Cubic time algorithms of amalgamating gene trees and building evolutionary scenarios

Biology Direct

Konstantin Yu Gorbunov²  Leonid Y Rusin¹  Lev I Rubanov²  Vassily A Lyubetsky² 

[1]Faculty of Biology, Moscow State University, Vorob’evy Gory 1/12, Moscow, 119991, Russia

[2]Institute for Information Transmission Problems, The Russian Academy of Sciences (Kharkevich Institute), Bolshoy Karetny per. 19, Moscow, 127994, Russia

关键词: Time slices;    Gene gain;    Horizontal gene transfer;    Gene loss;    Gene duplication;    Evolutionary events;    Supertree;    Tree reconciliation;    Tree amalgamation;    Species tree;    Tree inference;    Fast algorithms;    Phylogenetics;   

Others  :  795171
DOI  :  10.1186/1745-6150-7-48

 received in 2012-09-07, accepted in 2012-12-11,  发布年份 2012

【 摘 要 】

Background

A long recognized problem is the inference of the supertree S that amalgamates a given set {G_j} of trees G_j, with leaves in each G_j being assigned homologous elements.

We ground on an approach to find the tree S by minimizing the total cost of mappings α_j of individual gene trees G_j into S. Traditionally, this cost is defined basically as a sum of duplications and gaps in each α_j. The classical problem is to minimize the total cost, where S runs over the set of all trees that contain an exhaustive non-redundant set of species from all input G_j.

Results

We suggest a reformulation of the classical NP-hard problem of building a supertree in terms of the global minimization of the same cost functional but only over species trees S that consist of clades belonging to a fixed set P (e.g., an exhaustive set of clades in all G_j). We developed a deterministic solving algorithm with a low degree polynomial (typically cubic) time complexity with respect to the size of input data.

We define an extensive set of elementary evolutionary events and suggest an original definition of mapping β of tree G into tree S. We introduce the cost functional c(G, S, f ) and define the mapping β as the global minimum of this functional with respect to the variable f, in which sense it is a generalization of classical mapping α.

We suggest a reformulation of the classical NP-hard mapping (reconciliation) problem by introducing time slices into the species tree S and present a cubic time solving algorithm to compute the mapping β. We introduce two novel definitions of the evolutionary scenario based on mapping β or a random process of gene evolution along a species tree.

Conclusions

Developed algorithms are mathematically proved, which justifies the following statements. The supertree building algorithm finds exactly the global minimum of the total cost if only gene duplications and losses are allowed and the given sets of gene trees satisfies a certain condition. The mapping algorithm finds exactly the minimal mapping β, the minimal total cost and the evolutionary scenario as a minimum over all possible distributions of elementary evolutionary events along the edges of tree S.

The algorithms and their effective software implementations provide useful tools in many biological studies. They facilitate processing of voluminous tree data in acceptable time still largely avoiding heuristics. Performance of the tools is tested with artificial and prokaryotic tree data.

Reviewers

This article was reviewed by Prof. Anthony Almudevar, Prof. Alexander Bolshoy (nominated by Prof. Peter Olofsson), and Prof. Marek Kimmel.

【 授权许可】

   
2012 Lyubetsky et al.; licensee BioMed Central Ltd.

【 图 表 】

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