Math 4600, Homework 10
1. Probablity of mutation of a certain gene in one generation is 0.2%. Probablity of a correction of such mutation is 0.1%. We are watching 1000 mutationfree organisms each producing one offspring.
a) What is the average number of mutations among the offsprings?
b) What is the probability that 3 or more of the offsprings will have the mutation?
c) (computing) Find the distribution of the number of mutations after 1 generation, plot (numerically) the distribution function
d) (computing) What will be the probability to have a mutation after 2 generations? in the long run? find corresponding distributions of the number of
mutations. Plot the disribution functions.
2. HIV RNA consists of 104 bases. Suppose that new data give an estimate
of about 106 of new fully functional viral particles produced per day, and that
probability of an error during replication is 2 ∗ 10−5 per base per replication
cycle.
a) What is the total expected number of errors per replication cycle?
b) Find probability to get a mutant with mutation in exactly 1 base (use appropriate approximation as needed). What number of these mutants do you expect
to find among the population of viruses produced on a given day?
c) Number of all possible 1-base mutants is 3 ∗ 104 . Let’s say that one of these
mutations leads to antibiotic resistance. Assuming that all viruses produced on
a given day are different (or as diverse as they can be), what is the probability
that antibiotic resistance will appear on a given day?
d) Repeat b-c with two-base-mutations. Number of all possible two base mutants is approximately 4.5 ∗ 108 .
e) What are your conclusions on development of antibiotic resistance with these
data?
3. Consider Gompertz equation
dN
= −bN ln(N/K).
dt
a) Solve this equation with N (0) = N0 using the substitution u = ln(N/K), to
obtain the solution N (t) = K exp(−Ae−bt ), where A = − ln(N0 /K)
b) (computing) Plot the solution as a function of time for K = 1, N0 = 0.1,
b = 1. Illustrate with a plot and describe in words what happens to the tumor
dynamics if the growth rate b is varied. Compare with the logistic dynamics
Ṅ = −bN (N − K)
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