Screening synteny blocks in pairwise genome comparisons through integer programming

<p>Abstract</p> <p>Background</p> <p>It is difficult to accurately interpret chromosomal correspondences such as true orthology and paralogy due to significant divergence of genomes from a common ancestor. Analyses are particularly problematic among lineages that have repeatedly experienced whole genome duplication (WGD) events. To compare multiple "subgenomes" derived from genome duplications, we need to relax the traditional requirements of "one-to-one" syntenic matchings of genomic regions in order to reflect "one-to-many" or more generally "many-to-many" matchings. However this relaxation may result in the identification of synteny blocks that are derived from ancient shared WGDs that are not of interest. For many downstream analyses, we need to eliminate weak, low scoring alignments from pairwise genome comparisons. Our goal is to objectively select subset of synteny blocks whose total scores are maximized while respecting the duplication history of the genomes in comparison. We call this "quota-based" screening of synteny blocks in order to appropriately fill a quota of syntenic relationships within one genome or between two genomes having WGD events.</p> <p>Results</p> <p>We have formulated the synteny block screening as an optimization problem known as "Binary Integer Programming" (BIP), which is solved using existing linear programming solvers. The computer program QUOTA-ALIGN performs this task by creating a clear objective function that maximizes the compatible set of synteny blocks under given constraints on overlaps and depths (corresponding to the duplication history in respective genomes). Such a procedure is useful for any pairwise synteny alignments, but is most useful in lineages affected by multiple WGDs, like plants or fish lineages. For example, there should be a 1:2 ploidy relationship between genome A and B if genome B had an independent WGD subsequent to the divergence of the two genomes. We show through simulations and real examples using plant genomes in the rosid superorder that the quota-based screening can eliminate ambiguous synteny blocks and focus on specific genomic evolutionary events, like the divergence of lineages (in cross-species comparisons) and the most recent WGD (in self comparisons).</p> <p>Conclusions</p> <p>The QUOTA-ALIGN algorithm screens a set of synteny blocks to retain only those compatible with a user specified ploidy relationship between two genomes. These blocks, in turn, may be used for additional downstream analyses such as identifying true orthologous regions in interspecific comparisons. There are two major contributions of QUOTA-ALIGN: 1) reducing the block screening task to a BIP problem, which is novel; 2) providing an efficient software pipeline starting from all-against-all BLAST to the screened synteny blocks with dot plot visualizations. Python codes and full documentations are publicly available <url>http://github.com/tanghaibao/quota-alignment</url>. QUOTA-ALIGN program is also integrated as a major component in SynMap <url>http://genomevolution.com/CoGe/SynMap.pl</url>, offering easier access to thousands of genomes for non-programmers.</p

Local improvement algorithms for a path packing problem: A performance analysis based on linear programming

Given a graph, we wish to find a maximum number of vertex-disjoint paths of length 2. We propose a series of local improvement algorithms for this problem, and present a linear-programming based method for analyzing their performance

Nonoverlapping Local Alignments (Weighted Independent Sets of Axis Parallel Rectangles)

We consider the following problem motivated by an application in computational molecular biology. We are given a set of weighted axis-parallel rectangles such that for any pair of rectangles and either axis, the projection of one rectangle does not enclose that of the other. Define a pair to be independent if their projections in both axes are disjoint. The problem is to find a maximum-weight independent subset of rectangles. We show that the problem is NP-hard even in the uniform case when all the weights are the same. We analyze the performance of a natural local-improvement heuristic for the general problem and prove a performance ratio of 3.25. We extend the heuristic to the problem of finding a maximum-weight independent set in (d + 1)-claw free graphs, and show a tight performance ratio of d \Gamma 1 + 1 d . A performance ratio of d 2 was known for the heuristic when applied to the uniform case. Our contributions are proving the hardness of the problem and providing a tight..

Interval scheduling and colorful independent sets

Numerous applications in scheduling, such as resource allocation or steel manufacturing, can be modeled using the NP-hard Independent Set problem (given an undirected graph and an integer k, find a set of at least k pairwise non-adjacent vertices). Here, one encounters special graph classes like 2-union graphs (edge-wise unions of two interval graphs) and strip graphs (edge-wise unions of an interval graph and a cluster graph), on which Independent Set remains NP-hard but admits constant-ratio approximations in polynomial time. We study the parameterized complexity of Independent Set on 2-union graphs and on subclasses like strip graphs. Our investigations significantly benefit from a new structural "compactness" parameter of interval graphs and novel problem formulations using vertex-colored interval graphs. Our main contributions are: 1. We show a complexity dichotomy: restricted to graph classes closed under induced subgraphs and disjoint unions, Independent Set is polynomial-time solvable if both input interval graphs are cluster graphs, and is NP-hard otherwise. 2. We chart the possibilities and limits of effective polynomial-time preprocessing (also known as kernelization). 3. We extend Halld\'orsson and Karlsson (2006)'s fixed-parameter algorithm for Independent Set on strip graphs parameterized by the structural parameter "maximum number of live jobs" to show that the problem (also known as Job Interval Selection) is fixed-parameter tractable with respect to the parameter k and generalize their algorithm from strip graphs to 2-union graphs. Preliminary experiments with random data indicate that Job Interval Selection with up to fifteen jobs and 5*10^5 intervals can be solved optimally in less than five minutes.Comment: This revision does not contain Theorem 7 of the first revision, whose proof contained an erro

Discovery of Unconventional Patterns for Sequence Analysis: Theory and Algorithms

The biology community is collecting a large amount of raw data, such as the genome sequences of organisms, microarray data, interaction data such as gene-protein interactions, protein-protein interactions, etc. This amount is rapidly increasing and the process of understanding the data is lagging behind the process of acquiring it. An inevitable first step towards making sense of the data is to study their regularities focusing on the non-random structures appearing surprisingly often in the input sequences: patterns. In this thesis we discuss three incarnations of the pattern discovery task, exploring three types of patterns that can model different regularities of the input dataset. While mask patterns have been designed to model short repeated biological sequences, showing a high conservation of their content at some specific positions, permutation patterns have been designed to detect repeated patterns whose parts maintain their physical adjacency but not their ordering in all the pattern occurrences. Transposons, instead, model mobile sequences in the input dataset, which can be discovered by comparing different copies of the same input string, detecting large insertions and deletions in their alignment