-)Griffiths-Tavare method.





Stephens and Donnelly’s (2000) IMS candidate

Kingman case: Choose individual uniformly:

If type is unique in sample, remove “outmost” mutation,

if at least two individuals with this type, merge two lines.

(optimal proposal distribution for infinitely-many-alleles model,HUW (2008))

Heuristic extension to Λ case:

(t,n) →

(si (t),n

)w.p. ∝ 1 if ni = 1, xi0 unique, si (xi ) 6= xj∀j(

ri (t, ri (n + ej )) w.p. ∝ 1 if ni = 1, xi0 unique, si (xi ) 6= xj(t,n − (k − 1)ei

)w.p. ∝ ni qni

(k) if 2 ≤ k ≤ ni ,

where qni(k) =

qn,n−k+1Pnil=2 qn,n−l+1

, jump probabilities of block countingprocess.






5.0 5.5 6.0 6.5 7.0

−2.

5−

2.0

−1.

5−

1.0

−0.

50.

0

Griffiths & TavaréStephens & Donnelly

log10(

b sdest

imate

)

log10(#runs)





Hobolth, Uyenoyama & Wiuf’s (2008) idea

Sample of size n where exactly one mutation is visible (in d copies).

0 n-d

1 d

p(1)(r ,Λ)(n, d) = P(r ,Λ)

{most recent event involves in-dividual bearing mutation

}

Probability can be computed

Kingman case: explicit formula (HUW (2008))

Λ case: numerically, using recursion





Hobolth, Uyenoyama & Wiuf’s (2008) idea contd.

For a general sample (t,n)0 0

1 2

4 2 5 4

2 1 3 3

iput

ui ,m =

{

where mutation m is present in dm individuals.







1 2

4 2 5 4

2 1 3 3

i

carrying:

0 4

1 = d11 8

put

ui ,m =

{p

(1)(r ,Λ)(n, dm) · ni

dmif i bears m








1 2

4 2 5 4

2 1 3 3

i

carrying:

0 4

1 8

0 8

5 4

put

ui ,m =

{p

(1)(r ,Λ)(n, dm) · ni

dmif i bears m








1 2

4 2 5 4

2 1 3 3

i

carrying:

0 4

1 8

0 8

5 4

not carrying:

0 11

2 1

put

ui ,m =

{p

(1)(r ,Λ)(n, dm) · ni

dmif i bears m

(1 − p

(1)(r ,Λ)(n, dm)

)· ni

n−dmotherwise,








1 2

4 2 5 4

2 1 3 3

i

carrying:

0 4

1 8

0 8

5 4

not carrying:

0 11

2 1

0 9

3 3

0 10

4 2

put

ui ,m =

{p

(1)(r ,Λ)(n, dm) · ni

dmif i bears m

(1 − p

(1)(r ,Λ)(n, dm)

)· ni

n−dmotherwise,








1 2

4 2 5 4

2 1 3 3

i

carrying:

0 4

1 8

0 8

5 4

not carrying:

0 11

2 1

0 9

3 3

0 10

4 2

put

ui ,m =

{p

(1)(r ,Λ)(n, dm) · ni

dmif i bears m

(1 − p

(1)(r ,Λ)(n, dm)

)· ni

n−dmotherwise,

where mutation m is present in dm individuals. Propose type i

according to

qΛ-HUW

(i |(t,n)

)∝

{∑m ui ,m if i is allowed to act

0 otherwise.






If proposed type i

is singleton: remove “outmost” mutation,

has ni ≥ 2: merger inside type i .

Kingman case: merge two lines

Λ-case: propose ℓ+ 1-merger w.p. ∝ Pr,Λ

{0 n

1 do − l

∣∣∣∣0 n

1 do

}






5.0 5.5 6.0 6.5 7.0

−2.

5−

2.0

−1.

5−

1.0

−0.

50.

