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11. Radiation Hybrid Mapping
                              240
                              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.
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