Math 4600, Homework 12
1. (Computing) Many biological compounds (DNA, proteins, etc.) are polymers - compounds made up of
repeating units. Under the right conditions, amino acids in a solution will spontaneously (without the aid
of an enzyme) bind and unbind to one another to form polymers. Spontaneous binding and unbinding
was probably a crucial step in the origin of life. Let i be the number of amino acids in such a polymer.
Over the course of each minute, i
• is a random variable that can take on the values 1 though 5
• decreases by 1 with probability δ (except when i = 1)
• increases by 1 with probability (1 − 5i )
a. Choose δ = 0.2. Write down a transition matrix P for this process.
b. Find and plot the equilibrium distribution of i. Don’t forget to normalize. Which lengths are most
prevalent at equilibrium?
c. Verify this equilibrium distribution by writing a stochastic simulation of this process. The ion channel
code has been posted.
d. What is the theoretical expected waiting time until i = 5, given that initially i = 1? Do this by
reinterpreting i = 5 as an absorbing state, then finding the time until it is reached.
2. (Computing) In the model for random genetic drift in a population of bacteria, consider the case when
each cell has room for 4 plasmids (N = 4). The original cells (at time 0) have half of their plasmids as
type P and half of the plasmids of type Q. Let pi [j] be the probability that a cell has j type P plasmids
at generation i, so p0 [2] = 1.
a. Write down element Pi,j of the transition matrix P describing this process. This should be an
(N + 1) × (N + 1) matrix
b. Find the 1st , 5th , and 50th iterates of this process using the transition matrix. Make bargraph plots
of these with the “barplot” function in R.
c. Find the theoretical equilibrial distributions using the equation v ∗ = P 0 v ∗ , where P 0 = transpose(P ).
You should find two v ∗ vectors that have eigenvalue 1; the equilibrium distribution is a normalized
sum of these.
1