Lecture 20: Introduction to Neutral TheoryNovember 5, 2012

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Mutation introductionMutation-reversion equilibriumMutation and selectionMutation 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

22

211 qspsi

mi 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:

542900 1014.74 xn 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

21 )1()

211(

21

t

eet f

NNf

Probability of sampling same

allele twice Probability of sampling two

alleles identical by descent due to

inbreeding in ancestors

Probability neither allele

mutates

Expected Heterozygosity with Mutation-Drift Equilibrium under IAM

At equilibrium ft = ft-1=feq

Previous equation reduces to:

21421

e

eq Nf

Ignoring μ2

144

e

ee N

NH

Remembering that H=1-f:

4Neμ is called the population mutation

rate, also referred to as θ

21 )1()

211(

21

t

eet f

NNf

141

eeq Nf

Ignoring 2μ

Expected Heterozygosity with Mutation-Drift Equilibrium under IAM

At equilibrium:

11

141

ee Nf

1

eH

Remembering that H = 1-f:

set 4Neμ = θ

Equilibrium Heterozygosity under IAM Frequencies of

individual alleles are constantly changing

Balance between loss and gain is maintained

4Neμ>>1: mutation predominates, new mutants persist, H is high

4Neμ<<1: drift dominates: new mutants quickly eliminated, H is lowFraser et al. 2004 PNAS 102: 1968

2

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?

144

e

ee N

NH

IAM:

)18(11

ee N

H

SMM:

Which should have higher produce He,the Infinite Alleles Model, or the

Stepwise Mutation Model, given equal Ne and μ?

144

e

ee N

NH

IAM:

)18(11

ee N

H

SMM:

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?Avise 2004

Observed

1

eH

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

Ne of X expected to be 25% less than Ne of autosomes:

θX/θA=0.75

AutosomesX

X chromosome

Correct model for microsatellite evolution is a

combination of IAM and Stepwise

Why is Θ higher for autosomes?

Observed ratio of ΘX/ΘA was 0.8 for Infinite Alleles Model and 0.71 for Stepwise model

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

t

Expected Time Until Fixation of a New Mutation:Since μ is number of substitutions per unit

time

Time since divergence should therefore be the reciprocal of the estimated mutation rate

Probability of creation of new alleles

Probability of fixation of new alleles

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

Time to fixation of a new mutation is much longer than time to loss

Generations from birth to

fixation

Time between fixation events

Fate of Alleles in Mutation-Drift-Selection Balance

Purifying Selection

Neutrality

Balancing Selection/Overdominance

Which case will have the most alleles on average

at any given time?What will this depend

upon?Highest HE?

Assume you take a sample of 100 alleles from a large (but finite) population in

mutation-drift equilibrium.

A.

Number of Observations of Allele

Num

ber o

f Alle

les

2

4

6

8

10

2 4 6 8 10

B.

2 4 6 8 10

C.

2 4 6 8 10

What is the expected distribution of allele frequencies in your sample under

neutrality and the Infinite Alleles Model?

Allele Frequency Distributions 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

Depends on f, effective population size, and mutation rate

Hartl and Clark 2007

Black: Predicted from Neutral Theory

White: Observed (hypothetical)

Ewens Sampling Formula

i

10

2

3

12

0

)(N

i ikE

3211)(

3

0

12

0

i

N

i iikE

.

Probability the i-th sampled allele is new given i alleles already sampled:

Probability of sampling a new allele on the first sample:

eH

1

Probability of observing a new allele after sampling one allele:

Probability of sampling a new allele on the third and fourth samples:

12...

211

N

Expected number of different alleles (k) in a sample of 2N alleles is:

Example: Expected number of alleles in a sample of 4:

eN4Population mutation rate: index of variability of population: