Introduction CSBPes and Fleming-Viot processes Time change relation Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias {Birkner 1 , Steinr¨ ucken 2 } 1 University of Munich and 2 TU Berlin Joint project with Jochen Blath 2 Eindhoven, 25th/27th March 2009
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Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias
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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
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Outline
Introduction
Beta(2 − α,α)-coalescents
Dawson-Watanabe and Fleming-Viot processes, time-changerelation
(a random subsample of the sample described in Arnason, Genetics 2004)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
The Great Obsession of population geneticists (J. Gillespie)
What evolutionary forces could have lead to such
divergence between individuals of the same species?
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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.
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Genealogical point of view
Sample n (≪ N) individuals from the ‘present generation’
past
present
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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.
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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).
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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.
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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.
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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).
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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).
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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).
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.
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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).
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.
Form of rates stems from λn,k = λn+1,k + λn+1,k+1 (consistencycondition).
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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).
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.
Form of rates stems from λn,k = λn+1,k + λn+1,k+1 (consistencycondition).Note: Λ = δ0 corresponds to Kingman’s coalescent.
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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.
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
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]).
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
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
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
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
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
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.
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
TheoremHeuristics (discont. case)
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