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Fundamentals of Population Genetics III

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Title: Fundamentals of Population Genetics III


1
Fundamentals of Population Genetics III
  • Bio 5488

2
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

3
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?

4
I 1000 U 0.0001 Start as Homozygous At
allele A
5
Mutation and Drift
  • 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?

6
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?

7
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.

8
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)

9
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)

10
I 1000 U 0.0001 Start as Homozygous At
allele A
11
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.

12
A clarification allele versus mutation for high
mutation rates
13
Summary
  • K u
  • Heq 4Nu/(14Nu)

14
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.

15
Substitutions as a function of time
From Gillespie p. 33
16
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)

17
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.

18
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?

19
Conclusions
  • Mutation selection balance
  • Fixation Rate is independent of population size.
  • Molecular clock
  • Neutral Hypothesis of Population Genetics (and
    nearly neutral hypothesis)
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