Duplication-Loss Genome Alignment: Complexity and Algorithm

Abstract

Abstract. Recently, an Alignment approach for the comparison of two genomes, based on an evolutionary model restricted to Duplications and Losses, has been presented. An exact linear programming algorithm has been developed and successfully applied to the Transfer RNA (tRNA) repertoire in Bacteria, leading to interesting observation on tRNA shift of identity. Here, we explore a direct dynamic programming approach for the Duplication-Loss Alignment of two genomes, which proceeds in two steps: (1) (The Dynamic Programming step) Outputs a best candidate alignment between the two genomes and (2) (Minimum Label Alignment problem) Finds an evolutionary scenario of minimum duplication-loss cost that is in agreement with the alignment. We show that the Minimum Label Alignment is APX-hard, even if the number of occurrences of a gene inside a genome is bounded by 5. We then develop a heuristic which is a thousands of times faster than the linear programming algorithm and exhibits a high degree of accuracy on simulated datasets. The heuristic has been implemented in JAVA and is available on request.

Gene order alignment on trees with multiOrthoAlign

“…In a recent set of papers [ 17 - 19 ] we related the comparison of two gene orders to an alignment problem: find an alignment between the two gene orders that can be interpreted by a minimum number of evolutionary events (rearrangements and content-modifying operations). Although alignments are a priori simpler to handle than rearrangements, this problem has been shown NP-hard for the duplication-loss model of evolution [ 17 , 18 , 20 ]. Exact exponential-time algorithms based on linear programming [ 19 , 20 ] and a polynomial-time heuristic based on dynamic programming [ 17 ] have been developed for this model.…”

“…Although alignments are a priori simpler to handle than rearrangements, this problem has been shown NP-hard for the duplication-loss model of evolution [ 17 , 18 , 20 ]. Exact exponential-time algorithms based on linear programming [ 19 , 20 ] and a polynomial-time heuristic based on dynamic programming [ 17 ] have been developed for this model. Recently [ 21 ], we developed OrthoAlign (alignment of orthologs), a time-efficient heuristic for the gene order alignment problem, that extends the dynamic programming approach to a model including rearrangements (inversions and transpositions).…”

Aligning and Labeling Genomes under the Duplication-Loss Model

“…The second problem stems from a direct approach to Duplication-Loss Alignment which is in two steps: (1) Compute a best candidate labeled alignment between the two genomes that may be unfeasible for the Duplication-Loss model and (2) (Minimum Relabeling Alignment problem) Find an evolutionary scenario of minimum duplication-loss cost that is in agreement with the alignment. A similar approach has been proposed in [2], where first it is computed an unlabeled alignment of two genomes, and then the alignment is explained with an evolutionary scenario of minimum duplication-loss.…”

“…Notice that in practice genes have few occurrences inside a genome, so it is interesting to understand how the complexity of the problem is influenced by this parameter. We then show in Section 4 that Minimum Relabeling Alignment is equivalent to Minimum Feedback Vertex Set on Direct Graph, hence showing that the problem is APX-hard and that (1) it is fixedparameter tractable, when the parameter is the cost of the relabeling, (2) it is approximable within factor O(log |X | log log |X |), where X is the aligned genome considered by Minimum Relabeling Alignment.…”

Operon-based approach for the inference of rRNA and tRNA evolutionary histories in bacteria

“…As a consequence, there is simply not enough signal in the sequences themselves to identify orthology relationships. This alignment problem was then shown to be NP-hard for the duplication and losses model of evolution [13][14][15]. The exact algorithm proposed in [11], based on integer linear programming (ILP), was designed to solve the 2-Small Phylogeny Problem (2-SPP), which is to find a common ancestor A of two gene orders X and Y that minimizes the number of events on each of the two branches.…”