topological concordance of gene trees and …nakhleh/comp571/presentations/jatin.pdf• horizontal...
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Topological Concordance of Gene
Trees and Species TreeTrees and Species Tree
jatin narula
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Gene Trees and Species Tree
Lineage
• Gene Loss and
Duplication
• Horizontal Gene • Horizontal Gene
Transfer and
Recombination
• Stochastic Factors
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Topological Concordance
Species Tree
Gene Tree
Topologically Concordant
Takahata Congruent
Takahata Congruence => Topological Concordance
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Topological Concordance for Multiple
Lineages
Collapsed gene tree is both
topologically concordant and
Takahata congruent
Topologically Concordant Neither
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Monophyletically Concordant
Monophyletically ConcordantAll species are Monophyletic but gene
tree is not topologically concordant
Monophyletic + Topologically Concordant = Monophyletically Concordant
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Speciodendric GenesOrthology : Genes whose homology was the result of speciation and subsequent
descent, with no duplication
Speciodendric
Orthologous
Speciodendricity : Gene Tree constructed from all copies of the gene in all species is
topologically concordant
Speciodendric
XenologousParalogous
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The Problem
‘‘conditioned on the species tree topology and
assuming no gene exchange between species, what
is the probability that a tree of orthologous genes is is the probability that a tree of orthologous genes is
topologically concordant with a species tree?’’
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Takahata Concordance Probability
P(r lineages derive from m lineages at time T3)*P(s lineages derive from n lineages at time T3 )
P(Takahata Congruence) =
*P(at least one interspecific coalescence occurs during this process, and that the most
recent interspecific coalescence joins a lineage from species A and a lineage from
species B)
*P(m+n lineages at T3 derive from k lineages at time T3+T2)
grm(T3)*gsn(T3)*gm+n,k(T2) *FkA,B(m,n,0)∑ ∑ ∑
m s m+n
m=1 n=1 k=1
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Topological Concordance Probability
P(Topological Concordance) =
P(Takahata Concordance) + P(no interspecific coalescences happen in two species
phase)*P(most recent interspecific coalescence happens in the one species phase)*P(this
coalescence joins the ancestral lineages of A and B)
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Key Determinants of Topological Concordance :
T2
Small T2
Large T2
Concordance Highly Likely
Trifurcation
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Key Determinants of Topological Concordance :
T3
Small T3
Large T3
Concordance Unlikely
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Key Determinants of Topological Concordance :
Sample Sizes
Large T3Large T3
Small T3
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Probability of Speciodendricity
P(Speciodendricity) = P(Topological Concordance|Sample sizes = no. of copies of gene
in respective species)
≈ P(Topological Concordance|Sample sizes = ∞)
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Maximal Useful Sample Sizes
T3
Humans and Chimpanzees : Humans and Chimpanzees :
1.6 – 93.3
Humans and Neanderthals :
0.5 – 10
Modern Human Groups :
~0.05
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Estimation of Parameters T2, T3 and N
Known Species Tree Topology
Choose multiple independent loci
Construct Gene trees assuming
values of T , Tvalues of T2, T3
Estimate Likelihood of
Parameter Values
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Extension to Four or More SpeciesBalanced Unbalanced
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Summary
• Likelihood functions for observed gene tree conditioned on proposed species tree
• Inference of most likely Species History
• Estimation of optimal sample sizes that maximize concordance probabilityconcordance probability
• Estimation of divergence times and ancestral population sizes
• Identification of Speciation Genes
• Assumes equality and stability of population sizes
• Ignores gene exchange, mistaken orthology and other stochastic effects that cause discordance
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References
• Rosenberg N A. Theoretical Population Biology 61, 225-247
(2002).
• Takahata N. and Nei M. Genetics 110, 325–344 (1985).
• Hudson R. R. Evolution 37, 203–217 (1983).
• Takahata N. Genetics 122, 957–966 (1989).• Takahata N. Genetics 122, 957–966 (1989).
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gij(T)
probability that two lineages coalesce in the immediately preceding generation =
probability that they share a parent = 1/N
PC(t generations) = (1-1/N)t-1(1/N) ≈ exp(-t/N)/N (for large N)
Let T = t/N
Probability that p lineages coalesce to p-1 at T
fp-1(T) = p*(p-1)*exp(-T)/2
Tp-1 is the waiting time for p lineages to coalesce to p-1 with distribution fp-1
Define Sij = ∑p=jp=i-1TpDefine Sij = ∑p=j Tp
Then P(i lineages converge to j in T) = P(Sij = T) = gij(T)
(Hudson 1983, Takahata and Nei 1985)
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FkA,B(m,n,l)
probability that during the coalescence of m+n+l lineages from A, B and C, at least one
interspecific coalescence occurs during this process, and that the most recent
interspecific coalescence joins a lineage from species A and a lineage from species B.
FkA, B(a, b, c)=0 if a+b+c ≤ k.
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Probability of Monophyletic Concordance