Krister M. Swenson

Genomic distances under deletions and insertions

As more and more genomes are sequenced, evolutionary biologists are becoming increasingly interested in evolution at the level of whole genomes, in scenarios in which the genome evolves through insertions, deletions, and movements of genes along its chromosomes. In the mathematical model pioneered by Sankoff and others, a unichromosomal genome is represented by a signed permutation of a multi-set of genes; Hannenhalli and Pevzner showed that the edit distance between two signed permutations of the same set can be computed in polynomial time when all operations are inversions. El-Mabrouk extended that result to allow deletions and a limited form of insertions (which forbids duplications). In this paper we extend El-Mabrouks work to handle duplications as well as insertions and present an alternate framework for computing (near) minimal edit sequences involving insertions, deletions, and inversions. We derive an error bound for our polynomial-time distance computation under various assumptions and present preliminary experimental results that suggest that performance in practice may be excellent, within a few percent of the actual distance.

Approximating the true evolutionary distance between two genomes

As more and more genomes are sequenced, evolutionary biologists are becoming increasingly interested in evolution at the level of whole genomes, in scenarios in which the genome evolves through insertions, duplications, deletions, and movements of genes along its chromosomes. In the mathematical model pioneered by Sankoff and others, a unichromosomal genome is represented by a signed permutation of a multiset of genes; Hannenhalli and Pevzner showed that the edit distance between two signed permutations of the same set can be computed in polynomial time when all operations are inversions. El-Mabrouk extended that result to allow deletions and a limited form of insertions (which forbids duplications); in turn we extended it to compute a nearly optimal edit sequence between an arbitrary genome and the identity permutation. In this paper we generalize our approach to compute distances between two arbitrary genomes, but focus on approximating the true evolutionary distance rather than the edit distance. We present experimental results showing that our algorithm produces excellent estimates of the true evolutionary distance up to a (high) threshold of saturation; indeed, the distances thus produced are good enough to enable the simple neighbor-joining procedure to reconstruct our test trees with high accuracy.

Sorting Signed Permutations by Inversions in O(nlogn) Time

The study of genomic inversions (or reversals) has been a mainstay of computational genomics for nearly 20 years. After the initial breakthrough of Hannenhalli and Pevzner, who gave the first polynomial-time algorithm for sorting signed permutations by inversions, improved algorithms have been designed, culminating with an optimal linear-time algorithm for computing the inversion distance and a subquadratic algorithm for providing a shortest sequence of inversions--also known as sorting by inversions. Remaining open was the question of whether sorting by inversions could be done in O (n logn ) time. In this paper, we present a qualified answer to this question, by providing two new sorting algorithms, a simple and fast randomized algorithm and a deterministic refinement. The deterministic algorithm runs in time O (n logn + kn ), where k is a data-dependent parameter. We provide the results of extensive experiments showing that both the average and the standard deviation for k are small constants, independent of the size of the permutation. We conclude (but do not prove) that almost all signed permutations can be sorted by inversions in O (n logn ) time.

Hurdles Hardly Have to Be Heeded

As data about genomic architecture accumulates, genomic rearrangements have attracted increasing attention. One of the main rearrangement mechanisms, inversions (also called reversals), was characterized by Hannenhalli and Pevzner and this characterization in turn extended by various authors. The characterization relies on the concepts of breakpoints, cycles, and obstructions colorfully named hurdles and fortresses. In this paper, we study the probability of generating a hurdle in the process of sorting a permutation if one does not take special precautions to avoid them (as in a randomized algorithm, for instance). To do this we revisit and extend the work of Caprara and of Bergeron by providing simple and exact characterizations of the probability of encountering a hurdle in a random permutation. Using similar methods we, for the first time, find an asymptotically tight analysis of the probability that a fortress exists in a random permutation.

Gene tree correction guided by orthology

BackgroundReconciled gene trees yield orthology and paralogy relationships between genes. This information may however contradict other information on orthology and paralogy provided by other footprints of evolution, such as conserved synteny.ResultsWe explore a way to include external information on orthology in the process of gene tree construction. Given an initial gene tree and a set of orthology constraints on pairs of genes or on clades, we give polynomial-time algorithms for producing a modified gene tree satisfying the set of constraints, that is as close as possible to the original one according to the Robinson-Foulds distance. We assess the validity of the modifications we propose by computing the likelihood ratio between initial and modified trees according to sequence alignments on Ensembl trees, showing that often the two trees are statistically equivalent.AvailabilitySoftware and data available upon request to the corresponding author.

