#note Population Genetics A Concise Guide (9)

Page 16

Problem 1.7 Derive the Hardy-Weinberg law for a sex-linked locus. Let the initial frequency of A₁ in the females be pf and in the males, p_m. Follow the two alleles frequencies in successive generations until you understand the allele-frequency dynamics. Then, jump ahead and find the equilibrium genotype frequencies in females and males. Finally, graph the male and female allele frequencies over several generations for a population that is started with all A₁A₁ females (p_f=1) and A₂ males (p_m=0).

Note:

Males inherit their X chromosome always from their mother, so the frequency of A₁ in males is always equal to the frequency in females in the previous generation. while females inherit an X chromosome from their father, and the other X chromosome from their mother, so the frequency in females is always the average of the male an female frequencies in the previous generation.

Generation	Females	Males	Female – Male
1	pf	pm	pf – pm
2	(pf + pm)/2	pf	-(pf – pm)/2
3	(pf + (pf + pm)/2)/2	(pf + pm)/2	(pf – pm)/4

Start with p_f=1 and p_m=0, the first 7 generations:

Generation	Females	Males	Female – Male
1	1	0	1
2	0.5	1	-0.5
3	0.75	0.5	0.25
4	0.625	0.75	-0.125
5	0.6875	0.625	0.0625
6	0.65625	0.6875	-0.03125
7	0.671875	0.65625	0.015625

Graph by Python:

import matplotlib.pyplot as plt
import numpy as np

# Females
x = np.array([1,2,3,4,5,6,7])
y = np.array([1,0.5,0.75,0.625,0.6875,0.65625,0.671875])
plt.scatter(x, y)
plt.plot(x, y, '-o')

#Males
x = np.array([1,2,3,4,5,6,7])
y = np.array([0,1,0.5,0.75,0.625,0.6875,0.65625])
plt.scatter(x, y)
plt.plot(x, y, '-o')

plt.show()

Output: