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. 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 µ ~ 10-5 to 10-8 ν ~ 10-6 to 10-9 mutations/ gene. p0 and q0 - initial gene frequencies of the A1 and A2 alleles. under recurrent mutation model, each generation the frequency of A2 increases by p0µ and decreases by q0ν. After one generation q1 = q0 + p0µ - q0ν The change in gene frequency Δq = p0µ - q0ν 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. 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 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 4900 or 10542. This is the basis of the IAM - each mutation occurs independent of site, and so an infinite number of alleles are possible. 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 AiAi. -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 - remember that the inbreeding coefficient is the probability that a genotype is autozygous (and hence homozygous) -in an ideal population. Ft = 1/2N + (1 - 1/2N)Ft-1 -in a population under mutation pressure, probability that two alleles are IBD is dependent on mutation rate µ. Thus 2 2 Ft = (1/2N) (1−µ) + (1 - 1/2N)(1−µ) Ft-1 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) The parameter 4Nµ (or more commonly 4Neµ) occurs often in population genetics and is referred to as Θ. 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 The neutral theory (or mutation-drift balance) provides a powerful null hypothesis in evolutionary genetics. 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 (Ne) Genotypes A1A1 A1A2 A2A2 relative fitness 1 1 + 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. 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. - 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 Fixation Probability under Selection - under selection. P = (1 - e-4Nsq)/(1 - e-4Ns) if q = 1/2N 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. 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 = (rate of allele formation)(fixation probability) = (2N gametes)(µ mutation rate/gamete)(1/2N) K = µ 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 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 <t> = 4Ne generations Under selection <t> = (2/s)ln(2Ne) Thus, if N = 106 and generation time is 2 years -under neutrality <t> = 8 million years -with selected mutations, if s = 0.01 <t> = 5800 years HETEROZYGOSITY UNDER NEUTRALITY The heterozygosity H of neutral mutations is given under the infinite alleles model <H> = 4Ne/(1 + 4Neµ) = Θ/(1 + Θ).