Observation: we continue to observe large amounts of genetic variation in natural populations

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1 MUTATION AND GENETIC VARIATION Observation: we continue to observe large amounts of genetic variation in natural populations Problem: How does this variation arise and how is it maintained. Here, we look at whether mutation alone can explain this variation.

2 RECURRENT MUTATION MODEL A simple model to look at the effect of mutation on gene frequencies - examine an ideal population in which all the HWE assumptions hold except that mutations occur µ A1 A2 ν µ = forward mutation rate/generation ν = reverse mutation rate/generation

3 µ ~ 10-5 to 10-8 ν ~ 10-6 to 10-9 mutations/ gene.

4 p 0 and q 0 - initial gene frequencies of the A1 and A2 alleles. under recurrent mutation model, each generation the frequency of A2 increases by p 0 µ and decreases by q 0 ν. After one generation q 1 = q 0 + p 0 µ - q 0 ν The change in gene frequency Δq = p 0 µ - q 0 ν

5 Each generation, the change in allele frequency is negligible over the long term, an equilibrium is achieved - loss of A2 alleles is balanced by the gain. At equilbrium, Δq = <p>µ - <q>ν = 0 where <p> and <q> are equilibrium allele frequencies.

6 Substituting <p> = 1 - <q> <q> = µ/(µ + ν) <p> = ν/(µ + ν) - equilibrium allele frequencies are independent of original gene frequencies - dependent only on mutation rates. -time to equilibrium, however, is dependent on initial allele frequencies

7 INFINTE ALLELES MODEL This mutation model is particularly useful in considering DNA sequences. Consider a gene 900 bps in length. -each nucleotide site can have one of four states - total number of possible alleles for this gene is or This is the basis of the IAM - each mutation occurs independent of site, and so an infinite number of alleles are possible.

8 In the IAM -two alleles that are identical by state must also be identical by descent -thus, homozygous genotypes must be autozygous -to measure homozygosity, we need to measure autozygosity. Consider the genotype A i A i. -two alleles are IBD if neither allele has mutated in the course of one generation. - mutation rate is µ, then Pr(IBD) = (1 µ)(1 µ) = (1 µ) 2

9 - remember that the inbreeding coefficient is the probability that a genotype is autozygous (and hence homozygous) -in an ideal population. F t = 1/2N + (1-1/2N)F t-1 -in a population under mutation pressure, probability that two alleles are IBD is dependent on mutation rate µ. Thus F t = (1/2N) (1 µ) 2 + (1-1/2N)(1 µ) 2 F t-1

10 Thus, every generation random genetic drift (and inbreeding) increases autozygosity but mutations decrease autozygosity. - equilibrium is reached such that <F> = 1/(1 + 4Nµ) and the equilbrium heterozygosity <H> <H> = 1 - <F> = 4Nµ /(1 + 4Nµ)

11 The parameter 4Nµ (or more commonly 4N e µ) occurs often in population genetics and is referred to as Θ.

12 maintenace of genetic variation: mutation-drift balance and neutral theory of evolution - proposed in the 1960s to explain large amount of genetic variation observed in natural populations. -neutral theory has the following assumptions: (1) primary selective force is purifying selection (2) most alleles observed are selectively neutral - one allele is not favored over another (3) variation we observe in populations arises from balance between mutation and random genetic drift

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14 The neutral theory (or mutation-drift balance) provides a powerful null hypothesis in evolutionary genetics.

15 FIXATION PROBABILITY - the probability for fixation of a neutral allele. - consider two alleles A1 and A2 -probability that allele A2 will be fixed is dependent on its initial frequency its selective advantage or disadvantage (s) the effective population size (N e )

16 Genotypes A1A1 A1A2 A2A2 relative fitness s 1 + 2s Under additive selection model, Kimura(1962) showed that probability of fixation of allele A2 over time is P = (1 - e -4Nsq )/(1 - e -4Ns ) where q is the initial frequency of A2.

17 Fixation Probability under Neutrality Note: since e -x ~ 1 -x when x 0 When Ns 0, then P = 4Nsq/4Ns P = q The condition Ns 0 is called neutral limit (when selection is weak), or s << 1/N. -probability of fixation is dependent only on initial frequency of the allele.

18 - population of size N composed solely of A1 alleles. - A2 allele arises through a mutational event. - population of size N (with 2N alleles) there is now 1 A2 allele. q = 1/2N P = (1 - e -4Ns/2N )(1 - e -4Ns ) Under neutrality, Ns 0 P = (4Ns/2N)/4Ns P = 1/2N

19 Fixation Probability under Selection - under selection. if q = 1/2N P = (1 - e -4Nsq )/(1 - e -4Ns ) P = (1 - e -2s )/(1 - e -4Ns ) If absolute value of s 0 and N is very large P = 2s Under selection, the fixation probability is dependent on magnitude of selection.

20 RATES OF FIXATION rate at which a new neutral mutation is fixed in population is equal to neutral mutation rate. K = rate of fixation of alleles K = µ = (rate of allele formation)(fixation probability) = (2N gametes)(µ mutation rate/gamete)(1/2n)

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22 Under selection, K = (2N)(µ)(2s) = 4Nµs Thus, under selection ( Ns > 1) the rate of gene substitution is greater than under neutrality. - under neutrality, since rate of gene substitution is equal to µ, the average time between consecutive fixations is 1/µ. -higher the mutation rate, the smaller time between fixations

23 TIME TO FIXATION How long does it take for a neutral mutation to be fixed? Kimura showed that the mean time to fixation under neutrality is Under selection <t> = 4Ne generations <t> = (2/s)ln(2Ne)

24 Thus, if N = 10 6 and generation time is 2 years -under neutrality <t> = 8 million years -with selected mutations, if s = 0.01 <t> = 5800 years

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26 HETEROZYGOSITY UNDER NEUTRALITY The heterozygosity H of neutral mutations is given under the infinite alleles model <H> = 4N e /(1 + 4N e µ) = Θ/(1 + Θ)