Page 245 - Applied Probability

11. Radiation Hybrid Mapping
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on the human chromosome, the less likely it is that irradiation will cause a
break between them. Thus, close loci will tend to be concordantly retained
or lost in the hybrid cells, while distant loci will tend to be independently
retained or lost. The retention patterns from the various hybrid clones
therefore give important clues for determining locus order and for estimat-
ing the distances between adjacent loci for a given order.
11.2 Models for Radiation Hybrids
The breakage phenomenon for a particular human chromosome can be rea-
sonably modeled by a Poisson process. The preliminary evidence of Cox
et al. [10] suggests that this Poisson breakage process is roughly homoge-
neous along the chromosome. For their data on human chromosome 21, Cox
et al. [10] found that 8,000 rads of radiation produced on average about
four breaks per cell. The intensity λ characterizing the Poisson process is
formally defined as the breakage probability per unit length. Assuming a
4
4
length of 4×10 kilobases (kb) for chromosome 21, λ ≈ 4/(4×10 )= 10 −4
breaks per kb when a cell is exposed to 8,000 rads [10].
For any two loci, the simple mapping function
1 − θ = e −λδ (11.1)
relates the probability θ of at least one break between the loci to the phys-
ical distance δ between them. When λδ is small, θ ≈ λδ. This is analogous
to the approximate linear relationship between recombination fraction and
map distance for small distances in genetic recombination experiments. In-
deed, except for minor notational differences, equation (11.1) is Haldane’s
mapping function for recombination without chiasma interference.
In addition to breakage, fragment retention must be taken into account
when analyzing radiation hybrid data. A reasonable assumption is that dif-
ferent fragments are retained independently. For the purposes of this exposi-
tion, we will make the further assumption that there is a common fragment
retention probability r. Boehnke et al. [5] consider at length more compli-
cated models for fragment retention. For instance, the fragment bearing
the centromere of the chromosome may be retained more often than other
fragments. This is biologically plausible because the centromere is involved
in coordination of chromosome migration during cell division. However,
these more complicated models appear to make little difference in ultimate
conclusions.
In a radiation hybrid experiment, a certain number of clones are scored
at several loci. For example, in the Cox et al. [1990] chromosome 21 data,
99 clones were scored at 14 loci. In some of the clones, only a subset
of the loci was scored. One of their typical clones can be represented as
(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, ?, 0, 0, 1). A “1” in a given position of this observa-
tion vector indicates that the corresponding human locus was present in