Page 270 - Applied Probability

12. Models of Recombination
258
that correctly capture the phenomenon of chiasma interference. This sec-
ond kind of interference involves the suppression of additional chiasmata in
the vicinity of a chiasma already formed. Although a fair amount is known
about the biochemistry and cytology of crossing-over, no one has suggested
a mechanism that fully explains chiasma interference [28, 30]. Until such
a mechanism appears, we must be content with purely phenomenological
models for the relatively small number of chiasmata per chromatid bundle.
Even if a satisfactory model is devised, calculation of gamete probabilities
under it may be very cumbersome. The models considered in this chapter
have the advantage of permitting exact calculation of multilocus gamete
probabilities.
12.2 Mather’s Formula and Its Generalization
Mather [19] discovered a lovely formula connecting the recombination frac-
tion θ separating two loci at positions a and b to the random number of
chiasmata N [a,b] occurring on the interval [a, b] of the chromatid bundle.
Mather’s formula
1
θ = Pr(N [a,b] > 0)
2
1
= [1 − Pr(N [a,b] = 0)] (12.1)
2
1
makes it clear that 0 ≤ θ ≤ and that θ increases as b increases for a fixed.
2
The genetic map distance d separating a and b is defined as 1 E(N [a,b] ), the
2
expected number of crossovers on [a, b] per gamete. The unit of distance is
the Morgan, in honor of Thomas Hunt Morgan. For short intervals, θ ≈ d
because E(N [a,b] ) ≈ Pr(N [a,b] > 0).
To prove (12.1), note first that a gamete is recombinant between two loci
a and b if and only if an odd number of crossovers occurs on the gamete
between a and b. Let r n be the probability that the gamete is recombinant
given that n chiasmata occur on the chromatid bundle between a and b.It
is clear that r 0 =0. For n> 0, we have the recurrence
1 1
r n = r n−1 + (1 − r n−1 ) (12.2)
2 2
because a gamete is recombinant after n crossovers if it is recombinant
after n − 1 crossovers and does not participate in crossover n,or ifitis
nonrecombinant after n − 1 crossovers and does participate in crossover n.
In view of recurrence relation (12.2), it follows that r n = 1 for all n> 0,
2
and this fact proves Mather’s formula (12.1).
As a simple application of Mather’s formula, suppose that the number
of chiasmata on the chromatid bundle between a and b follows a Poisson