Question

Create two different populations that are each individually in linkage disequilibrium, but when combined are in linkage equilibrium. Use two loci, each with two alleles. None of the populations are allowed to have any of the chromosome frequencies equal to zero. Demonstrate that each population satisfies these requirements (i.e., show that the two original populations are in LD and the combined population is in LE).

Answer 1

Assume
Population 1 (AB = 150 , Ab = 70 , aB = 50 , ab = 30)
Population 2 (AB = 60 , Ab = 70 , aB = 40 , ab = 30)
Combined Population (AB = 210 , Ab = 140 , aB = 90 , ab = 60)

Population 1
Total Population = 300
Haplotype || Frequency
AB || x11 || 0.500
Ab || x12 || 0.233
aB || x21 || 0.166
ab || x22 || 0.100

Allele || Frequency
A || p1 = x11 + x12 || 0.733
a || p2 = x21 + x22 || 0.266
B || q1 = x11 + x21 || 0.666
b || q2 = x12 + x22 || 0.333

For linkage equilibrium to exist,
D = x11 - p1q1 = x12 - p1q2 = x21 - p2q1 = x22 - p2q2 = 0
Since, x11 - p1q1 & x12 - p1q2 & x21 - p2q1 & x22 - p2q2 is not equal to 0 Verified, hence not in equilibrium

Population 2
Total Population = 200
Haplotype || Frequency
AB || x11 || 0.3
Ab || x12 || 0.35
aB || x21 || 0.2
ab || x22 || 0.15

Allele || Frequency
A || p1 = x11 + x12 || 0.65
a || p2 = x21 + x22 || 0.5
B || q1 = x11 + x21 || 0.5
b || q2 = x12 + x22 || 0.5

For linkage equilibrium to exist,
D = x11 - p1q1 = x12 - p1q2 = x21 - p2q1 = x22 - p2q2 = 0
Since, x11 - p1q1 & x12 - p1q2 & x21 - p2q1 & x22 - p2q2 is not equal to 0 Verified, hence not in equilibrium

Combined Population
Total Population = 500
Haplotype || Frequency
AB || x11 || 0.42
Ab || x12 || 0.28
aB || x21 || 0.18
ab || x22 || 0.12

Allele || Frequency
A || p1 = x11 + x12 || 0.7
a || p2 = x21 + x22 || 0.3
B || q1 = x11 + x21 || 0.6
b || q2 = x12 + x22 || 0.4

For linkage equilibrium to exist,
D = x11 - p1q1 = x12 - p1q2 = x21 - p2q1 = x22 - p2q2 = 0
Since, x11 - p1q1 = x12 - p1q2 = x21 - p2q1 = x22 - p2q2 = 0 Verified, hence in equilibrium