Computational Performance Evaluation of Two Integer Linear Programming Models for the Minimum Common String Partition Problem

In the minimum common string partition (MCSP) problem two related input strings are given. "Related" refers to the property that both strings consist of the same set of letters appearing the same number of times in each of the two strings. The MCSP seeks a minimum cardinality partitioning of one string into non-overlapping substrings that is also a valid partitioning for the second string. This problem has applications in bioinformatics e.g. in analyzing related DNA or protein sequences. For strings with lengths less than about 1000 letters, a previously published integer linear programming (ILP) formulation yields, when solved with a state-of-the-art solver such as CPLEX, satisfactory results. In this work, we propose a new, alternative ILP model that is compared to the former one. While a polyhedral study shows the linear programming relaxations of the two models to be equally strong, a comprehensive experimental comparison using real-world as well as artificially created benchmark instances indicates substantial computational advantages of the new formulation.Comment: arXiv admin note: text overlap with arXiv:1405.5646 This paper version replaces the one submitted on January 10, 2015, due to detected error in the calculation of the variables involved in the ILP model

Models and Algorithms for Sorting Permutations with Tandem Duplication and Random Loss

A central topic of evolutionary biology is the inference of phylogeny, i. e., the evolutionary history of species. A powerful tool for the inference of such phylogenetic relationships is the arrangement of the genes in mitochondrial genomes. The rationale is that these gene arrangements are subject to different types of mutations in the course of evolution. Hence, a high similarity in the gene arrangement between two species indicates a close evolutionary relation. Metazoan mitochondrial gene arrangements are particularly well suited for such phylogenetic studies as they are available for a wide range of species, their gene content is almost invariant, and usually free of duplicates. With these properties gene arrangements of mitochondrial genomes are modeled by permutations in which each element represents a gene, i. e., a specific genetic sequence. The mutations that shape the gene arrangement of genomes are then represented by operations that rearrange elements in permutations, so-called genome rearrangements, and thereby bridge the gap between evolutionary biology and optimization. Many problems of phylogeny inference can be formulated as challenging combinatorial optimization problems which makes this research area especially interesting for computer scientists. The most prominent examples of such optimization problems are the sorting problem and the distance problem. While the sorting problem requires a minimum length sequence of rearrangements that transforms one given permutation into another given permutation, i. e., it aims for a hypothetical scenario of gene order evolution, the distance problem intends to determine only the length of such a sequence. This minimum length is called distance and used as a (dis)similarity measure quantifying the evolutionary relatedness. Most evolutionary changes occurring in gene arrangements of mitochondrial genomes can be explained by the tandem duplication random loss (TDRL) genome rearrangement model. A TDRL consists of a duplication of a consecutive set of genes in tandem followed by a random loss of one copy of each duplicated gene. In spite of the importance of the TDRL genome rearrangement in mitochondrial evolution, its combinatorial properties have rarely been studied. In addition, models of genome rearrangements which include all types of rearrangement that are relevant for mitochondrial genomes, i. e., inversions, transpositions, inverse transpositions, and TDRLs, while admitting computational tractability are rare. Nevertheless, especially for metazoan gene arrangements the TDRL rearrangement should be considered for the reconstruction of phylogeny. Realizing that a better understanding of the TDRL model is indispensable for the study of mitochondrial gene arrangements, the central theme of this thesis is to broaden the horizon of TDRL genome rearrangements with respect to mitochondrial genome evolution. For this purpose, this thesis provides combinatorial properties of the TDRL model and its variants as well as efficient methods for a plausible reconstruction of rearrangement scenarios between gene arrangements. The methods that are proposed consider all types of genome rearrangements that predominately occur during mitochondrial evolution. More precisely, the main points contained in this thesis are as follows: The distance problem and the sorting problem for the TDRL model are further examined in respect to circular permutations, a formal concept that reflects the circular structure of mitochondrial genomes. As a result, a closed formula for the distance is provided. Recently, evidence for a variant of the TDRL rearrangement model in which the duplicated set of genes is additionally inverted have been found. Initiating the algorithmic study of this new rearrangement model on a certain type of permutations, a closed formula solving the distance problem is proposed as well as a quasilinear time algorithm that solves the corresponding sorting problem. The assumption that only one type of genome rearrangement has occurred during the evolution of certain gene arrangements is most likely unrealistic, e. g., at least three types of rearrangements on top of the TDRL rearrangement have to be considered for the evolution metazoan mitochondrial genomes. Therefore, three different biologically motivated constraints are taken into account in this thesis in order to produce plausible evolutionary rearrangement scenarios. The first constraint is extending the considered set of genome rearrangements to the model that covers all four common types of mitochondrial genome rearrangements. For this 4-type model a sharp lower bound and several close additive upper bounds on the distance are developed. As a byproduct, a polynomial-time approximation algorithm for the corresponding sorting problem is provided that guarantees the computation of pairwise rearrangement scenarios that deviate from a minimum length scenario by at most two rearrangement operations. The second biologically motivated constraint is the relative frequency of the different types of rearrangements occurring during the evolution. The frequency is modeled by employing a weighting scheme on the 4-type model in which every rearrangement is weighted with respect to its type. The resulting NP-hard sorting problem is then solved by means of a polynomial size integer linear program. The third biologically motivated constraint that has been taken into account is that certain subsets of genes are often found in close proximity in the gene arrangements of many different species. This observation is reflected by demanding rearrangement scenarios to preserve certain groups of genes which are modeled by common intervals of permutations. In order to solve the sorting problem that considers all three types of biologically motivated constraints, the exact dynamic programming algorithm CREx2 is proposed. CREx2 has a linear runtime for a large class of problem instances. Otherwise, two versions of the CREx2 are provided: The first version provides exact solutions but has an exponential runtime in the worst case and the second version provides approximated solutions efficiently. CREx2 is evaluated by an empirical study for simulated artificial and real biological mitochondrial gene arrangements

