Determine transposable genes when the orders of genes are different

Certain nucleotide sequences in DNA can change their positions. Such nucleotide sequences might be shorter than a general gene. When we restrict to nucleotide sequences that form complete genes, we can still find genes that change their relative locations in a genome. Thus for different individuals of the same species, the orders of genes might be different. Such spatial difference of gene orders might be affected by temporal difference of gene (mutation) orders, and can be used to explain the order of mutation problem in myeloproliferative neoplasm. A practical problem is to determine such transposable genes in given gene sequences. Through an intuitive rule, we transform the biological problem of determining transposable genes into a rigorous mathematical problem of determining the longest common subsequence. Given several number sequences, determining the longest common subsequence is a classical problem in computer science. Depending on whether the gene sequence is linear or circular, and whether genes have multiple copies, we classify the problem of determining transposable genes into different scenarios and design corresponding algorithms. Specifically, we study the situation where the longest common subsequence is not unique.