Reversal distance for partially ordered genomes

MOTIVATION The total order of the genes or markers on a chromosome inherent in its representation as a signed per-mutation must often be weakened to a partial order in the case of real data. This is due to lack of resolution (where several genes are mapped to the same chromosomal position) to missing data from some of the datasets used to compile a gene order, and to conflicts between these datasets. The available genome rearrangement algorithms, however, require total orders as input. A more general approach is needed to handle rearrangements of gene partial orders. RESULTS We formalize the uncertainty in gene order data by representing a chromosome from each genome as a partial order, summarized by a directed acyclic graph (DAG). The rearrangement problem is then to infer a minimal sequence of reversals for transforming any topological sort of one DAG to any one of the other DAG. Each topological sort represents a possible linearization compatible with all the datasets on the chromosome. The set of all possible topological sorts is embedded in each DAG by appropriately augmenting the edge set, so that it becomes a general directed graph (DG). The DGs representing chromosomes of two genomes are combined to produce a bicoloured graph from which we extract a maximal decomposition into alternating coloured cycles, and from which, in turn, an optimal sequence of reversals can usually be identified. We test this approach on simulated incomplete comparative maps and on cereal chromosomal maps drawn from the Gramene browser.