Page 255 - Applied Probability

11. Radiation Hybrid Mapping
242
view because only the presence or absence of markers is directly observ-
able. Suppose the chain is in state i at locus k. The probability t c,k (i, j)of a
transition from state i at locus k to state j at locus k +1 is of fundamental
importance.
To compute t c,k (i, j), consider first a haploid clone. In this situation the
chromosome copy number c = 1, and it is clear that
t 1,k (0, 0) = 1 − θ k r
t 1,k (0, 1) = θ k r
t 1,k (1, 0) = θ k (1 − r)
t 1,k (1, 1) = 1 − θ k (1 − r).
Employing these simple transition probabilities, we can write the following
general expression:
min{i,j}
i c − i
t c,k (i, j)= (11.13)
l j − l
l=max{0,i+j−c}
l
× t 1,k (1, 1) t 1,k (1, 0) i−l t 1,k (0, 1) j−l t 1,k (0, 0) c−i−j+l .
Formula (11.13) can be deduced by letting l be the number of markers
retained at locus k that lead via the same original chromosomes to markers
i

retained at locus k + 1. These l markers can be chosen in ways. Among
l
the i markers retained at locus k, the fate of the l markers retained at locus
k+1 and the remaining i−l markers not retained at locus k+1 is captured
l
by the product t 1,k (1, 1) t 1,k (1, 0) i−l in (11.13). For j total markers to be
retained at locus k + 1, the c − i markers not retained at locus k must lead
to j − l markers retained at locus k + 1. These j − l markers can be chosen
c−i j−l c−i−j+l
in ways. The product t 1,k (0, 1) t 1,k (0, 0) captures the fate
j−l
of the c − i markers not retained at locus k. Finally, the upper and lower
bounds on the index of summation l insure that none of the powers of the
t 1,k (u, v) appearing in (11.13) is negative.
In setting down the likelihood for the observations (X 1 ,... ,X m ) from a
single clone, it is helpful to define a set O i corresponding to each X i . This
set indicates the range of markers possible at locus i. Thus, let


 {0, 1,... ,c} for X i missing
O i = {0} for X i =0

{1,... ,c} for X i =1.
The sets O 1 ,... ,O m encapsulate the same information as (X 1 ,...,X m).
Owing to the Markovian structure of the model, the likelihood of the ob-
servation vector (X 1 ,... ,X m ) amounts to

m−1

c
j 1
P = ··· r (1 − r) c−j 1 t c,k (j k ,j k+1 ). (11.14)
j 1
j 1 ∈O 1 j m ∈O m k=1