Genome Evolution 2012, Exercise 2
Q1: Fixation under selection: (20 points)
Assume the Moran model with selection as formulated in class, with a population of
size 1000, of which a single individual has a beneficial allele (fitness=1+s, s>0).
a) Approximate the probability that at some point 3 individuals will have the
beneficial allele.
b) What is the fixation probability of a weakly beneficial allele assuming it is
shared among 3 individuals of this population?
c) Compute and plot (as a function of s) the ratio between the results of a) and
b) and discuss the ratio shortly with respect the overall (Kimura) probability
of fixation of a beneficial mutation.
Q2: Allele frequency distributions (20 points): In this question we will examine
whether natural selection affect the allele frequency distribution of polymorphic sites.
Simulate the Wright-Fisher model (similarly to exercise 1 Q1) for a haploid
population of size n=200 and one locus. Let it evolve for 1000 generations or until
fixation, starting from a homogenous population immediately after a mutation
occurred in one individual (na0=999 nA0=1). Run it for 10000 times, with each of the
following selection coefficients (s=-0.010, s=-0.005, s=0, s=0.005, s=0.010, 0.015 and
𝑝(0<n ≤50)
s=0.020). Plot your estimation of the frequency of rare alleles 𝑝(0<n A<200) as a
A
function of s. Discuss briefly the relation between this frequency and selection.
Q3: Coalescence given recurrent sweeps (20 points)
A population of 1000 individuals evolve asexually (according to our standard
assumptions - e.g. WF) and adapt to a new environment. Initially the fitness of all
individuals is 1. Mutations increase fitness by factors of 1.05 (ignore all other types
of mutations). Population size is fixed. What is the expected time to the MRCA in this
population? First argue theoretically based on results we covered, then demonstrate
it in a simulation (or do it the other way around, but find some theoretical
justification to what you find).
Q4: Single genome approximation (40 points)
Again, haploid population of 1000 individuals evolves asexually.
a) Each individual carry one of the alleles (A, a), with different selection
coefficients (sA=-0.05, sa=0.05, or fitness is 0.95 and 1.05). Assume the
mutation rate μ=10-6. Assume that under these conditions a single genome
approximation to the evolutionary process is valid so that this problem can
be modeled as continuous time Markov process on two states. Simulate
evolution by sampling random walks of length 10,000 in this continuous time
markov model and calculate the average fitness (average over time).
Hint:
Transition rate= μ*Kimura fixation probability
Time to transition ~Exp(transition rate)
b) Next assume selection work on a “quantitative trait”. Do this by generalizing
your simulation and assume 101 possible alleles A0, A1, … ,A100 with selection
coefficients sAi:=-0.05+0.1*(i/100) (for 0≤ i ≤100) . Calculate the average
fitness, when mutation rate is 10-5 or 10-6
c) Next assume evolution is manipulating two linked loci, one of them
determining the fitness as in (b), and the other (called the mutator locus)
determining the mutation rate but not affecting fitness directly. Assume the
rate of mutation between the two possible states of the mutator locus
(denoted M and m) is μ (m->M) = μ (m->M) := 10-10 and that the mutation
rate of locus A is 10-9 if the mutator locus is M and 10-8 if the mutator locus is
m.
Calculate the average fitness given these assumptions, and also estimate the
probability distribution of the mutator locus (i.e. what is Pr(m)). You can
continue to assume the single genome approximation and use the following
Markov model:
d) Now model a fluctuating environment as follows. Introduce catastrophes that
flip the fitness landscape completely by inversing the fitness
(f’=1/f, or sAi’:= -1*sAi for each i). For different rates of catastrophe (φ=10-10,
φ=10-9, φ=10-8) calculate the average fitness and estimate p(m).