Page 36 - Applied Probability

1. Basic Principles of Population Genetics
19
recessive disease allele A 1 . Mutation from A 2 to A 1 takes place at rate
µ. No backmutation is permitted. An entire population is screened
for carriers. If a husband and wife are both carriers, then all fetuses
of the wife are checked, and those who will develop the disease are
aborted. The couple compensates for such unsuccessful pregnancies,
so that they have an average number of normal children. Affected
children born to parents not at high risk likewise are compensated for
by the parents. These particular affected children are new mutations
and do not contribute to the next generation. Let u n and v n be the
frequency of people with genotypes A 1 /A 2 and A 2 /A 2 , respectively,
at generation n.
TABLE 1.3. Mating Outcomes under Genetic Screening
Mating Type Frequency A 1 /A 2 Offspring A 2 /A 2 Offspring
4
4
u 2 2 + µ 1 − µ
A 1 /A 2 × A 1 /A 2
n 3 9 3 9
1 3 1 3
A 1 /A 2 × A 2 /A 2 2u nv n + µ − µ
2 4 2 4
v 2 2µ 1 − 2µ
A 2 /A 2 × A 2 /A 2
n
(a) In Table 1.3, mathematically justify the mating frequencies ex-
2
actly and the offspring frequencies to order O(µ ). (Hint: Apply
k
the expansion (1 − x) −1 = ∞ x for |x| < 1.)
k=0
(b) Derive a pair of recurrence relations for u n+1 and v n+1 based
on the results of Table 1.3. Use the recurrence relations to show
that u n + v n = 1 for all n.
(c) Demonstrate that the recurrence relation for u n+1 has equilib-
√
rium value u ∞ = 6µ. This implies a frequency of approxi-

mately 3µ/2 for allele A 1 . (Hint: In the recurrence for u n+1 ,
substitute v n =1 − u n and take limits. Assume that u ∞ is of
√ 3/2
order µ and neglect all terms of order µ or smaller.)
(d) Find the function f(u) giving the recurrence u n+1 = f(u n).

Show that f (u ∞ ) ≈ 1 − 2 2µ/3.

(e) Discuss the implications of the above analysis for genetic screen-
ing. Consider the increase in the equilibrium frequency of the
disease allele and, in light of Problem 13, the speed at which
this increased frequency is attained.

1.8 References

[1] Bennet JH (1954) On the theory of random mating. Ann Eugen
18:311–317