0

Griffiths & TavaréStephens & DonnellyHobolth, Uyenoyama & Wiuf

log10(

b sdest

imate

)

log10(#runs)





Performance

Simulated 50 samples of size 15 with r = 2 and α = 1.5. Analysedwith r = 1 and α = 1.5. Time needed to get relative error below0.01:

05

1015

20

3.9 4.5 5.1 5.7 6.3 6.9 7.5 8.1 8.7

(a) measured in log10(# runs of MC)

05

1015

-0.7 -0.2 0.3 0.8 1.3 1.8 2.3 2.8 3.3

(b) measured in log10(seconds)




IllustrationOutlook

Dataset from Ward et al, Extensive Mitochondrial

Diversity Within a Single Amerindian Tribe, PNAS 1991

Analysis with Beta-Coalescent:

1

2

3

4

5

1.2 1.4 1.6 1.8 2.0

1e−2

5

1e−25

1e−2

4

1e−241e−23

1e−2

3

1e−22

1e−2

2

1e−211e−21

1e−202e−20

4e−206e−20

8e−209e−20

α

r

Mitochondrial control region from 55 female Nuu-Chah-Nulth,αML = 1.9, rML = 2.2(Sample as edited in Griffiths & Tavare, Stat. Sci., 1994)




IllustrationOutlook

Genetic variation at the mitochondrial cyt b locus of Atlantic cod

Carr & Marshall 1991 Pepin & Carr 1993 Arnason & Palsson 1996

0.5

1.0

1.5

2.0

2.5

3.0

1.2 1.4 1.6 1.8 2.0

−15.0

−12.

0

−12.0−11.0

−11.0

−10.0

−10.0

−9.0

−8.7 −8.5

−15

−14

−13

−12

−11

−10

−9

−8

α

r

0.5

1.0

1.5

2.0

2.5

3.0

1.2 1.4 1.6 1.8 2.0

−15.0

−13.0

−13.0−12.0

−12.0

−11.5

−11.5

−11.0 −10.5

−10.0 −9.8

−20

−18

−16

−14

−12

−10

α

r0.5

1.0

1.5

2.0

2.5

3.0

1.2 1.4 1.6 1.8 2.0

−20.0

−18.0

−15.0

−15.0−14.0−14.0

−13.5

−13.0−12.7

−22

−21

−20

−19

−18

−17

−16

−15

−14

−13

α

r

bαML = 1.4,brML = 0.8 bαML = 1.4,brML = 0.6 bαML = 1.65,brML = 0.7

Arnason et al 1998 Arnason et al 2000 Sigurgıslason & Arnason 2003

0.5

1.0

1.5

2.0

2.5

3.0

1.2 1.4 1.6 1.8 2.0

−18.0

−18.0−15.0

−15.0

−14.5

−14.5

−14.0−13.7

−22

−21

−20

−19

−18

−17

−16

−15

−14

α

r

0.5

1.0

1.5

2.0

2.5

3.0

1.2 1.4 1.6 1.8 2.0

−15.0

−12.

0

−12.0−11.0

−11.0

−10.0

−10.0

−9.0

−8.7 −8.5

−20

−18

−16

−14

−12

−10

−8

α

r

0.5

1.0

1.5

2.0

2.5

3.0

1.2 1.4 1.6 1.8 2.0

−15.0

−12.

0

−12.0−11.0

−11.0

−10.0

−10.0

−9.0

−8.8

−8.5

−20

−18

−16

−14

−12

−10

−8

α

r

bαML = 1.55,brML = 0.6 bαML = 1.4,brML = 0.8 bαML = 1.4,brML = 0.8




IllustrationOutlook

Summary & Outlook

Eldon & Wakeley, Genetics 2006, wrote

For many species, the coalescent with multiple mergers might

be a better null model than Kingman’s coalescent.




IllustrationOutlook

Summary & Outlook

Eldon & Wakeley, Genetics 2006, wrote

For many species, the coalescent with multiple mergers might

be a better null model than Kingman’s coalescent.

For panmictic fixed-size discrete generations populations,haploid neutral one-locus theory is “mathematicallycomplete”.

Tools for estimation exist, results point towards“non-Kingman-ness” in certain cases.

Statistical properties of estimators?

speed-up of computer-intensive methods?combinations between IS-methods possibleDouble-HUW: ask all pairs of mutations what to do

A good class of alternative models?




IllustrationOutlook

Thank you for your attention!

lambda-coalescents and population genetic inference...neutral mutation models computing likelihoods...

Documents