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Lab 6: Genetic Drift and Effective Population Size Goals 1. To calculate the probability of fixation or loss of an allele. 2. To estimate mean time until fixation of an allele. 3. To estimate effective population size affected by past cataclysms. 4. To learn how genetic drift and selection interact in populations of various Ne. Probability of fixation or loss 1. Genetic drift results from chance changes in allele frequencies that result from sampling of gametes from generation to generation in a finite population. 2. Exact probability of fixation of an allele is equal to the initial frequency of that allele in absence of selection. u ( p) p 0 3. Probability of fixation of an allele can be calculated empirically by using Monte Carlo simulations as implemented in Populus. Problem 1 (15 minutes). The frequencies of alleles A1 and A2 are p = 0.75 and q = 0.25, respectively. Use Populus to empirically estimate the probabilities of fixation and loss for each of these alleles. What do you think are the exact probabilities of fixation and loss for each allele? Do these probabilities depend on the population size? Problem 2 (15 minutes). Consider a population with the following genotype counts: Genotype counts Case A1A1 A1A2 A2A2 1 14 3 3 2 0 3 17 4 N (1 p ) ln(1 p ) T ( p) . p a) Use Populus to empirically estimate the mean time (in number of generations) until fixation for allele A1 for each case. b) Show the mean time until fixation of A1 calculated using the diffusion approximation (given above) for each case. c) Discuss the reasons for the differences (if any) between the two types of estimates. What are some of the assumptions underlying each method? Mean time until fixation of an allele depends on population size and initial frequency of that allele. Kimura and Ohta(1971)diffusion approximation : 4 N (1 p) ln(1 p) T ( p) . p A1A1 A1A2 A2A2 (N11) (N12) (N22) N p q T(p) in terms T(p) of N 16 2 2 20 0.85 0.15 26.78 1.34N 0 2 18 20 0.05 0.95 77.96 3.90N Problem 3 (15 minutes). The census population size of an isolated population of finches on the Galapagos Islands is as follows. What is the effective population size in 2010? Year Females Males 1930 1950 1970 1990 2010 120 15 250 1500 3500 250 290 350 2500 5000 Dr. Robert Rothman, Rochester Institute of Technology http://people.rit.edu/rhrsbi/GalapagosPages/DarwinFinch.html When time is discontinuous, a transition matrix can be used to determine the probability of fixation in the next generation. xij P(Yt 1 (2 N )! i 2 N i p' q' i | Yt j ) (2 N i)!i! p (1 s) pq(1 hs) p' 2 1 2 pqhs p s 2 Fitness Fitness in terms of s and h (adaptive Darwinian selection) Fitness in terms of s and h (purifying selection) A1A1 ω11 Genotype A1A2 ω12 A2A2 ω22 1+s 1 + hs 1 1 1 − hs 1−s • Using traditional setup for adaptive Darwinian (positive) selection • These can be easily converted to terms for purifying selection Let s D selectioncoefficient under Darwinian selection Let s P selectioncoefficient under purifyingselection Let h D heterozygous effect under Darwinian selection Let h P heterozygous effect under purifyingselection sD sP 1 sD hp 1 hD Problem 4 (15 minutes). If adaptive Darwinian selection (characterized by h = 0.5 and s = 0.25) is operating on a locus and the frequency of allele A1 at that locus is p = 0.25, predict whether A1 is more likely to get lost or to become fixed: (i) In a population with Ne = 10. (ii) In a population with Ne = 30. a) For each of the two cases, calculate the probability of fixation of A1 empirically (i.e., using Populus). b) If Ne affects the probabilities of fixation and loss of A1, explain why. If not, explain why not. Problem 5. GRADUATE STUDENTS ONLY: Starting with the conditions in Problem 4-a), calculate the probability that: a) The frequency of A1 becomes 0.1 in the next generation. b) A1 becomes fixed in the next generation. c) If the two transition probabilities differ dramatically, explain why. If not, explain why not.