Sorting genomes with rearrangements and segmental duplications through trajectory graphs

We study the problem of sorting genomes under an evolutionary model that includes genomic rearrangements and segmental duplications. We propose an iterative algorithm to improve any initial evolutionary trajectory between two genomes in terms of parsimony. Our algorithm is based on a new graphical model, the trajectory graph, which models not only the final states of two genomes but also an existing evolutionary trajectory between them. We show that redundant rearrangements in the trajectory correspond to certain cycles in the trajectory graph, and prove that our algorithm converges to an optimal trajectory for any initial trajectory involving only rearrangements

Representing and decomposing genomic structural variants as balanced integer flows on sequence graphs

The study of genomic variation has provided key insights into the functional role of mutations. Predominantly, studies have focused on single nucleotide variants (SNV), which are relatively easy to detect and can be described with rich mathematical models. However, it has been observed that genomes are highly plastic, and that whole regions can be moved, removed or duplicated in bulk. These structural variants (SV) have been shown to have significant impact on the phenotype, but their study has been held back by the combinatorial complexity of the underlying models. We describe here a general model of structural variation that encompasses both balanced rearrangements and arbitrary copy-numbers variants (CNV). In this model, we show that the space of possible evolutionary histories that explain the structural differences between any two genomes can be sampled ergodically

Sorting genomes with rearrangements and segmental duplications through trajectory graphs

We study the problem of sorting genomes under an evolutionary model that includes genomic rearrangements and segmental duplications. We propose an iterative algorithm to improve any initial evolutionary trajectory between two genomes in terms of parsimony. Our algorithm is based on a new graphical model, the trajectory graph, which models not only the final states of two genomes but also an existing evolutionary trajectory between them. We show that redundant rearrangements in the trajectory correspond to certain cycles in the trajectory graph, and prove that our algorithm converges to an optimal trajectory for any initial trajectory involving only rearrangements

Models and Algorithms for Comparative Genomics

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