Download Neutral Theory

Lecture 16: Introduction to Neutral
Theory
October 22, 2010
Last Time
Mutation-selection balance example
Effects of inbreeding
Effect of heterozygosity (h)
Mutation in finite populations
Infinite alleles mutation model
Today
Mutation-selection balance with
dominance
Infinite alleles versus stepwise
mutation models
Introduction to neutral theory
Molecular clock
Expectations for allele frequency
distributions under neutral theory
Mutation-Selection Balance with Dominance
 Dominance exposes alleles to selection, and therefore
acts to decrease equilibrium allele frequencies
qeq 

for h>>0
hs
 Complete Dominance of A2:
qeq 

s
 Recessive Case:
qeq 

s
Which qeq is larger?
Why?
Effect of dominance and selection on allele
frequency in mutation-selection balance (μ=10-5)
 Drastic effect of
dominance on
equilibrium
frequencies of
deleterious alleles
 Exposure to
selection in
heterozygotes
recessive case
Lab Exercise: Mutation and Selection
h=0, s=0.12 µ=1e-6, p=0.1
0.9
0.6
 Case with A1
dominant to A2 has
much more rapid
increase of A1 at low
frequency of A1
Frequency of A1
h=0.5, s=0.12 µ=1e-6, p=0.1
0.98
0.47
This was wrong in your handouts
Time
Dominant
Additive
30
0.6
0.47
90
0.9
0.98
pqq22  12   p11  12 
q 

Why Do A1 Alleles Increase More Rapidly Under
Dominance when p<0.5?
 Selection removes
A1 and A2 when
heterozygotes are
selected against
 In additive case, A1
alleles are penalized
by fitness effects
of A2 in
heterozygotes
 Balance of
heterozygotes and
homozygotes
determines relative
efficiency of
selection under
additivity
Equilibrium Allele Frequencies with Selection,
Mutation, and Drift
2N e
qeq  
s
when 2Ne μ<1 and
h=0
Compare to infinite
population expectation:
qeq 

s
 Degree of dominance removes population size from
the equation.
qeq 

hs
when
h  s
 Same as infinite population!
Equilibrium Heterozygosity under IAM
4Ne 
He 
4Ne   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
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  s2 q
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)
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
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
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
Autosomes
X
 Only 3 X chromosomes for
every 4 autosomes in the
population
 Ne of X expected to be 25% Infinite Alleles Model: Stepwise Model:

1
1
less than Ne of autosomes:   H e
 

1

1 He
2  (1  H e ) 2 
θX/θA=0.75
 Observed ratio was 0.8
for Infinite Alleles Model
and 0.71 for Stepwise
model
Molecular Clock
 If neutrality prevails, nucleotide divergence between two sequences
should be a function entirely of mutation rate
1
k  2 N
2N

 Time since divergence should therefore be the reciprocal of the
estimated mutation rate
Expected Time Until Substitution
t
1

Since μ is number of
substitutions per unit time
Molecular Clock
 If neutrality prevails, nucleotide divergence between two sequences
should be a function entirely of mutation rate
 Neutrality is not necessarily a
reasonable assumption for coding
sequences (genes)
 Different rates observed for
different genes
The main power of neutral theory is it provides a
theoretical expectation for genetic variation in
the absence of selection.
What is the expected distribution of allele
frequencies under the neutral model?
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
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:
  4Ne 

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
Expected number of different alleles (k) in a sample of 2N alleles is:
E (k ) 

 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




 i
1
Probability of sampling a new allele on the third and fourth samples:
2 N 1


 1   2   3
 He


 2
 3
Ewens Sampling Formula
E ( n) 
2 N 1

  i
i 0
 1



 ... 

 1   2
  2N 1
where E(n) is the expected number
of different alleles in a sample of N
diploid individuals, and  = 4Ne.
1
1
fe 

4Ne  1   1
 Predicts number of
different alleles that
should be observed in a
given sample size if
neutrality prevails under
Infinite Alleles Model
 Small θ, E(n) approaches 1
 Large θ, E(n) approaches
2N
 θ can be predicted from
number of observed
alleles for given sample
size
 Can also predict expected
homozygosity (fe) under
this model
Ewens-Watterson Test
 Compares expected homozygosity under the neutral
model to expected homozygosity under HardyWeinberg equilibrium using observed allele
frequencies
 Comparison of allele frequency distributions
 fe comes from infinite allele model simulations and
can be found in tables for given sample sizes and
observed allele numbers
f HW   p
2
i
Ewens-Watterson Test Example
 Drosophila pseudobscura
collected from winery
 Xanthine dehydrogenase
alleles
 15 alleles observed in 89
chromosomes
 fHW = 0.366
Hartl and Clark 2007
fe
 Generated fe by
simulation: mean 0.168
How would you interpret this result?
Expected Homozygosity fe
Most Loci Look Neutral According to
Ewens-Watterson Test
Hartl and Clark 2007