Page 271 - Applied Probability

12. Models of Recombination
259
distribution with mean λ. Then a gamete is recombinant with probability
1
1
−λ
−λ
(1 − e
(1 + e
). Haldane’s
) and nonrecombinant with probability
2
2
model, which postulates that the chiasma process is Poisson, turns out to
be crucial in generalizing Mather’s formula [17, 24]. To derive this gener-
alization, we employ a randomization technique pioneered by Schr¨odinger.
Consider a sequence of k+1 loci along a chromosome. The k+1 loci define
k adjacent intervals I 1 ,... ,I k . A gamete can be recombinant on some of
these intervals and nonrecombinant on others. For a subset S ⊂{1,...,k},
let y S denote the probability that the gamete is recombinant on each of the
intervals I i indexed by i ∈ S and nonrecombinant on each of the remaining
intervals I i indexed by i/∈ S. Under Haldane’s model, the numbers of
chiasmata falling on disjoint intervals are independent. Therefore,
1 1
y S = (1 − e −λ i ) (1 + e −λ i ), (12.3)
2 2
i∈S i/∈S
where λ i denotes the expected number of chiasmata on interval I i .
= n i of chias-
Next consider what happens when we fix the number N I i
mata occurring on each interval I i . The probability of the recombination
pattern dictated by S now changes to the conditional probability y n S ,
where n is the multi-index (n 1 ,...,n k ). We recover y S via
k
λ n i
i
y S = y n S e −λ i . (12.4)
n i !
n i=1
k
Equating formulas (12.3) and (12.4) and multiplying by e , we de-

λ i
i=1
duce that
k k
i
λ n i
e λ i =
y S y n S
n i !
i=1 n i=1
1 k

= (e λ i − 1) (e λ i +1). (12.5)
2
i∈S i/∈S
We extract the coefficient y nS by evaluating the partial derivative
k k

∂ i=1 n i
y S e λ i
∂λ n 1 ··· ∂λ n k
1 k i=1
at λ 1 = ··· = λ k = 0. Equating the two results from identity (12.5) yields
1 k

y n S = (1 − 1 {n i =0} ) (1+1 {n i =0} )
2
i∈S i/∈S
1 k

= (−1) |S∩T | ,
1
2 { i∈T n i =0}
T