Page 33 - Applied Probability

1. Basic Principles of Population Genetics
16
2
It follows that lim t→∞ u(t)= p .
0
(f) Finally, show that
lim s(t)
t→∞
lim v(t) = 1 − p 0
= 2p 0 (1 − p 0 )
t→∞
2
lim w(t) = (1 − p 0 ) .
t→∞
7. Consider three loci A—B—C along a chromosome. To model conver-
gence to linkage equilibrium at these loci, select alleles A i , B j , and
C k and denote their population frequencies by p i , q j , and r k . Let θ AB
be the probability of recombination between loci A and B but not
between B and C. Define θ BC similarly. Let θ AC be the probability of
simultaneous recombination between loci A and B and between loci
B and C. Finally, adopt the usual conditions for Hardy-Weinberg and
linkage equilibrium.
(a) Show that the gamete frequency P n (A i B j C k ) satisfies
P n (A i B j C k )=(1 − θ AB − θ BC − θ AC )P n−1 (A i B j C k )
+ θ AB p i P n−1 (B j C k )+ θ BC r k P n−1 (A i B j )
+ θ AC q j P n−1 (A i C k ).

(b) Define the function

L n (A i B j C k )= P n (A i B j C k ) − p i q j r k − p i [P n (B j C k ) − q j r k ]
− r k [P n (A i B j ) − p i q j ] − q j [P n (A i C k ) − p i r k ].

Show that L n (A i B j C k ) satisfies

L n(A i B j C k )=(1 − θ AB − θ BC − θ AC )L n−1 (A i B j C k ).

(Hint: Substitute for P n (B j C k ) − q j r k and similar terms using
the recurrence relation for two loci.)
(c) Argue that lim n→∞ L n (A i B j C k ) = 0. As a consequence, con-
clude that lim n→∞ P n (A i B j C k )= p i q j r k .
8. Consulting Problems 5 and 6, formulate a Moran model for approach
to linkage equilibrium at two loci. In the context of this model, show
that

P t (A i B j )= e −θt P 0 (A i B j )+(1 − e −θt )p i q j ,

where time t is measured continuously.