Problem 2. (11 P) Denote the frequency of the focal genotype at generation t by pt and the frequency of the reference genotype at generation t by qt = 1 — pt. Also denote the ratio of these frequencies at generation t by t = Pt/qt. (Note that the focal genotype is in the numerator and the reference genotype is in the denominator.) Suppose that the selection coefficient of the focal genotype relative to the reference genotype is s. Then the frequency ratio ot changes over time according to equation Φt = Φo est, (1) where do is the frequency ratio at generation 0. By taking the natural logarithm of both sides of equation (1), it can also be rewritten as (2) Equation (2) shows that the natural log of the frequency ratio changes linearly with time and the slope of this line is the selection coefficient s. Equation (2) gives us a convenient way of estimating the selection coefficient s by plotting frequency ratio against time. In(RifR/PAO1) 2.5 2.0 1.5 1.0 0.5 0 -0.5 -1.0 -1.5 0 In ot 0 0 = No antibiotic 0.5 µg/ml 1.0 μg/ml 1.5 μg/ml 2.0 μg/ml 10 In do + st. 20 30 Time, in generations 4 40 Figure 1: Log of the ratio of the frequency of RifR to the frequency of PAO1 as a function of time, in different concentrations of rifampicin. A single amnio acid change from aspartic acid to glycine at site 521 in the gene rpoB in the strain PAO1 of Pseudomonas aeruginosa causes it to be resistant to antibiotic rifampicin. Denote the resistant mutant strain by RifR. Whether this mutation is ben- eficial (increases fitness) or deleterious (decrea reases ess) depends on the concentration of rifampicin in the environment. Figure 1 shows how the frequency ratio ₺ of RifR to PAO1 changes over time at four different concentrations of rifampicin (open symbols) as well as in an antibiotic-free environment (black circles). 1. (5 P) Estimate the selection coefficient s of RifR relative to PAO1 in each of these environments. Fill in column 2 in Table 1. Table 1: Estimated selection coefficient for RifR Concentration, pg/ml 0 0.5 1.0 1.5 2.0 Selection coefficient B, D or N 2. (1 P) Determine whether the resistance mutation is beneficial (B), deleterious (D) or neutral (N) in each environment. Fill in column 3 in Table 1. 5 3. (3 P) Plot the selection coefficient of RifR (y-axis) against the rifampicin concen- tration (x-axis). Don't forget to label the axes and indicate the scale. 4. (1 P) Fit a line to the data in the plot that you have produced in the previous question. 5. (1 P) Determine the maximal rifampicin concentration at which the sensitive strain is expected to be favored by natural selection. 6