Evolutionary-Dynamics Homework 2 Solved

Exercises marked with a ” ” are programming exercises. These can be solved in a programming lan- guage of your choice. Please make sure to hand in your code along with your answers to these exercises.

Problem 1: Sequence space and Hamming distance

Consider an alphabet A of size |A| = B. For a binary alphabet, one has A = {0, 1} and B = 2, and for DNA, one has A = {A,T,C,G} and B = 4. We are studying sequences S ∈ AL of length L. Assume sequences are random with a uniform distribution,

1. (a)  How many unique binary and DNA sequences exists for L = 28? (1 point)
2. (b)  What is the average Hamming distance between two random binary sequences? What is the

   expected Hamming distance for two random DNA sequences? (1 point)

3. (c)  GivenabinarysequenceoflengthL,howmanysequencesexistataHammingdistancethreefrom

   it? How many at distance K with K ≤ L? Repeat the calculation for DNA sequences. (2 points)

Problem 2: Quasispecies

Consider the quasispecies equation with two genotypes 0,1 (i.e., binary sequences of length 1). Let the fitness of genotype 0 be f0 > 1, and the fitness of genotype 1 be f1 = 1. Moreover, genotypes are replicated error-free with probability q,

1. (a)  Write down the mutation-selection matrix W and find its eigenvalues.
2. (b)  To which eigenvalue corresponds the non-trivial equilibrium point?

   Hint: Perron-Frobenius theorem.

3. (c)  Examine the dynamics of the quasispecies equation and confirm the results obtained

   in (b). Assume that q = 0.6 and f0 = 1.5, and initial condition (0.65, 0.35).

4. (d)  What is the equilibrium point for f0 = f1 = 1?
5. (e)  Calculate the equilibrium point in the limit of low mutation rate (q ≈ 1).

(2 points) (1 point)

(1 point) (1 point) (1 point)

1