Page 272 - Applied Probability

12. Models of Recombination
260
where T ranges over all subsets of {1,... ,k} and |S ∩ T| indicates the
number of elements in the intersection S ∩ T. This last formula continues
to hold when the fixed counts n i are replaced by the random counts N I i
Taking expectations therefore produces
y S =E(y N S ) .
1 k

= (−1) |S∩T | Pr N I i =0 (12.6)
2
T i∈T
1 1 k
1 k k

= ··· (−1) i=1 s i t i Pr t i N I i =0 ,
2
t 1 =0 t k =0 i=1
where s i =1 {i∈S} and t i =1 {i∈T } are the obvious indicator functions.
This is the sought-after generalization of Mather’s formula [24, 26]. It col-
lectively expresses the multilocus gamete probabilities as the inverse Walsh
transform of the avoidance probabilities of the chiasma process.
Special cases of (12.6) are easy to construct. For instance, Mather’s for-
1
2
mula (12.1) can be restated as y {1} = [1 − Pr(N I 1 = 0)] for k = 1. The
1
2
probability of nonrecombination is y ∅ = [1 + Pr(N I 1 = 0)]. When k =2,
two of the relevant gamete probabilities are
y {1}
1
= [1 − Pr(N I 1 = 0)+Pr(N I 2 =0) − Pr(N I 1 + N I 2 =0)
4
(12.7)
y {1,2}
1
= [1 − Pr(N I 1 =0) − Pr(N I 2 = 0)+Pr(N I 1 + N I 2 = 0)].
4
12.3 Count-Location Model
The count-location model operates by first choosing the total number N of
chiasmata along the bundle of four chromatids [13, 23]. If we identify the
bundle with the unit interval [0, 1], then N = N [0,1] . Let q n =Pr(N = n)
be the distribution of N. Once the number of chiasmata is chosen, the
individual chiasmata are located independently along the bundle according
to some common continuous distribution F(t). If λ =E(N) in this setting,
1
then the map length of an interval [a, b] reduces to d = λ[F(b) − F(a)].
2
The recombination fraction θ of the interval can be expressed compactly
∞ n
via the generating function Q(s)= n=0 n s of N. Conditioning on the
q
value of N, we find that
1
θ = Pr(N [a,b] > 0)
2
∞
1
= q n [1 − Pr(N [a,b] =0 | N = n)]
2
n=0