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Expected time to coalescence and FST under a skewed offspring distribution

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8 Pages
English

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Expected time to coalescence and FST under a skewed offspring distribution among individuals in a population Bjarki Eldon (with John Wakeley) Mathematics and Informatics in Evolution and Phylogeny June 10-12, 2008

  • survivorship curves

  • coalescent timescale

  • singleton genetic variants

  • low genetic

  • migration

  • population sizes

  • modified moran model


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Expected time to coalescence andF
ST
under a skewed offspring distribution
among individuals in a population
Bjarki Eldon (with John Wakeley)
Mathematics and Informatics in Evolution and Phylogeny
June 10-12, 2008High variance in offspring distribution
• broadcast spawning and external fertilization
• type III survivorship curves
• very large population sizes
• low genetic variation
• large number of singleton genetic variants
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0 100
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orsesLifeISurvivorshiptISpanI
survivIPIIStandard reproduction models
have finite variance in offspring number
Moran model of overlapping generations:
a single randomly chosen individual
produces one offspring
2
N
Coalescent timescale: generations
2A modified Moran model
A special case of the models of Pitman (1999) and Sagitov (1999)
The number U of offspring is a random variable
The probabilityG of an x-merger (2≤x≤n):
n,x

u N−u
N
X
x n−x

G = P (u)
n,x U
N
u=2
n

−γ

1−φN /2 if u = 2


P (u) =
U



−γ
φN /2 if u =ψN, 0<ψ< 1Population subdivision with migration
Conservative migration between finite number D of subpopulations
gives convergence to the structured coalescent
if Nm is finite as N→∞

γ 2
N ≡ min N ,N , γ > 0
γ
N is the coalescence timescale;
γ
m N <∞ as N→∞
γ γ
m is rescaled migration;
γ
2
λ =I +φψ I , φ> 0, 0<ψ< 1
γ γ≥2 γ≤2
λ is the rate of coalescence of two lines
γTime to coalescence for two lines sampled from
same (T ) or different (T ) subpopulations
0 1
D D D−1
E(T )= < + =E(T )
0 1
λ λ N m
γ γ γ γ
2 γ 2
λ =I +φψ I , N ≡ min(N ,N )
γ γ≥2 γ≤2 γIndicators of population subdivision - F and N
ST ST
F defined in terms of probabilities of identity
ST
N defined in terms of average numbers of pairwise differences
ST
1 1
F = , N =
ST ST
2
N m D
N m D θ/2 D γ γ
γ γ
1+
1+ +
2
λ D−1
λ (D−1) λ D−1 γ
γ γ
In a Wright-Fisher population:
1 1
F = , N =
ST ST
2
D
D D
1+4Nm
1+4Nm +θ
2
D−1
(D−1) D−1

2 γ 2
λ =I +φψ I , N ≡ min N ,N
γ γ≥2 γ≤2 γSUMMARY
(i) multiple mergers coalescent processes may
better apply to some marine organisms
(ii) coalescent times are shorter than in the
standard coalescent
(iii) patterns indicating population subdivision can
be observed in DNA sequence data even if the
usual migration rate Nm is very, very large