Algorithms for determining transposons in gene sequences

Some genes can change their relative locations in a genome. Thus for different individuals of the same species, the orders of genes might be different. Such jumping genes are called transposons. A practical problem is to determine transposons in given gene sequences. Through an intuitive rule, we transform transposons from a biological concept into a rigorous mathematical object. Depending on whether the gene sequence is linear (each sequence has a fixed head and tail) or circular (we can choose any gene as the head, and the previous one is the tail), and whether genes have multiple copies, we classify the problem of determining transposons into four scenarios: (1) linear sequences without replicated genes; (2) circular sequences without replicated genes; (3) linear sequences with replicated genes; (4) circular sequences with replicated genes. With the help of graph theory, we design fast algorithms for different scenarios. We also derive some results that might be of theoretical interests in combinatorics.