Generalized Hultman Numbers and the Distribution of Multi-break Distances
Genome rearrangements can be modeled by k-breaks, which break a genome at k positions and glue the resulting fragments in a new order. In particular, reversals, translocations, fusions, and fissions are modeled as 2-breaks, and transpositions are modeled as 3-breaks. While k-break rearrangements for \(k>3\) have not been observed in evolution, they are used in cancer genomics to model chromothripsis, a catastrophic event of multiple breakages happening in a genome simultaneously.
It is known that the k-break distance between two genomes (i.e., the minimal number of k-breaks needed to transform one genome into the other) can be computed in terms of cycle lengths of the breakpoint graph of these genomes. In the current work, we address the combinatorial problem of enumeration of genomes at a given k-break distance from a fixed genome. More generally, we enumerate genome pairs, whose breakpoint graph has a fixed distribution of cycle lengths.
The work is supported by the National Science Foundation under the grant No. IIS-1462107. The work of NA is also partially supported by RFBR grant 13-01-12422-ofi-m.
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