Page 313 - Applied Probability

14. Poisson Approximation
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gene in the clone. Genes are usually detected by electrophoresis of their
protein products or by annealing of an appropriate DNA probe directly to
part of the gene. Third, each of the 24 distinct human chromosomes should
be either absent from a clone or cytologically or biochemically detectable
in the clone. Chromosomes can be differentiated cytologically by size, by
the position of their centromeres, and by their distinctive banding patterns
under appropriate stains. It is also possible to distinguish chromosomes by
in situ hybridization of large, fluorescent DNA probes or by isozyme assays
that detect unique proteins produced by genes on the chromosomes.
In this application of the Chen-Stein method, we consider the informa-
tion content of a panel of somatic cell hybrids [11]. Let n denote the number
of hybrid clones in a panel. Since the Y chromosome bears few genes of in-
terest, hybrids are usually created from human female cells. This gives a
total of 23 different chromosome types—22 autosomes and the X chromo-
some. Figure 14.1 depicts a hybrid panel with n = 9 clones. Each row of
this panel corresponds to a particular clone. Each of the 23 columns cor-
responds to a particular chromosome. A 1 in row i and column j of the
panel indicates the presence of chromosome j in clone i. A 0 indicates the
absence of a chromosome in a clone. An additional test column of 0’s and
1’s is constructed when each clone is assayed for the presence of a given
human gene. Barring assay errors or failures of one of the major assump-
tions, the test column will uniquely match one of the columns of the panel.
In this case the gene is assigned to the corresponding chromosome.

FIGURE 14.1. A Somatic Cell Hybrid Panel
0101000100000010110111 1
1010110010000100101011 1
0111101000001001101101 1
1110011001010001110010 1
0001111000111110100011 0
0111111111100000100000 0
0010101101110000111110 0
0001011100010111101010 1
1000110001011010101100 1

If two columns of a panel are identical, then gene assignment becomes
ambiguous for any gene residing on one of the two corresponding chro-
mosomes. Fortunately, the columns of the panel in Figure 14.1 are unique.
This panel has the unusual property that every pair of columns differs in at
least three entries. This level of redundancy is useful. If a single assay error
is made in creating a test column for a human gene, then the gene can still
be successfully assigned to a particular human chromosome because it will
differ from one column of the panel in one entry and from all other columns
of the panel in at least two entries. This consideration suggests that built-in