Page 273 - Applied Probability

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12. Models of Recombination
∞
1

n
=
q n {1 − [1 − F(b)+ F(a)] }
2
n=0
1
1
−1
− Q(1 − 2λ
d).
=
2
2
Thus, the count-location model yields a map function θ = M(d) giving
the recombination fraction θ of an interval purely in terms of the corre-
sponding map distance d.
Gamete probabilities are easily computed in the count-location model
based on formula (12.6). Indeed, if I i =[a i ,b i ] and w i = F(b i ) − F(a i ),
then the gamete probability y S can be expressed in either of the equivalent
forms
1 k

= (−1) |S∩T | w i )
y S Q(1 −
2
T i∈T
∞ k
1
n
= q n (−1) |S∩T | (1 − w i ) .
2
n=0 T i∈T
Further algebraic simplification of y S is possible but will not be pursued
here [24].
Example 12.3.1 Haldane’s Model
Haldane’s Poisson model corresponds to the chiasma count distribution
n −λ
q n = λ e /n! with generating function Q(s)= e −λ(1−s) [11]. In this case,
the conversion between map distance d and recombination fraction θ is
mediated by the pair of functions mentioned in equations (7.8) and (7.8)
of Chapter 7. Although Haldane’s model is widely used, it unrealistically
entails no chiasma interference. This defect is partially compensated for by
its computational simplicity. See, for instance, Problem 1 of this chapter
and Trow’s formula in Problem 1 of Chapter 7.
Other simple choices for the chiasma count distribution include the bi-
nomial and truncated Poisson distributions with generating functions
1 s
r
Q(s) = +
2 2
e −λ(1−s) − e −λ
Q(s) = ,
1 − e −λ
respectively [12, 29]. The latter choice is particularly useful because it incor-
porates the empirical observation that almost all chromatid bundles display
at least one chiasma.
12.4 Stationary Renewal Models
Renewal processes provide another important class of recombination