An efficiently computed lower bound on the number of recombinations in phylogenetic networks: Theory and empirical study

Abstract

Phylogenetic networks are models of sequence evolution that go beyond trees, allowing biological operations that are not tree-like. One of the most important biological operations is recombination between two sequences. An established problem [J. Hein, Reconstructing evolution of sequences subject to recombination using parsimony, Math. Biosci. 98 (1990) 185-200; J. Hein, A heuristic method to reconstruct the history of sequences subject to recombination, J. Molecular Evoluation 36 (1993) 396-405;Y. Song, J. Hein, Parsimonious reconstruction of sequence evolution and haplotype blocks: finding the minimum number of recombination events, reconstruction of root-unknown phylogenetic networks with constrained and structured recombination, J. Comput. Systems Sci. 70 (2005) 381-398] is to find a phylogenetic network that derives an input set of sequences, minimizing the number of recombinations used. No efficient, general algorithm is known for this problem. Several papers consider the problem of computing a lower bound on the number of recombinations needed. In this paper we establish a new, efficiently computed lower bound. This result is useful in methods to estimate the number of needed recombinations, and also to prove the optimality of algorithms for constructing phylogenetic networks under certain conditions [D. Gusfield, S. Eddhu, C. Langley, Optimal, efficient reconstruction of phylogenetic networks with constrained recombination, J. Bioinform. Comput. Biol. 2(1) (2004) 173-213; D. Gusfield, Optimal, efficient reconstruction of root-unknown phylogenetic networks with constrained and structured recombination, J. Comput. Systems Sci. 70 (2005) 381-398; D. Gusfield, Optimal, efficient reconstruction of root-unknown phylogenetic networks with constrained recombination, Technical Report, ]. The lower bound is based on a structural, combinatorial insight, using only the site conflicts and incompatibilities, and hence it is fundamental and applicable to many biological phenomena other than recombination, for example, when gene conversions or recurrent or back mutations or cross-species hybridizations cause the phylogenetic history et al. / Discrete Applied Mathematics 155 (2007) 806 -830 807 to deviate from a tree structure. In addition to establishing the bound, we examine its use in more complex lower bound methods, and compare the bounds obtained to those obtained by other established lower bound methods.

A Decomposition Theory for Phylogenetic Networks and Incompatible Characters

“…The algorithmic problem of reconstructing a history of recombination events (with mutations), or determining the minimum number of recombination nodes needed in a phylogenetic network (for both rooted and unrooted problems), has been studied in a number of papers (Gusfield et al, 2007;Hein, 1990Hein, , 1993Hein, 2003, 2004;Wang et al, 2001;Myers and Griffiths, 2003;Hudson and Kaplan, 1985;Kececioglu and Gusfield, 1998;Gusfield, 2005a;Gusfield et al, 2004b,a;Nakhleh et al, 2003Nakhleh et al, , 2004Moret et al, 2004;Bansal, 2004, 2006a;Lyngso et al, 2005;Song et al, 2006;Myers, 2003).…”

“…The incompatibility graph in Figure 2 has two components. Previously (Gusfield et al, 2004b;Gusfield, 2005a;Gusfield et al, 2007Gusfield et al, ,2004aBafna and Bansal, 2004), the non-trivial connected components of the conflict and incompatibility graphs were shown to be very informative, used both to derive efficient algorithms and to expose combinatorial structure in phylogenetic networks. The structural importance of the non-trivial connected components is further developed in the main result, presented next.…”

“…The main tools that we used in Gusfield et al (2004bGusfield et al ( , 2007; Gusfield (2005a); Bafna and Bansal (2004); Song (2006b) and other papers were two graphs representing "incompatibilities" and "conflicts" between sites. We introduce these graphs here.…”

Constructing the Simplest Possible Phylogenetic Network from Triplets

“…Gusfield et al gave a biological justification for level-1 networks (which they call "galled trees") [9]. Minimising reticulations has been very well studied in the framework where the input consists of (binary) sequences [10] [11][23] [24]. For example, Wang et al considered the problem of finding a "perfect phylogeny" with a minimum number of reticulations and showed that this problem is NP-hard [25].…”

A new algorithm to construct phylogenetic networks from trees

“…Here, the task is to construct a rooted phylogenetic network from a set of binary sequences. The third is horizontal gene transfer networks, where the goal is to explain the discrepancies between gene trees and a species tree (e.g., Gusfield et al, 2007;Semple, 2007;Nakleh, 2009;van Iersel et al, 2010). The fourth is the construction of rooted phylogenetic networks from triplets (van Iersel et al, 2008;van Iersel and Kelk, 2011).…”