  Exchange type chromosome aberrations (ETCAs) are rearrangements of the genome
that occur when chromosomes break and the resulting fragments rejoin with other
fragments from other chromosomes. ETCAs are commonly observed in cancer cells
and in cells exposed to radiation. The frequency of these chromosome
rearrangements is correlated with their spatial proximity, therefore it can be
used to infer the three dimensional organization of the genome. Extracting
statistical significance of spatial proximity from cancer and radiation data
has remained somewhat elusive because of the sparsity of the data. We here
propose a new approach to study the three dimensional organization of the
genome using algebraic statistics. We test our method on a published data set
of irradiated human blood lymphocyte cells. We provide a rigorous method for
testing the overall organization of the genome, and in agreement with previous
results we find a random relative positioning of chromosomes with the exception
of the chromosome pairs \{1,22\} and \{13,14\} that have a significantly larger
number of ETCAs than the rest of the chromosome pairs suggesting their spatial
proximity. We conclude that algebraic methods can successfully be used to
analyze genetic data and have potential applications to larger and more complex
data sets.
