1. Show the result claimed in the previous section (6.2). Suppose that
crossovers occur as a Poisson process rate 1. Given a
crossover occurs somewhere on a chromosome with genetic length d Morgans,
show that its location X is uniformly distributed over the chromosome.
(X ~ U(0,d)).
[You have this one in your notes from 394, in a different
context.]
2. (Harder). Given two crossovers occur somewhere on a chromosome with
genetic length d Morgans,
show that their locations X and Y are independently and
uniformly distributed over the chromosome.
(X,Y ~ U(0,d); X,Y independent.).
(This can be argued in various ways -- some of which are simple, but not
100% convincing.)
3. Let X1 be the number of crossovers in an interval of chromosome of
genetic length d1. By definition
E(X1)=d1.
Let X2 be the number of crossovers in an interval of chromosome of
genetic length d2. By definition
E(X2)=d2.
Suppose the two intervals are disjoint.
Show that the expected number of crossovers in the two intervals combined is
d1+d2. That is; genetic distance is additive.
Note that to show this you do not need to assume that X1 and
X2 are independent, or that either has a Poisson distribution.
The result is true when X1 and X2 have an arbitrary
joint probability mass function
P(X1 =m, X2=n) = pm,n, m,n=0,1,2,3,....
4. Returning to Haldane's model, consider a large chromosome, such as
Chromosome-1. This chromosome has length about 3 Morgans.
(a) What is the probability that there are no crossovers on Chromosome-1
in a meiosis?
(b) What is the expected number of crossovers, m?
(c) What is the probability of m or more crossovers?
(d) What is the probability of recombination between the two ends of the
chromosome? (see 6.1)
(e) What is the probability there is recombination between each end and
the mid-point of the chromosome? (as always, we measure "midpoint" etc,
in terms of genetic distance)
5. Repeat question 4(a)-(e), for a short chromosome, such as
Chromosome-20, which has length about 0.55 Morgans.
(Haldane's model is not such a good model for short chromosomes, for
reasons we may get to discuss later.)
6. In plant breeding, one objective is to make sure that a gene for
disease resistance (or whatever) which has been inserted into a parent
plant is transmitted to the seeds which are to be sold. Suppose we can
test whether the gene is in a given seed (non-destructively), but we want
to know how big a piece of chromosome is being carried along with it.
This will be the piece extending to the first crossover each side of gene.
(a) In a given meiosis, what is the distribution of the distance to the
first crossover "to the right" of the given gene? What is the expected
distance? (Ignore the problem that might get to end of chromosome.)
(b) In a given meiosis, what is the distribution of the distance to the
first crossover "to the left" of the given gene? What is the expected
distance? What is the expected length of the piece.
(c) Now we are going to transmit this gene over k generations. The
length to the right/left of the gene will just be the minimum of the
distances in each of the k meioses (why?). What is the distribution of
the piece of original chromosome to the right/left of the gene, after k
generations?
(d) What is the expected length of the piece? How would you find the pdf
of the length of the piece?