Gene tree correction for reconciliation and species tree inference

BackgroundReconciliation is the commonly used method for inferring the evolutionary scenario for a gene family. It consists in “embedding” inferred gene trees into a known species tree, revealing the evolution of the gene family by duplications and losses. When a species tree is not known, a natural algorithmic problem is to infer a species tree from a set of gene trees, such that the corresponding reconciliation minimizes the number of duplications and/or losses. The main drawback of reconciliation is that the inferred evolutionary scenario is strongly dependent on the considered gene trees, as few misplaced leaves may lead to a completely different history, with significantly more duplications and losses.ResultsIn this paper, we take advantage of certain gene trees’ properties in order to preprocess them for reconciliation or species tree inference. We flag certain duplication vertices of a gene tree, the “non-apparent duplication” (NAD) vertices, as resulting from the misplacement of leaves. In the case of species tree inference, we develop a polynomial-time heuristic for removing the minimum number of species leading to a set of gene trees that exhibit no NAD vertices with respect to at least one species tree. In the case of reconciliation, we consider the optimization problem of removing the minimum number of leaves or species leading to a tree without any NAD vertex. We develop a polynomial-time algorithm that is exact for two special classes of gene trees, and show a good performance on simulated data sets in the general case.

OMG! orthologs in multiple genomes: competing graph-theoretical formulations

From the set of all pairwise homologies, weighted by sequence similarities, among a set of genomes, we seek disjoint orthology sets of genes, in which each element is orthogonal to all other genes (on a different genome) in the same set. In a graph-theoretical formulation, where genes are vertices and weighted edges represent homologies, we suggest three criteria, with three different biological motivations, for evaluating the partition of genes produced by deletion of a subset of edges: i) minimum weight edge removal, ii) minimum degree-zero vertex creation, and iii) maximum number of edges in the transitive closure of the graph after edge deletion. For each of the problems, all either proved or conjectured to be NP-hard, we suggest approximate and heuristic algorithms of finding orthology sets satisfying the criteria, and show how to incorporate genomes that have a whole genome duplication event in their immediate lineage. We apply this to ten flowering plant genomes, involving 160,000 different genes in given pairwise homologies. We evaluate the results in a number of ways and recommend criterion iii) as best suited to applications to multiple gene order alignment.

Uncovering hidden phylogenetic consensus

Many of the steps in phylogenetic reconstruction can be confounded by “rogue” taxa, taxa that cannot be placed with assurance anywhere within the tree—whose location within the tree, in fact, varies with almost any choice of algorithm or parameters. Phylogenetic consensus methods, in particular, are known to suffer from this problem. In this paper we provide a novel framework in which to define and identify rogue taxa. In this framework, we formulate a bicriterion optimization problem that models the net increase in useful information present in the consensus tree when certain taxa are removed from the input data. We also provide an effective greedy heuristic to identify a subset of rogue taxa and use it in a series of experiments, using both pathological examples described in the literature and a collection of large biological datasets. As the presence of rogue taxa in a set of bootstrap replicates can lead to deceivingly poor support values, we propose a procedure to recompute support values in light of the rogue taxa identified by our algorithm; applying this procedure to our biological datasets caused a large number of edges to change from “unsupported” to “supported” status, indicating that many existing phylogenies should be recomputed and reevaluated to reduce any inaccuracies introduced by rogue taxa.

Inversion-based genomic signatures

BackgroundReconstructing complete ancestral genomes (at least in terms of their gene inventory and arrangement) is attracting much interest due to the rapidly increasing availability of whole genome sequences. While modest successes have been reported for mammalian and even vertebrate genomes, more divergent groups continue to pose a stiff challenge, mostly because current models of genomic evolution support too many choices.ResultsWe describe a novel type of genomic signature based on rearrangements that characterizes evolutionary changes that must be common to all minimal rearrangement scenarios; by focusing on global patterns of rearrangements, such signatures bypass individual variations and sharply restrict the search space. We present the results of extensive simulation studies demonstrating that these signatures can be used to reconstruct accurate ancestral genomes and phylogenies even for widely divergent collections.ConclusionFocusing on genome triples rather than genomes pairs unleashes the full power of evolutionary analysis. Our genomic signature captures shared evolutionary events and thus can form the basis of a robust analysis and reconstruction of evolutionary history.

A framework for orthology assignment from gene rearrangement data

Gene rearrangements have been used successfully in phylogenetic reconstruction and comparative genomics, but usually under the assumption that all genomes have the same gene content and that no gene is duplicated. While these assumptions allow one to work with organellar genomes, they are too restrictive for nuclear genomes. The main challenge in handling more realistic data is how to deal with gene families, specifically, how to identify orthologs. While searching for orthologies is a common task in computational biology, it is usually done using sequence data. Sankoff first addressed the problem in 1999, introducing the notion of exemplar, but his approach uses an NP-hard optimization step to discard all but one member (the exemplar) of each gene family, losing much valuable information in the process. We approach the problem using all available data in the gene orders and gene families, provide an optimization framework in which to phrase the problem, and present some preliminary theoretical results.