By Mireya Diaz

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The Coalescent Theory

• By Mireya Diaz
• Department of Epidemiology and Biostatistics for
  EECS 458

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Agenda

• Basic concepts of population genetics
• The coalescent theory
• Coalescent process of two sequences
• Coalescent time
• Statistical inference
• Applications reconstruction of human
  evolutionary history
• Future venues

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Basic Concepts in Population Genetics
Mutation
Random genetic drift
Selection
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Basic Concepts in Population Genetics

• Mutation limited role in evolution due to its
  slow effect, however contributes to the
  maintenance of alleles in the population
• Locus with 2 allelles A1 (p(n)) and A2
  (q(n)1-p(n))
• Non-overlapping generations
• A1-gtA2 at rate u and A2-gtA1 at rate v (u, v
  10-5, 10-6)
• Allele can mutate most once/generation
• if initial gene freq. of A1p(0)

As n-gt8
equilibrium
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Basic Concepts in Population Genetics

• Random genetic drift change in gene frequency
  due to random sampling of gametes from a finite
  population. Important for small size populations
• Each generation 2N gametes sampled at random
  from parent generation
• y(n) gametes of type A1, in absence of
  mutation and selection

Wright-Fisher model

• One allele will be lost

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Basic Concepts in Population Genetics

• Selection can act at different stages of the
  life of an organism (e.g. differential fecundity,
  viability)
• Locus with 2 alleles A1, A2
• Three genotypes A1 A1 (w11), A1 A2 (w12), A2A2
  (w22)
• with fitness wij, relative survival chances of
  zygotes of genotype AiAj
• Under Hardy-Weinberg equilibrium

If w11gtw12gtw22 -gt A1 becomes fixed
w11ltw12ltw22 -gt A2 becomes fixed w11,w22ltw12
-gt overdominance, stable polymorphism w12lt
w11,w22 -gt underdominance, unstable
polymorphism, A1 or A2 becomes fixed
f(0)
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The Coalescent Theory

• Stochastic process continuous-time Markov
  process
• Large population approximation of Wright-Fisher
  model, and other neutral models
• Probability model for genealogical tree of random
  sample of n genes from large population
• Most significant progress in theoretical
  population genetics (past 2 decades). Cornerstone
  for rigorous statistical analysis of molecular
  data from populations
• Need of inferring the past from samples taken
  from present population
• Seminal work Kingman, J Appl Prob 19A27, 1982

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The Coalescent Theory Key Idea

• Start with a sample and trace backwards in time
  to identify EVENTS in the past since the Most
  Recent Common Ancestor (MRCA) in the sample
• Consider sample of n sequences of a DNA region
  for a population
• Assume no recombination between sequences
• N sequences are connected by a single
  phylogenetic tree (genealogy) where the rootMRCA

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The Coalescent Theory Usefulness

• Sample-based theory
• By-product development of highly-efficient
  algorithms for simulation of samples under
  various population genetics models
• Particularly suitable for molecular data
• Estimate parameters of evolutionary models (vs.
  history of specific locus phylogenetics)

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The Coalescent Process of Two Sequences

• Consider diploid organisms
• Wright-Fisher model
• Sequence in a population at a generation random
  sample with replacement from those in the
  previous generation
• Mutations at locus of interest selectively
  neutral (do not affect reproductive success, all
  individuals likely to reproduce, all lineages
  equally likely to coalesce)
• P(coalescence at previous generation)?
• P1/2N, Neffective population size

• For haploid structures, use N rather than 2N

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The Coalescent Tree
Genealogical relationship of sample of genes

• Topology is independent of branch lengths
• Branch lengths are independent, exponential rvs
  (waiting time between coalescent events)
• Topology is generated by randomly picking
  lineages to coalesce -gt all topologies are
  equally likely

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The Coalescent Time

• Assume mutations in a given period Poisson
• mean time 2N generation between two sequences
• mean mutations in two sequences
• ? 4Nm (m mutation rate seq/generations)
• Underlying assumption randomly mating
• ( organisms with high mobility)
• Coalescent time time between two successive
  coalescent events
• Exponential variable, mean 2/k(k-1)
• k ancestral sequences between the two events

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Coalescent Tree Parameters
Expected total branch length of the tree
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The Coalescent Theory Statistical Inference

• Mutation rate
• Age of MRCA
• Recombination rate
• Ancestral population size
• Migration rate

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Reconstruction of Human Evolutionary History

• Goal estimate times of evolutionary events
  (major migrations), demographic history
  (population bottlenecks, expansions)
• Haploid sequences mtDNA, Y chromosome
• Case study recent common ancestry of human Y
  chromosome
• Source Thomson et al. PNAS 2000 977360-5
• Estimations expected time to MRCA and ages of
  certain mutations
• Data 53-70 chromosomes, sequences variation at
  three genes (SMCY, DBY, DFFRY) in Y chromosome

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Recent common ancestry of Y chromosome

• For ages of major events need mutation rate
  estimate (SN substitution)
• Substitutions between chimpanzee and human
  sequences
• Mutation rate per site per year No.
  subst./2TsplitL
• Tsplit time since chimp and human split (5M
  years ago)
• Assumptions selective neutrality of all changes
  on Y since divergence

Summary of gene characteristics from sample
Source Table 1 from article () in no.
polymorphisms after removal of length variants,
repeat sequences, indels
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GENETREE Analysis

• Software www.stats.ox.ac.uk/stephens/group/softw
  are.html
• Estimate mean number of mutations ? 2Nem
• Ne effective number of Y chromosomes in
  population
• m mutation rate per gene per generation
• Also expected ages of mutation, time since MRCA
• Assumptions coalescent process,
  infinitely-many-sites mutation (mutation rate low
  enough -gt e/occurs at new site)
• Four insertions, three deletions, two repeat
  mutations (different rates from SN substitutions)
• Only one segregating site in SMCY appeared to
  have mutated gt1 -gt data fit infinitely-many sites
  model

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Recent common ancestry of Y chromosome
MRCA distribution under constant population
MRCA distribution under exponential population
growth
1Expected age in Ne generations. 2Value in years
Ne25
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GENETREE Analysis
Expected ages of mutations in tree Mutation 1
47,000 (35,000 89,000) male movement out of
Africa Mutation 2 40,000 (31,000 79,000)
beginning of global expansion
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Future Venues

• Population genetics models incorporation of
  migration, population growth, recombination,
  natural selection
• Longitudinal analysis
• Evolutionary analysis of quantitative trait loci
  (QTL)
• Properties of CT
• Accuracy of coalescent approximation under
  combinations of population size, sample size,
  mutation rate
• Properties of estimators under MCMC

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References

• Handbook of Statistical Genetics, 2nd edition,
  Vol.2
• Nature 2002 3380-390
• Theoretical Population Biology 1999 561-10.