The Distance and Median Problems in the Single-Cut-Or-Join Model with Single-Gene Duplications

Background. In the field of genome rearrangement algorithms, models accounting for gene duplication lead often to hard problems. For example, while computing the pairwise distance is tractable in most duplication-free models, the problem is NP-complete for most extensions of these models accounting for duplicated genes. Moreover, problems involving more than two genomes, such as the genome median and the Small Parsimony problem, are intractable for most duplication-free models, with some exceptions, for example the Single-Cut-or-Join (SCJ) model. Results. We introduce a variant of the SCJ distance that accounts for duplicated genes, in the context of directed evolution from an ancestral genome to a descendant genome where orthology relations between ancestral genes and their descendant are known. Our model includes two duplication mechanisms: single-gene tandem duplication and the creation of single-gene circular chromosomes. We prove that in this model, computing the directed distance and a parsimonious evolutionary scenario in terms of SCJ and single-gene duplication events can be done in linear time. We also show that the directed median problem is tractable for this distance, while the rooted median problem, where we assume that one of the given genomes is ancestral to the median, is NP-complete. We also describe an Integer Linear Program for solving this problem. We evaluate the directed distance and rooted median algorithms on simulated data. Conclusion. Our results provide a simple genome rearrangement model, extending the SCJ model to account for single-gene duplications, for which we prove a mix of tractability and hardness results. For the NP-complete rooted median problem, we design a simple Integer Linear Program. Our publicly available implementation of these algorithms for the directed distance and median problems allow to solve efficiently these problems on large instances

The Rooted SCJ Median with Single Gene Duplications

The median problem is a classical problem in genome rearrangements. It aims to compute a gene order that minimizes the sum of the genomic distances to  k>=3  given gene orders. This problem is intractable except in the related Single-Cut-or-Join and breakpoint rearrangement models. Here we consider the rooted median problem, where we assume one of the given genomes to be ancestral to the median, which is itself ancestral to the other genomes. We show that in the Single-Cut-or-Join model with single gene duplications, the rooted median problem is NP-hard. We also describe an Integer Linear Program for solving this problem, which we apply to simulated data, showing high accuracy of the reconstructed medians

Gene Orthology Inference via Large-Scale Rearrangements for Partially Assembled Genomes

Recently we developed a gene orthology inference tool based on genome rearrangements (Journal of Bioinformatics and Computational Biology 19:6, 2021). Given a set of genomes our method first computes all pairwise gene similarities. Then it runs pairwise ILP comparisons to compute optimal gene matchings, which minimize, by taking the similarities into account, the weighted rearrangement distance between the analyzed genomes (a problem that is NP-hard). The gene matchings are then integrated into gene families in the final step. Although the ILP is quite efficient and could conceptually analyze genomes that are not completely assembled but split in several contigs, our tool failed in completing that task. The main reason is that each ILP pairwise comparison includes an optimal capping that connects each end of a linear segment of one genome to an end of a linear segment in the other genome, producing an exponential increase of the search space. In this work, we design and implement a heuristic capping algorithm that replaces the optimal capping by clustering (based on their gene content intersections) the linear segments into m ? 1 subsets, whose ends are capped independently. Furthermore, in each subset, instead of allowing all possible connections, we let only the ends of content-related segments be connected. Although there is no guarantee that m is much bigger than one, and with the possible side effect of resulting in sub-optimal instead of optimal gene matchings, the heuristic works very well in practice, from both the speed performance and the quality of computed solutions. Our experiments on real data show that we can now efficiently analyze fruit fly genomes with unfinished assemblies distributed in hundreds or even thousands of contigs, obtaining orthologies that are more similar to FlyBase orthologies when compared to orthologies computed by other inference tools. Moreover, for complete assemblies the version with heuristic capping reports orthologies that are very similar to the orthologies computed by the optimal version of our tool. Our approach is implemented into a pipeline incorporating the pre-computation of gene similarities

Exact reconciliation of undated trees

Reconciliation methods aim at recovering macro evolutionary events and at localizing them in the species history, by observing discrepancies between gene family trees and species trees. In this article we introduce an Integer Linear Programming (ILP) approach for the NP-hard problem of computing a most parsimonious time-consistent reconciliation of a gene tree with a species tree when dating information on speciations is not available. The ILP formulation, which builds upon the DTL model, returns a most parsimonious reconciliation ranging over all possible datings of the nodes of the species tree. By studying its performance on plausible simulated data we conclude that the ILP approach is significantly faster than a brute force search through the space of all possible species tree datings. Although the ILP formulation is currently limited to small trees, we believe that it is an important proof-of-concept which opens the door to the possibility of developing an exact, parsimony based approach to dating species trees. The software (ILPEACE) is freely available for download