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11. Radiation Hybrid Mapping
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that the same clones can be used to map any chromosome of interest.
Although in principle one could employ heterozygous markers, geneticists
forgo this temptation and score only the presence, and not the number of
markers per locus in a diploid clone. Finally, just as with haploid clones,
one can pool diploid clones to achieve sampling units with an arbitrary
even number of chromosomes.
We now present methods for analyzing polyploid radiation hybrids with
c chromosomes per clone or sampling unit [15]. For the sake of brevity, we
use the term “clone” to mean either a haploid clone, a diploid clone, or a
fixed number of pooled haploid or diploid clones. Our analysis will assume
that the breakage and fragment retention processes are independent among
chromosomes and that typing can reveal only the presence and not the
number of markers per locus in a clone.
11.7 Maximum Likelihood Under Polyploidy
Again let X =(X 1 ,... ,X m ) denote the observation vector for a single
clone. If no markers are observed at the ith locus, then X i = 0. If one or
c
more markers are observed, then X i = 1. Because (1 − r) is the probabil-
ity that all c copies of a given marker are lost, the single-locus polyploid
likelihood reduces to the Bernoulli distribution
c i
Pr(X 1 = i)= [1 − (1 − r) ] (1 − r) c(1−i) .
The two-locus polyploid likelihoods
Pr(X 1 =0,X 2 =0) = [(1 − r)(1 − θr)] c
Pr(X 1 =1,X 2 = 0) = Pr(X 1 =0,X 2 =1)
c
=(1 − r) − [(1 − r)(1 − θr)] c (11.9)
c
Pr(X 1 =1,X 2 =1) = 1 − 2(1 − r) +[(1 − r)(1 − θr)] c
generalize the two-locus haploid likelihoods (11.5). The expression for the
first probability Pr(X 1 =0,X 2 = 0) in (11.9) is a direct consequence of
the independent fate of the c chromosomes during fragmentation and re-
tention. Considering a given chromosome, the marker at locus 1 is lost
with probability 1 − r. Conditional on this event, the marker at locus 2
must also be lost. This second event occurs with probability 1−θr since its
complementary event occurs only when there is a break between the two
loci and the fragment bearing the second locus is retained. The stated ex-
pression for Pr(X 1 =0,X 2 = 1) in (11.9) can be computed by subtracting
c
Pr(X 1 =0,X 2 = 0) from the probability (1−r) that all c markers are lost
at locus 1. Finally, Pr(X 1 =1,X 2 = 1) is most easily computed by sub-
tracting the three previous probabilities in (11.9) from 1 and simplifying.