Coalescent

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Coalescent

• Introduction

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Classical Population Genetics

• How do evolutionary forces affect future change
  in allele and genotyping frequencies.
• Mutation
• Natural selection
• Random genetic drift
• Migration and population growth

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Coalescent Theory

• What past evolutionary forces have acted to give
  us the scope and distribution of genetic
  variation we see today?
• The major difference
• Classical population genetics focuses on
  properties of the entire population
• While coalescent theory on those of a sample from
  the population

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divergence
coalescent
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Terminology

• Gene Genealogy , Genealogy
• The phylogeny of a sample from a population.
• The Most Recent Common Ancestor (MRCA)
• The root of the genealogy
• Coalescent event
• When two alleles coalesce, it is called a
  coalescent event.
• The Coalescent time
• The number of generations between successive
  coalescent events
• The Infinite Allele Model
• Each mutation create an allele that has never be
  present in the population.

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The Wright-Fisher model The genes of next
generation are random sample with replacement
from the gene pool of current generation.
Current Generation
P1/(2N)
Next Generation
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The coalescence of two sequence
t-1 generation (2N sequence)
P1/(2N)
P1-1/(2N)
t generation (2N sequence)
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The probability that the two sequence came from a
single ancestral sequence sequence t2t1
generation ago is
(1-p2)(1-p2)p2(1-p2)tp2
(1- )t
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Continuous approximation Advantage Make the
mathematics simple and simulation faster. Note
that e-x1-x when x is small. Therefore the
distribution of t2 can be approximation by a
exponential distribution.
-
Properties
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Generalization i-coalescent time The
distribution of ti
Properties
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The expected total of a sample genealogy
is where
i
i
i
e
where
e
e
e
e
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Mutations in a genealogy µ the mutation rate per
sequence per generation, The probability that
number ?of mutations in branch of length ?is
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The number of segregating sites Let K be the
total number of mutation that occur in a
genealogy. Then given that the total tree length
is T, K is a Poisson variable with mean equal to
Tµ .That is
Consider all possible values for T, we have
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The age of the most recent common ancestor
Since
We have
Note when simple size (n) is large, E(T) is
approximately equal to 4N.
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Major population genetics models that have been
studied

• The neutral Wright-Fisher model
• Recombination
• Selection
• Population subdivision and Migration
• Population growth
• Partially selfing

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Traditional summary statistics

• Heterozygosity
• The probability that two randomly selected
  alleles are different.
• Number of alleles
• The number of distinct alleles in a sample.
• Proportion of polymorphic loci
• The proportion of loci that are polymorphic.
  Traditionally a locus is said to be polymorphic
  if the most frequent allele is less than , say
  90 in the sample.

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• Measure more suitable for DNA polymorphism
• The number of segregating site K
• The number of nucleotide sites that are variable.
• The mean number of nucleotide differences ?
• The average number of nucleotide differences
  between a pair of sequences is intuitive and has
  interesting statistical properties
• Frequency spectrum
• Mutations in a genealogy can be partitioned into
  different categories.

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The mean nucleotide difference between two
sequences
Where dij is the number of differences between
sequence i and j.
Seq1 AAGCTTTCC Seq2 AAGCATTCC Seq3 AACCATTCC
d121 d132 d231
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The mean value of dij is the same as the expected
number of segregating sites in a sample of two
sequences.So
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Estimating ?
The quantity ?4Nµ is the most important
parameter for the evolution of a DNA region in a
population. Wattersons estimator is defined as
Tajimas esimator is difined as