Download Computational Biology B Graphs and genes, genetical drift and

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.
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