From Gene Trees to Species Trees

This paper studies various algorithmic issues in reconstructing a species tree from gene trees under the duplication and the mutation cost model. This is a fundamental problem in computational molecular biology. Our main results are as follows. 1. A linear time algorithm is presented for computing all the losses in duplications associated with the least common ancestor mapping from a gene tree to a species tree. This answers a problem raised recently by Eulenstein et al. (1998). 2. The complexity of nding an optimal species tree from gene trees is studied. The problem is proved to be NP-hard for the duplication cost and for the mutation cost. Further, the concept of reconciled trees was introduced by Goodman et al. and formalized by Page for visualizing the relationship between gene and species trees. We show that constructing an optimal reconciled tree for gene trees is also NP-hard. Finally, we consider a general reconstruction problem and show it to be NP-hard even for the well-known nearest neighbor interchange distance. 3. A new and e ciently computable metric is de ned based on the duplication cost. We show that the problem of nding an optimal species tree from gene trees is NP-hard under this new metric but it can be approximated within factor 2 in polynomial time. Using this approximation result, we propose a heuristic method for nding a species tree from gene trees with uniquely labeled leaves under the duplication cost. Our experimental tests demonstrate that, when the number of species is larger than 15 and gene trees are close to each other, our heuristic method is signi cantly better than the existing program in Page's GeneTree 1.0 that starts the search from a random tree.