Page 34 - Applied Probability

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1. Basic Principles of Population Genetics
9. To verify convergence to linkage equilibrium for a pair of X-linked
loci A and B, define P nx (A i B j ) and P ny (A i B j ) to be the frequencies
of the A i B j haplotype at generation n in females and males, respec-
tively. For the sake of simplicity, assume that both loci are in Hardy-
Weinberg equilibrium and that the alleles A i and B j have frequencies
p i and q j .If z n denotes the column vector [P nx (A i B j ),P ny (A i B j )] t
and θ the female recombination fraction between the two loci, then
demonstrate the recurrence relation
1

2
z n = Mz n−1 + θp i q j (1.5)
1
under the usual equilibrium conditions, where the matrix
1 [1 − θ] 1

M = 2 2 .
1 − θ 0
Show that equation (1.5) can be reformulated as w n = Mw n−1 for
t
t
w n = z n − p i q j 1 , where 1 =(1, 1) . Solve this last recurrence and
show that lim n→∞ w n = 0. (Hints: The matrix power M n can be
simplified by diagonalizing M. Show that the eigenvalues ω 1 and ω 2
of M are distinct and less than 1 in absolute value.)
10. Consider an autosomal dominant disease in a stationary population.
If the fitness of normal A 2 /A 2 people to the fitness of affected A 1 /A 2
people is in the ratio 1 − s : 1, then show that the average num-
ber of people ultimately affected by a new mutation is 1−s . (Hints:
−s
An A 2 /A 2 person has on average 2 children while an A 1 /A 2 person
has on average 2 children, half of whom are affected. Write and
1−s
solve an equation counting the new mutant and the expected num-
ber of affecteds originating from each of his or her mutant children.
Remember that s< 0.)
11. Consider a model for the mutation-selection balance at an X-linked
locus. Let normal females and males have fitness 1, carrier females
fitness t x , and affected males fitness t y . Also, let the mutation rate
from the normal allele A 2 to the disease allele A 1 be µ in both sexes.
It is possible to write and solve two equations for the equilibrium
frequencies p ∞x and p ∞y of carrier females and affected males.

(a) Derive the two approximate equations
1
p ∞x ≈ 2µ + p ∞x t x + p ∞y t y
2
1
p ∞y ≈ µ + p ∞x t x
2
assuming the disease is rare.