Transcript Neutral Theory

Lecture 20: Introduction to Neutral
Theory
November 5, 2012
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Last Time
Mutation introduction
Mutation-reversion equilibrium
Mutation and selection
Mutation and drift
Today
Introduction to neutral theory
Molecular clock
Expectations for allele frequency
distributions under neutral theory
Classical-Balance
 Fisher focused on the dynamics of allelic forms of
genes, importance of selection in determining
variation: argued that selection would quickly
homogenize populations (Classical view)
 Wright focused more on processes of genetic drift
and gene flow, argued that diversity was likely to be
quite high (Balance view)
 Problem: no way to accurately assess level of
genetic variation in populations! Morphological traits
hide variation, or exaggerate it.
Molecular Markers
 Emergence of enzyme electrophoresis in mid 1960’s
revolutionized population genetics
 Revealed unexpectedly high levels of genetic
variation in natural populations
 Classical school was wrong: purifying selection does
not predominate
 Initially tried to explain with Balancing Selection
 Deleterious homozygotes create too much fitness
burden
i  1  s1 p  s2q
2
2
  i
m
for m loci
The rise of Neutral Theory
 Abundant genetic variation exists, but perhaps not
driven by balancing or diversifying selection:
selectionists find a new foe: Neutralists!
 Neutral Theory (1968): most genetic mutations are
neutral with respect to each other
 Deleterious mutations quickly eliminated
 Advantageous mutations extremely rare
 Most observed variation is selectively neutral
 Drift predominates when s<1/(2N)
Infinite Alleles Model (Crow and Kimura Model)
 Each mutation creates a completely new allele
 Alleles are lost by drift and gained by mutation: a
balance occurs
 Is this realistic?
 Average human protein contains about 300 amino acids
(900 nucleotides)
 Number of possible mutant forms of a gene:
n4
900
 7.14x10
542
If all mutations are equally probable, what is
the chance of getting same mutation twice?
Infinite Alleles Model (IAM: Crow and Kimura
Model)
 Homozygosity will be a function of mutation and
probability of fixation of new mutants
 1

1
2
ft  
 (1 
) f t 1 (1   )
2Ne
 2Ne

Probability of
Probability of
sampling same allele
sampling two alleles
twice
identical by descent
due to inbreeding in
ancestors
Probability neither
allele mutates
Expected Heterozygosity with Mutation-Drift
Equilibrium under IAM
 1

1
ft  
 (1 
) f t 1 (1   ) 2
2Ne
 2Ne

 At equilibrium ft = ft-1=feq
 Previous equation reduces to:
Ignoring μ2
1  2
f eq 
4 N e   1  2
Ignoring 2μ
1
f eq 
4Ne   1
 Remembering that H=1-f:
4Ne 
He 
4Ne   1
4Neμ is called the
population mutation rate,
also referred to as θ
Expected Heterozygosity with Mutation-Drift
Equilibrium under IAM
 At equilibrium:
1
1
fe 

4Ne   1   1
set 4Neμ = θ
 Remembering that H = 1-f:
He 

 1
Equilibrium Heterozygosity under IAM
4N em
q
He =
=
4N em +1 q +1
 Frequencies of individual
alleles are constantly
changing
 Balance between loss and
gain is maintained
 4Neμ>>1: mutation
predominates, new
mutants persist, H is
high
2
Fraser et al. 2004 PNAS 102: 1968
 4Neμ<<1: drift
dominates: new mutants
quickly eliminated, H is
low
Stepwise Mutation Model
 Do all loci conform to Infinite Alleles Model?
 Are mutations from one state to another equally
probable?
 Consider microsatellite loci: small insertions/deletions
more likely than large ones?
SMM:
1
He  1
(8 N e   1)
IAM:
4Ne 
He 
4Ne   1
Which should have higher produce He,the
Infinite Alleles Model, or the Stepwise
Mutation Model, given equal Ne and μ?
SMM:
1
He  1
(8 N e   1)
IAM:
4Ne 
He 
4Ne   1
Plug numbers into the equations to see how
they behave.
e.g, for Neμ = 1, He = 0.66 for SMM and 0.8 for
IAM
Expected Heterozygosity Under Neutrality
 Direct assessment of
neutral theory based on
expected heterozygosity
if neutrality
predominates (based on
a given mutation model)
 Allozymes show lower
heterozygosity than
expected under strict
neutrality
 Why?
He 

