Fundamentals of Population Genetics II

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Fundamentals of Population Genetics II

• Bio 5488

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A good referencePopulation Genetics A concise
guide. John Gillespie
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Review of last time

• Wright Fisher Model
• Decay of Ht. Ht H0(1-1/2N)t
• Simple mathematical constructs can provide
  non-obvious insights
• Power of an analytic attack

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Department of Corrections

• Probability of fixation of Superman
• How does this change with population size.

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I 50 A 100 R 99 G 1 fG 3fR
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I 50 A 100 R 99 G 1 fG 3fR
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I 1000 A 2000 R 1999 G 1 fG 3fR
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An interesting point

• You are given two alleles, R and G. If the
  fraction of total alleles that are R is p in
  generation 0, what is the expected value of p,
  the fraction of total alleles in generation 1?

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We can ask two questions?

• Given a population size N (2N alleles) and a
  mutation rate u, what is the heterozygosity of
  the population?
• Given a population size N (2N alleles) and a
  mutation rate u, what is the rate at which new
  mutations (not allele) are fixed by chance?

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First some finer points about our model with
mutation Wright Fisher v0.3.

• Gillespie p 27-34, but especially p. 31
• What is a locus?
• Large (infinite) string of nucleotides
• Each evolves independently (free recombination)
• Mutation versus allele?

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Postulate that an equilibrium exists

• Ht1 Ht 0
• Our approach is to use the work we did on Monday
  to derive an expression for Ht1 Ht and set
  this to zero. Then we solve this equation.
• We start with the equation we derived last
  lecture
• Ht1 Ht (1-1/2N) or
• Ht1 Ht - Ht1/2N
• Now we include the effects of mutation.
• Ht1 Ht - Ht1/2N (1- Ht)1-(1-u)2
• Use a well known mathematical approximation
  (1-u)2 1-2u for ultlt1.

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The derivation continues.

• Ht1 Ht - Ht1/2N (1- Ht)1-(1-2u)
• Ht1 Ht (1- Ht)2u - Ht1/2N
• Ht1 - Ht (1- Ht)2u - Ht1/2N
• We did it. Now set to zero
• 0 (1- Heq)2u - Heq1/2N
• Heq/2N -2u Heq 2u
• Heq(1 4Nu) 4Nu
• Heq 4Nu/(14Nu)

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Lets analyze this

• Heq 4Nu/(14Nu)
• Let Nu be big compared to 1. Then the population
  is almost always heterozygous.
• (mutations occur before drift can remove)
• Let Nu be much less than 1. Then the population
  has little variation. (drift removes variation
  before a new mutation occurs)

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I 1000 U 0.0001 Start as Homozygous At
allele A
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What about our second question?

• K Rate of fixation or substitution of a
  mutation?
• How many mutations enter a population per
  generation? Simple 2Nu.
• What is the probability of a mutation fixing?
  Simple 1/2N.
• What is fixation rate k? K u!!!!!!! No
  dependance on N. This is very important.

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A clarification allele versus mutation for high
mutation rates
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Summary

• K u
• Heq 4Nu/(14Nu)

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We could now do some biology, but first.

• A nagging problem.
• What if the population size changes?
• Consider Heterozygosity Ht1 Ht (1-1/2Nt)

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See board
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How are we doing?
From Gillespie, p36.
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Harmonic Mean versus Arithmetic Mean

• 11,21,1000,21,4000,45,6000,12
• Arithmetic Mean (1121100021400045600012)/8
  1389
• Harmonic Mean 27
• Harmonic Mean is smaller (small values have more
  important effects!)

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On to some biology Kimura and Ohta. A classic
paper.

• Lets set the stage. Predominant view
  mutations come in two flavors. Rare deleterious
  mutations and very rare advantageous mutations.
• The concept of a molecular clock (depends on
  neutrality)
• Heterozygosity of proteins
• Measured by electrophoresis
• Different vertebrates different mutation rates,
  but it corresponds to evolutionary distance.

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Substitutions as a function of time
From Gillespie p. 33
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Estimate mutation rate. Measure Heterozygosity
for mouse. Does population size make sense?

• Heq 4Nu/(14Nu)
• u 10-7
• H 0.1
• Therefore Neff 3e5 (algebra on board)

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What about time versus generation?

• Kimura and Ohta ignore at first.
• But eventually leads to nearly neutral
  hypothesis.
• Non-coding regions follow expected relationship.
  Our molecular clock.

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Some problems we can now answer

• A new species of monkey.
• How does this fit in to our evolutionary picture?
  What can we infer about its population history?
• How many genomes do we need to sequence?
• How many mice are in the Warren Alpert Building
  at HMS?

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Conclusions

• Mutation selection balance
• Fixation Rate is independent of population size.
• Molecular clock
• Neutral Hypothesis of Population Genetics (and
  nearly neutral hypothesis)