Mitochondrial genome diversity in closely related species provides an excellent platform for investigation of chromosome architecture and its evolution by means of comparative genomics. In this study, we determined the complete mitochondrial DNA sequences of eight Candida species and analyzed their molecular architectures. Our survey revealed a puzzling variability of genome architecture, including circular- and linear-mapping and multipartite linear forms. We propose that the arrangement of large inverted repeats identified in these genomes plays a crucial role in alterations of their molecular architectures. In specific arrangements, the inverted repeats appear to function as resolution elements, allowing genome conversion among different topologies, eventually leading to genome fragmentation into multiple linear DNA molecules. We suggest that molecular transactions generating linear mitochondrial DNA molecules with defined telomeric structures may parallel the evolutionary emergence of linear chromosomes and multipartite genomes in general and may provide clues for the origin of telomeres and pathways implicated in their maintenance.

We study three classical problems of genome rearrangement--sorting, halving, and the median problem--in a restricted double cut and join (DCJ) model. In the DCJ model, introduced by Yancopoulos et al., we can represent rearrangement events that happen in multichromosomal genomes, such as inversions, translocations, fusions, and fissions. Two DCJ operations can mimic transpositions or block interchanges by first extracting an appropriate segment of a chromosome, creating a temporary circular chromosome, and then reinserting it in its proper place. In the restricted model, we are concerned with multichromosomal linear genomes and we require that each circular excision is immediately followed by its reincorporation. Existing linear-time DCJ sorting and halving algorithms ignore this reincorporation constraint. In this article, we propose a new algorithm for the restricted sorting problem running in O(n log n) time, thus improving on the known quadratic time algorithm. We solve the restricted halving problem and give an algorithm that computes a multilinear halved genome in linear time. Finally, we show that the restricted median problem is NP-hard as conjectured.

We study the complexity of rearrangement problems in the generalized breakpoint model of Tannier et al. and settle several open questions. We improve the algorithm for the median problem and show that it is equivalent to the problem of finding maximum cardinality nonbipartite matching (under linear reduction). On the other hand, we prove that the more general small phylogeny problem is NP-hard. Surprisingly, we show that it is already NP-hard (or even APX-hard) for a quartet phylogeny. We also show that in the unichromosomal and the multilinear breakpoint model the halving problem is NP-hard, refuting the conjecture of Tannier et al. Interestingly, this is the first problem that is harder in the breakpoint model than in the double cut and join or reversal models.

Let G = (V, E) be a directed acyclic graph with two distinguished vertices s, t, and let F be a set of forbidden pairs of vertices. We say that a path in G is safe, if it contains at most one vertex from each pair {u, v} ∈ F. Given G and F, the path avoiding forbidden pairs (PAFP) problem is to find a safe s-t path in G.We systematically study the complexity of different special cases of the PAFP problem defined by the mutual positions of fobidden pairs. Fix one topological ordering ≺ of vertices; we say that pairs {u, v} and {x, y} are disjoint,The PAFP problem is known to be NP-hard in general or if no two pairs are disjoint; we prove that it remains NP-hard even when no two forbidden pairs are nested. On the other hand, if no two pairs are halving, the problem is known to be solvable in cubic time. We simplify and improve this result by showing an O(M(n)) time algorithm, where M(n) is the time to multiply two n × n boolean matrices.

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