Page 282 - Applied Probability

12. Models of Recombination
270
Example 12.5.1 Chi-Square Model
In the chi-square model [1, 5, 10, 21, 22, 33], a fixed number of points is
skipped. If every rth point is a χ point, then s i =1 {i=r} and the equilibrium
distribution π is uniform on {0, 1,...,r − 1}. The discrete renewal density
u n =1 {n=0 mod r} . The expressions for q 0j (t) and p ij (t) simplify to
& −λt
e j =0
q 0j (t) = (λt) r−j −λt
(r−j)! e j> 0
and
∞
(λt) i−j −λt (λt) mr+i−j −λt
p ij (t)=1 {j≤i} e + e .
(i − j)! (mr + i − j)!
m=1
This model tends to fit data well.
12.6 Chiasma Interference

Chiasma interference can be roughly divided into count interference and
position interference. Count interference arises when the total number
of chiasmata on a chromosome follows a non-Poisson distribution. Position
interference arises when the formation of one chiasma actively discourages
the formation of other chiasmata nearby. The count-location model exhibits
count interference but not position interference. Stationary renewal mod-
els exhibit both types of interference and therefore are somewhat better
equipped to capture the subtleties of recombination data.
Traditionally, geneticists have measured interference by coincidence co-
efficients. The coincidence coefficient C(I 1 ,I 2 ) of two adjacent intervals I 1
and I 2 is defined as the ratio of the probability of recombination on both
intervals to the product of their individual recombination fractions. Based
on equations (12.1) and (12.7), this ratio is
1 = 0)]
C(I 1 ,I 2 ) = 4 [1 − Pr(N I 1 =0) − Pr(N I 2 = 0)+Pr(N I 1 + N I 2 .
1 1
2
2 [1 − Pr(N I 1 = 0)] [1 − Pr(N I 2 = 0)]
The conditions C(I 1 ,I 2 ) < 1 and C(I 1 ,I 2 ) > 1 are referred to as posi-
tive and negative interference, respectively. Positive interference occurs
when

=0), (12.15)
Pr(N I 1 + N I 2 =0) < Pr(N I 1 =0) Pr(N I 2
and negative interference occurs when the reverse inequality obtains. Hal-
dane’s model gives equality and exhibits no interference.