On Computing Breakpoint Distances for Genomes with Duplicate Genes

A fundamental problem in comparative genomics is to compute the distance between two genomes in terms of its higher level organization (given by genes or syntenic blocks). For two genomes without duplicate genes, we can easily define (and almost always efficiently compute) a variety of distance measures, but the problem is NP-hard under most models when genomes contain duplicate genes. To tackle duplicate genes, three formulations (exemplar, maximum matching, and any matching) have been proposed, all of which aim to build a matching between homologous genes so as to minimize some distance measure. Of the many distance measures, the breakpoint distance (the number of nonconserved adjacencies) was the first one to be studied and remains of significant interest because of its simplicity and model-free property. The three breakpoint distance problems corresponding to the three formulations have been widely studied. Although we provided last year a solution for the exemplar problem that runs very fast on full genomes, computing optimal solutions for the other two problems has remained challenging. In this article, we describe very fast, exact algorithms for these two problems. Our algorithms rely on a compact integer-linear program that we further simplify by developing an algorithm to remove variables, based on new results on the structure of adjacencies and matchings. Through extensive experiments using both simulations and biological data sets, we show that our algorithms run very fast (in seconds) on mammalian genomes and scale well beyond. We also apply these algorithms (as well as the classic orthology tool MSOAR) to create orthology assignment, then compare their quality in terms of both accuracy and coverage. We find that our algorithm for the any matching formulation significantly outperforms other methods in terms of accuracy while achieving nearly maximum coverage.

A fast and exact algorithm for the exemplar breakpoint distance

Bernard Moret, Mingfu Shao

A fundamental problem in comparative genomics is to compute the distance between two genomes. For two genomes without duplicate genes, we can easily compute a variety of distance measures in linear time, but the problem is NP-hard under most models when genomes contain duplicate genes. Sankoff proposed the use of exemplars to tackle the problem of duplicates genes and gene families: each gene family is represented by a single gene (the exemplar for that family), chosen so as to optimize some metric. Unfortunately, choosing exemplars is itself an NP-hard problem. In this paper, we propose a very fast and exact algorithm to compute the exemplar breakpoint distance, based on new insights in the underlying structure of genome rearrangements and exemplars. We evaluate the performance of our algorithm on simulation data and compare its performance to the best effort to date (a divide-and-conquer approach), showing that our algorithm runs much faster and scales much better. We also devise a new algorithm for the generalized breakpoint distance problem, which can then be applied to assign orthologs. We compare our algorithm with the state-of-the-art method MSOAR by assigning orthologs among five well annotated mammalian genomes, showing that our algorithm runs much faster and is slightly more accurate than MSOAR.

Springer2015

Models and Algorithms for Comparative Genomics

Mingfu Shao

The deluge of sequenced whole-genome data has motivated the study of comparative genomics, which provides global views on genome evolution, and also offers practical solutions in deciphering the functional roles of components of genomes. A fundamental computational problem in whole-genome comparison is to infer the most likely large-scale events~(rearrangements and content-modifying events) of given genomes during their history of evolution. Based on the principle of parsimony, such inference is usually formulated as the so called edit distance problems~(for two genomes) or median problems~(for multiple genomes), i.e., to compute the minimum number of certain types of large-scale events that can explain the differences of the given genomes. In this dissertation, we develop novel algorithms for edit distance problems and median problems and also apply them to analyze and annotate biological datasets. For pairwise whole-genome comparison, we study the most challenging cases of edit distance problems---the given genomes contain duplicate genes. We proposed several exact algorithms and approximation algorithms under various combinations of large-scale events. Specifically, we designed the first exact algorithm to compute the edit distance under the DCJ~(double-cut-and-join) model, and the first exact algorithm to compute the edit distance under a model including DCJ operations and segmental duplications. We devised a

(1.5 + \epsilon)

-approximation algorithm to compute the edit distance under a model including DCJ operations, insertions, and deletions. We also proposed a very fast and exact algorithm to compute the exemplar breakpoint distance. For multiple whole-genome comparison, we study the median problem under the DCJ model. We designed a polynomial-time algorithm using a network flow formulation to compute the so called adequate subgraphs---a central phase in computing the median. We also proved that an existing upper bound of the median distance is tight. These above algorithms determine the correspondence between functional elements~(for instance, genes) across genomes, and thus can be used to systematically infer functional relationships and annotate genomes. For example, we applied our methods to infer orthologs and in-paralogs between a pair of genomes---a key step in analyzing the functions of protein-coding genes. On biological whole-genome datasets, our methods run very fast, scale up to whole genomes, and also achieve very high accuracy.

EPFL2015

Models and Algorithms for Comparative Genomics

Mingfu Shao

(1.5 + \epsilon)

EPFL2015