Educación y Sociedad de la Información y del Conocimiento

We remark that the high speed and good scaling properties of our algorithms make them practical for research on large-scale genomic evolution, but also for improved orthology

assignment, as the exemplar concept finds broad applicability in comparative genomics. The performance of our algorithms can be further improved. We expect additional structure can be discovered and turned into constraints for the ILP formulation, thereby reducing the search space for the ILP solver. We are also studying the use of a set of PSSSs (rather than a single PSSS) to define candidates for fixing in the optimal substructure, because it is possible that several PSSSs as a group pass the test, while any single one of these PSSSs fails.

3

DCJ Distance with Duplicate Genes

In this chapter, we give the first algorithm to compute the DCJ distance with the maximum matching strategy in the presence of duplicate genes. To achieve this, we first reduce this problem to the problem of finding the optimal consistent decomposition of the corresponding adjacency graph, then formulate the latter problem as an integer linear program (ILP). We also provide an efficient preprocessing approach to reduce the ILP formulation while preserving optimality. Finally, we apply our method to assign orthologs and also compare its performance with MSOAR on both simulated and biological datasets.

3.1 Problem Statement

In this chapter, we study two genomes with the same gene content: each gene family has the same number of genes in both genomes. We say a bijection between G1and G2is valid if it specifies n homologous gene pairs, where n is the number of genes in each genome. If G1 and G2contain only singleton gene families (exactly one gene in each family in each genome), then there is a unique valid bijection between G1and G2, and the DCJ distance between G1 and G2can be computed in linear time [11]. If G1and G2contain gene families with multiple genes in each genome, then there are many valid bijections between G1and G2. Different valid bijections define different one-to-one correspondences between homologous genes, yielding possibly different DCJ distances between G1and G2. In this chapter, we study the DCJ distance problem with the maximum matching strategy: given two genomes G1and G2 with the same gene content, find a valid bijection between G1and G2that minimizes the DCJ distance induced by this bijection. We denote the DCJ distance with the maximum matching strategy between G1and G2as d(G1,G2). This problem is NP-hard, which can be proved by a reduction from the NP-hard problem of breakpoint graph decomposition [59].

We now introduce the data structure of adjacency graph, which plays a central role in comput- ing the DCJ distance. We first introduce two more notations. If genes a and b are homologous, we say its corresponding extremities (ahand bh, atand bt) are also homologous. The set of

same gene content, and let S1and S2be the extremity sets of G1and G2, respectively. The

adjacency graph with respect to G1and G2can be written as AG = (V,E), where V = S1∪S2and

E is composed of two types of edges, black edges and gray edges. Two extremities in different

extremity sets (one is in S1and the other one is in S2) are connected by one black edge if they are homologous, and two extremities in the same extremity set are connected by one gray edge if they form an existing adjacency. Figure 3.1(a) gives an example.

We say that a cycle (or path) in the adjacency graph is alternating if any two adjacent edges in this cycle (or path) consist of one black edge and one gray edge. The length of a cycle (or path) is defined as the number of its black edges. A decomposition of the adjacency graph is a set of vertex-disjoint alternating cycles and paths that cover all vertices and all gray edges. We say a decomposition is consistent if for any two homologous genes a and b, either both (ah,bh) and

(at,bt) are in this decomposition, or neither of them is in this decomposition. Figure 3.1(b)

and 3.1(c) give two examples of consistent decompositions.

Given two genomes G1and G2with the same gene content, there is a natural one-to-one correspondence between the set of all possible valid bijections from G1to G2and the set of all possible consistent decompositions of the adjacency graph. In fact, if one valid bijection is given, which maps gene a in G1to a homologous gene b in G2, then we can keep the black edges (ah,bh) and (at,bt) in the decomposition. We do the same thing for every pair of genes

specified by this valid bijection; this process culminates in a consistent decomposition. On the other hand, if we are given a consistent decomposition of the corresponding adjacency graph, we can collect all homologous gene pairs (a,b) indicated by black edges (ah,bh) and (at,bt),

which form a valid bijection from G1to G2. Given a consistent decomposition with c cycles and

a1t a1hb1t b1ha2t a2hc1t c1h a3_h a3t b2h b2t c2h c2t a4t a4h (a) a1t a1hb1t b1ha2t a2hc1t c1h a3_h a3t b2h b2t c2h c2t a4t a4h (b) a1t a1hb1t b1ha2t a2hc1t c1h a3_h a3t b2h b2t c2h c2t a4t a4h (c)

Figure 3.1 – An example of adjacency graph and its two consistent decompositions. Genome 1

contains one linear chromosome, (a1,b1, a2,c1), and genome 2 also contains one linear chro- mosome (−a3_,−b2_,−c2_{, a}4_{). Genes in the same gene family are represented by the same label,} and distinguished by different superscripts. All black edges are represented by long thin lines, and all gray edges are represented by short thick lines. (a) The corresponding adjacency graph, in which head extremities are represented by circles, while tail extremities are represented by diamonds. (b) A consistent decomposition with 2 odd-length paths, whose corresponding valid bijection maps a1 to a3and a2 to a4. (c) Another consistent decomposition with 2 odd-length paths and 1 cycle, whose corresponding valid bijection maps a1_{to a}4_{and a}2_{to a}3_.

o odd-length paths, exactly (|V |/4 − c − o/2) DCJ operations are needed to transform G1into

G2[11]. Thus, we can write d(G1,G2) = minD∈D(|V |/4−cD−oD/2) = |V |/4−maxD∈D(cD+oD/2),

whereD is the space of all consistent decompositions, and cDand oDare the numbers of

cycles and odd-length paths in a decomposition D, respectively. This formula transforms our DCJ distance problem into the maximum cycle decomposition problem, which asks for a consistent decomposition of the adjacency graph such that the number of cycles plus half the number of odd-length paths in this decomposition is maximized.