Crossovers are points at which, in a given meiosis, the points at which the chromosome which goes to the gamete switches from being a copy of either one of the parent's chromosomes to being a copy of the other.
A recombination occurs between two loci (positions) on a chromosome if the genes which are copied to the gamete at those two loci derive from the two different parental chromosomes. Thus there is a recombination between two loci on a chromosome if there is an odd number of crossovers between them.
Haldane assumed that crossovers occur as a Poisson process. He defined the unit of genetic distance, the Morgan, as the expected number of crossovers between two loci. Since expectations are additive, so is this measure of distance along a chromosome (see exercises 6.3). [This is always true, even without the assumption that crossovers follow a Poisson process.] If genetic distance is measured in Morgans, the rate of the Poisson process is 1. In fact, it has become more standard to measure genetic distance in centiMorgans (cM); in that case the Poisson process has rate parameter 0.01.
What is the probability of an odd number of events in a given interval
length d in
a Poisson process rate 1: here is a useful trick to give us the answer.
The number of crossovers Xis Po(d): P(X=k) = exp(-d) dk/k!
We want to sum P(X=k) over k=1, 3, 5, 7, 9, ....
exp (d) = 1 + d + d2/2! + d3/3! +
d4/4! + ...
exp(-d) = 1 + (-d) + (-d)2/2! + (-d)3/3! +
(-d)4/4! + ...
= 1 - d + d2/2! - d3/3! +
d4/4! + ...
So exp(d)-exp(-d) = 2( d + d3/3! + d5/5! + ....)
= 2.exp(d).P(X odd)
So P(recombination) = P(X odd) = (1/2) (1- exp(-2d)).
Note, when d is 0, the probability of recombination is 0. When d is small, the probability of recombination is approximately d. When d is large, the probability of recombination is approximately 1/2. Recall from last quarter, that when loci are very close, probabilities of recombination are small. When loci are far apart, they segregate almost independently; the recombination frequency is almost 1/2. When they are on different chromosomes (d infinite), they do segregate independently; recombination frequency =1/2. Note that under Haldane's model, the recombination frequency is always between 0 and 1/2, as also it seems to be in real life.