 1
Observed
Avise 2004
Neutral Expectations and Microsatellite Evolution
 Comparison of Neμ (Θ) for
216 microsatellites on
human X chromosome
versus 5048 autosomal loci
 Only 3 X chromosomes for
every 4 autosomes in the
population
Why is Θ higher for
 Ne of Xautosomes?
expected to be 25%
less than Ne of autosomes:
θX/θA=0.75
 Observed ratio of ΘX/ΘA
was 0.8 for Infinite
Alleles Model and 0.71 for
Stepwise model
Autosomes
X
X chromosome
Correct model for
microsatellite evolution
is a combination of
IAM and Stepwise
Sequence Evolution
 DNA or protein sequences in different taxa trace
back to a common ancestral sequence
 Divergence of neutral loci is a function of the
combination of mutation and fixation by genetic
drift
 Sequence differences are an index of time since
divergence
Molecular Clock
 If neutrality prevails, nucleotide divergence between two sequences
should be a function entirely of mutation rate
1
k = 2N m
=m
2N
Probability of
creation of new
alleles
Probability of
fixation of new
alleles
 Time since divergence should therefore be the reciprocal of the
estimated mutation rate
Expected Time Until Fixation of a New Mutation:
t
1

Since μ is number of
substitutions per unit time
Variation in Molecular Clock
 If neutrality prevails, nucleotide divergence between two sequences
should be a function entirely of mutation rate
 So why are rates of substitution so different for different classes
of genes?
The main power of neutral theory is it provides a
theoretical expectation for genetic variation in
the absence of selection.
Fate of Alleles in Mutation-Drift Balance
Generations from
birth to fixation
Time between
fixation events
 Time to fixation of a new mutation is much longer than
time to loss
Fate of Alleles in Mutation-Drift-Selection
Balance
Purifying Selection
Which case will have the
most alleles
Highest
on H
average
at
E?
What will this depend
upon?
any given time?
Neutrality
Balancing
Selection/Overdomina
nce
Assume you take a sample of 100 alleles from a
large (but finite) population in mutation-drift
equilibrium.
What is the expected distribution of allele
frequencies in your sample under neutrality and
the Infinite Alleles Model?
Number of Alleles
A.
B.
C.
10
8
6
4
2
2
4
6
8
10
2
4
6
8
10
Number of Observations of Allele
2
4
6
8
10
Allele Frequency Distributions
Black: Predicted from Neutral
Theory
White: Observed (hypothetical)
 Neutral theory allows a
prediction of frequency
distribution of alleles
through process of birth
and demise of alleles
through time
 Comparison of observed to
expected distribution
provides evidence of
departure from Infinite
Alleles model
Hartl and Clark 2007
 Depends on f, effective
population size, and
mutation rate
Ewens Sampling Formula
Population mutation rate: index of variability of population:
  4 Ne 


 i
Probability the i-th sampled allele is new given i alleles already sampled:
Probability of sampling a new allele on the first sample: 

 0
Probability of observing a new allele after sampling one allele:
.

1

 1
Probability of sampling a new allele on the third and fourth samples:
Expected number of different alleles (k) in a sample of 2N alleles is:
E (k ) 
2 N 1

  i
 1
i 0



 1   2
 ... 

  2N 1
Example: Expected number of alleles in a sample of 4:
E (k ) 
2 N 1

  i
i 0
3

i 0

 i
1





 1   2   3
 He


 2
 3