Efficient Algorithms for Genomic Duplication Models

Predicting the Evolution of Syntenies—An Algorithmic Review

Syntenies are genomic segments of consecutive genes identified by a certain conservation in gene content and order. The notion of conservation may vary from one definition to another, the more constrained requiring identical gene contents and gene orders, while more relaxed definitions just require a certain similarity in gene content, and not necessarily in the same order. Regardless of the way they are identified, the goal is to characterize homologous genomic regions, i.e., regions deriving from a common ancestral region, reflecting a certain gene co-evolution that can enlighten important functional properties. In addition of being able to identify them, it is also necessary to infer the evolutionary history that has led from the ancestral segment to the extant ones. In this field, most algorithmic studies address the problem of inferring rearrangement scenarios explaining the disruption in gene order between segments with the same gene content, some of them extending the evolutionary model to gene insertion and deletion. However, syntenies also evolve through other events modifying their content in genes, such as duplications, losses or horizontal gene transfers, i.e., the movement of genes from one species to another. Although the reconciliation approach between a gene tree and a species tree addresses the problem of inferring such events for single-gene families, little effort has been dedicated to the generalization to segmental events and to syntenies. This paper reviews some of the main algorithmic methods for inferring ancestral syntenies and focus on those integrating both gene orders and gene trees.

Evolution through segmental duplications and losses: a Super-Reconciliation approach

The classical gene and species tree reconciliation, used to infer the history of gene gain and loss explaining the evolution of gene families, assumes an independent evolution for each family. While this assumption is reasonable for genes that are far apart in the genome, it is not appropriate for genes grouped into syntenic blocks, which are more plausibly the result of a concerted evolution. Here, we introduce the Super-Reconciliation problem which consists in inferring a history of segmental duplication and loss events (involving a set of neighboring genes) leading to a set of present-day syntenies from a single ancestral one. In other words, we extend the traditional Duplication-Loss reconciliation problem of a single gene tree, to a set of trees, accounting for segmental duplications and losses. Existency of a Super-Reconciliation depends on individual gene tree consistency. In addition, ignoring rearrangements implies that existency also depends on gene order consistency. We first show that the problem of reconstructing a most parsimonious Super-Reconciliation, if any, is NP-hard and give an exact exponential-time algorithm to solve it. Alternatively, we show that accounting for rearrangements in the evolutionary model, but still only minimizing segmental duplication and loss events, leads to an exact polynomial-time algorithm. We finally assess time efficiency of the former exponential time algorithm for the Duplication-Loss model on simulated datasets, and give a proof of concept on the opioid receptor genes.

Polynomial-Time Algorithms for Phylogenetic Inference Problems involving duplication and reticulation

A common problem in phylogenetics is to try to infer a species phylogeny from gene trees. We consider different variants of this problem. The first variant, called Unrestricted Minimal Episodes Inference, aims at inferring a species tree based on a model with speciation and duplication where duplications are clustered in duplication episodes. The goal is to minimize the number of such episodes. The second variant, Parental Hybridization, aims at inferring a species network based on a model with speciation and reticulation. The goal is to minimize the number of reticulation events. It is a variant of the well-studied Hybridization Number problem with a more generous view on which gene trees are consistent with a given species network. We show that these seemingly different problems are in fact closely related and can, surprisingly, both be solved in polynomial time, using a structure we call "beaded trees". However, we also show that methods based on these problems have to be used with care because the optimal species phylogenies always have a restricted form. To mitigate this problem, we introduce a new variant of Unrestricted Minimal Episodes Inference that minimizes the duplication episode depth. We prove that this new variant of the problem can also be solved in polynomial time.

Reconciling multiple genes trees via segmental duplications and losses

Reconciling gene trees with a species tree is a fundamental problem to understand the evolution of gene families. Many existing approaches reconcile each gene tree independently. However, it is well-known that the evolution of gene families is interconnected. In this paper, we extend a previous approach to reconcile a set of gene trees with a species tree based on segmental macro-evolutionary events, where segmental duplication events and losses are associated with cost δ and λ , respectively. We show that the problem is polynomial-time solvable when δ ≤ λ (via LCA-mapping), while if δ > λ the problem is NP-hard, even when λ = 0 and a single gene tree is given, solving a long standing open problem on the complexity of multi-gene reconciliation. On the positive side, we give a fixed-parameter algorithm for the problem, where the parameters are δ / λ and the number d of segmental duplications, of time complexity O ⌈ δ λ ⌉ d · n · δ λ . Finally, we demonstrate the usefulness of this algorithm on two previously studied real datasets: we first show that our method can be used to confirm or raise doubt on hypothetical segmental duplications on a set of 16 eukaryotes, then show how we can detect whole genome duplications in yeast genomes.

Inferring duplication episodes from unrooted gene trees

Background

One of evolutionary molecular biology fundamental issues is to discover genomic duplication events and their correspondence to the species tree. Such events can be reconstructed by clustering single gene duplications inferred by reconciling a set of gene trees with a species tree.

Results

Here we propose the first solutions to the genomic duplication problem in which every reconciliation with the minimal number of single gene duplications is allowed and the method of clustering called minimum episodes under the assumption that input gene trees are unrooted.

Conclusions

We showed new theoretical properties of unrooted reconciliation for the duplication cost and apply them to design several exact and heuristic algorithms for solving the problem. Our evaluation study on empirical dataset confirmed several genomic duplication events from the literature and demonstrate that algorithms can be successfully applied.