19-11-20 Neutral theory 2 (2019 abridged): Neutral theory of - - PDF document

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Neutral theory 2 (2019 abridged): Neutral theory of molecular evolution

Jack King and Thomas Jukes: Independently arrived at same conclusion as Kimura Published (1969) under the provocative title “Non-Darwinian evolution” I cannot over emphasize how radical this idea was at that time. “Neutralist – Selectionist debate” Motoo Kimura:

• troubled by cost Haldane’s dilemma:
• 1 substitution every 300 generations
• troubled by Zukerkandl and Pauling’s (1965) molecular clock:
• 1 substitution every 2 years

Published a model of neutral evolution in 1968

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Neutral theory 1. Mutation 2. Polymorphism 3. Substitution

Neutral theory: connected these is a new (radical) way 1.0 allele frequency A C G T

k = rate of nucleotide substitution at a site per generation [year] k = new mutations × probability of fixation

v v neutral theory of molecular evolution (Kimura 1968)

the number of new mutations arising in a diploid population

v

2Nµ

the fixation probability of a new mutant by drift

12N

the substitution (fixation) rate, k

k = 2Nµ ×1 2N

k = µ

the elegant simplicity of neutral theory:

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Neutral theory of molecular evolution

Neutral theory: the rate of evolution is independent of effective population size

• mutation-drift equilibrium (steady-state between gain and loss)
• assumes (i) neutrality and (ii) constant mutation rate
• polymorphism is simply a phase of evolution (mutation, polymorphism and

substitution are not separate processes)

2sij 1− e

−4 Nsij

• prob. of fixation:

k = µ

neutral theory: Evolution by natural selection: Kimura 1957 & 1962

• rate depends on mutation rate and population size and intensity of selection
• generalization of Wright’s (1938) work

Remember the genetic drift lecture…

• rate to fixation [under drift] slows with increasing in Ne
• ultimate fate is fixation or loss
• Larger Ne yield larger residence time of a polymorphism in a population

If we run this simulation long enough it will go to fixation or loss; it just takes much longer

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Ne = 5000 Generations = 30 Generations = 500 Generations = 1000 Generations = 50

Allele frequency

1

t

Ne = small

1

Allele frequency

t

Ne = large

Time to fixation (t) of new alleles in populations with different effective sizes. Note that most new mutations are lost from the population due to drift and those mutations are NOT shown. The time to fixation (as an average) is longer in populations with large size.

A slice in time for each population is shown by a dotted vertical line ( ). Note that at such a slice in time the population with larger effective size is more polymorphic as compared with the smaller population.

The average time to fixation is 4Ne generations

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The average time between neutral substitutions is the reciprocal of k (µ)

Mean time between mutation ev ents (1/ µ 2Ne) is much shorter in the larger population because number

• f new mutations is on average =

µ × ´ 2Ne (for diploid organisms). The mutation rate ( µ ) is the same in both populations, but numbers differ because of differences in population size (2Ne). Allele frequency

1

N

e

= small

1

Allele frequency

N

e

= large

Mutation event

m ean 1/ µ µ 2Ne m ean 1/ µ 2 µ 2Ne

1 2 3 k=3 1 2 3 k=3

The population attains an equilibrium substitution rate (k = µ) In words: Large populations: high number of new mutants each generation (2Ne is high) but probability of fixation is low (1/ 2Ne) Small populations: lower number of new mutants each generation (2Ne is lower), but each has a higher probability of fixation (1/ 2Ne is larger)

k = µ

neutral theory:

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The average time between neutral substitutions is the reciprocal of k (µ)

Mean time between mutation ev ents (1/ µ 2Ne) is much shorter in the larger population because number

• f new mutations is on average =

µ × ´ 2Ne (for diploid organisms). The mutation rate ( µ ) is the same in both populations, but numbers differ because of differences in population size (2Ne). Allele frequency

1

N

e

= small

1

Allele frequency

N

e

= large

Mutation event

m ean 1/ µ µ 2Ne m ean 1/ µ 2 µ 2Ne

1 2 3 k=3 1 2 3 k=3

“molecular clock”

Important interpretation: Under neutral theory, the mean rate of evolution is constant over time. This is the so-called “molecular clock”. The mean waiting time until a substitution also constant over time. The actual “waiting time” is a random variable because mutation and drift are

• stochastic. (the clock is “sloppy”)

If evolution is in state i, then we wait an exponentially distributed amount of time with a constant mean rate:

waiting time = − 1 λ loge(u)

• 1. λ!=!rate!=!qii!
• 2. u!=!uniform(0,1)!random!number!

A

time

start observing the process here 1st event 2nd event 3rd event

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The average time between neutral substitutions is the reciprocal of k (µ)

Mean time between mutation ev ents (1/ µ 2Ne) is much shorter in the larger population because number

• f new mutations is on average =

µ × ´ 2Ne (for diploid organisms). The mutation rate ( µ ) is the same in both populations, but numbers differ because of differences in population size (2Ne). Allele frequency

1

N

e

= small

1

Allele frequency

N

e

= large

Mutation event

m ean 1/ µ µ 2Ne m ean 1/ µ 2 µ 2Ne

1 2 3 k=3 1 2 3 k=3

It looks like there is more variability within the population when Ne is large The population attains an equilibrium (steady-state) polymorphism

He = 4Neµ/(1+4Neµ) θ = 4Neµ

this result assumes an “infinite alleles model” population geneticists are obsessed with the µ parameter

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The population attains an equilibrium polymorphism

Expected equilibrium levels of heterozygosity at a locus as a function of the parameter θ. Heterozygosity will be higher in larger populations. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 Heterozygosity (H)

θ

H: Substantial “standing polymorphism” with NO LOAD1

The population attains an equilibrium polymorphism

Expected equilibrium levels of heterozygosity at a locus as a function of the parameter θ. Heterozygosity will be higher in larger populations. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 Heterozygosity (H)

θ

H: Steady-state heterozygosity that results when increase in alleles due to mutation is exactly balanced by their loss due to drift.

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The average time between neutral substitutions is the reciprocal of k (µ)

Mean time between mutation ev ents (1/ µ 2Ne) is much shorter in the larger population because number

• f new mutations is on average =

µ × ´ 2Ne (for diploid organisms). The mutation rate ( µ ) is the same in both populations, but numbers differ because of differences in population size (2Ne). Allele frequency

1

N

e

= small

1

Allele frequency

N

e

= large

Mutation event

m ean 1/ µ µ 2Ne m ean 1/ µ 2 µ 2Ne

1 2 3 k=3 1 2 3 k=3

we “see” how there could be a difference in steady state heterozygosity when Ne is large vs small

Neutral theory of molecular evolution

• 1. We can compute expectations for long-term

mutation + drift equilibrium

• 2. The standing level of polymorphism is

dependent on effective population size

• 3. The rate of evolution is independent of

effective population size (constant, on ave.)