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Computational Biology B Graphs and genes, genetical drift and diffusion Examples sheet 2 1. Linkage equilibrium in an infinitely large population. Consider two loci (A and B) in a diploid population. Assume that there are two alleles at each locus A1 and A2 as well as B1 and B2 . Assume that recombination occurs between the loci at rate r per pair of individuals per generation. Assume random mating and Mendelian inheritance. In the limit of an infinitely large population, derive the deterministic recursions for the haplotype frequences fAi Bj . Derive the recursion for D ≡ fA1 B1 − pq where p and q are the allelic frequencies of A1 and B1 respectively. Show that D tends to zero (linkage equilibrium) as time tends to infinity. Discuss your result. What do you expect happens in a large but finite population? 2. Fisher-Wright dynamics without mutation. Consider a population of size N . Suppose that at a given locus A there are two alleles A1 and A2 and assume that a new generation is obtained from the previous one by random sampling with replacement. Implement the algorithm and answer the following questions: 1. What is the probability that fixation of, say, A1 is obtained (depends on initial conditions)? 2. How long does it take, on average, to reach fixation? 3. Time to the MRCA Consider samples of size n N of individuals of the population you simulate. Plot the excepted time to the most recent common ancestor (MRCA) for the sample versus sample size n. 4. Fisher-Wright dynamics with mutation. Allow for mutation in the above computer program. Suppose that mutations A1 → A2 and reverse happen with the same rate 0 < u 1. 1. Calculate the average population heterozygosity as a function of θ ≡ 2N u. 2. Determine the distribution of the population heterozygosity in the stationary state. 1