Maximum Parsimony Problem Research Articles

The tree alignment problem.

BackgroundThe inference of homologies among DNA sequences, that is, positions in multiple genomes that share a common evolutionary origin, is a crucial, yet difficult task facing biologists. Its computational counterpart is known as the multiple sequence alignment problem. There are various criteria and methods available to perform multiple sequence alignments, and among these, the minimization of the overall cost of the alignment on a phylogenetic tree is known in combinatorial optimization as the Tree Alignment Problem. This problem typically occurs as a subproblem of the Generalized Tree Alignment Problem, which looks for the tree with the lowest alignment cost among all possible trees. This is equivalent to the Maximum Parsimony problem when the input sequences are not aligned, that is, when phylogeny and alignments are simultaneously inferred.ResultsFor large data sets, a popular heuristic is Direct Optimization (DO). DO provides a good tradeoff between speed, scalability, and competitive scores, and is implemented in the computer program POY. All other (competitive) algorithms have greater time complexities compared to DO. Here, we introduce and present experiments a new algorithm Affine-DO to accommodate the indel (alignment gap) models commonly used in phylogenetic analysis of molecular sequence data. Affine-DO has the same time complexity as DO, but is correctly suited for the affine gap edit distance. We demonstrate its performance with more than 330,000 experimental tests. These experiments show that the solutions of Affine-DO are close to the lower bound inferred from a linear programming solution. Moreover, iterating over a solution produced using Affine-DO shows little improvement.ConclusionsOur results show that Affine-DO is likely producing near-optimal solutions, with approximations within 10% for sequences with small divergence, and within 30% for random sequences, for which Affine-DO produced the worst solutions. The Affine-DO algorithm has the necessary scalability and optimality to be a significant improvement in the real-world phylogenetic analysis of sequence data.

Parsimony Score of Phylogenetic Networks: Hardness Results and a Linear-Time Heuristic

Phylogenies-the evolutionary histories of groups of organisms-play a major role in representing the interrelationships among biological entities. Many methods for reconstructing and studying such phylogenies have been proposed, almost all of which assume that the underlying history of a given set of species can be represented by a binary tree. Although many biological processes can be effectively modeled and summarized in this fashion, others cannot: recombination, hybrid speciation, and horizontal gene transfer result in networks of relationships rather than trees of relationships. In previous works, we formulated a maximum parsimony (MP) criterion for reconstructing and evaluating phylogenetic networks, and demonstrated its quality on biological as well as synthetic data sets. In this paper, we provide further theoretical results as well as a very fast heuristic algorithm for the MP criterion of phylogenetic networks. In particular, we provide a novel combinatorial definition of phylogenetic networks in terms of "forbidden cycles," and provide detailed hardness and hardness of approximation proofs for the "small" MP problem. We demonstrate the performance of our heuristic in terms of time and accuracy on both biological and synthetic data sets. Finally, we explain the difference between our model and a similar one formulated by Nguyen et al., and describe the implications of this difference on the hardness and approximation results.

A Subdivision Approach to Maximum Parsimony

Determining an optimal phylogenetic tree using maximum parsimony, also referred to as the Steiner tree problem in phylogenetics, is NP hard. Here we provide a new formulation for this problem which leads to an analytical and linear time solution when the dimensionality (sequence length, or number of characters) is at most two. This new formulation of the problem provides a direct link between the maximum parsimony problem and the maximum compatibility problem via the intersection graph. The solution for the “two character case” has numerous practical applications in phylogenetics, some of which are discussed.

Progressive Tree Neighborhood applied to the Maximum Parsimony Problem

The Maximum Parsimony (MP) problem aims at reconstructing a phylogenetic tree from DNA sequences while minimizing the number of genetic transformations. To solve this NP-complete problem, heuristic methods have been developed, often based on local search. In this article, we focus on the influence of the neighborhood relations. After analyzing the advantages and drawbacks of the well-known Nearest Neighbor Interchange (NNI), Subtree Pruning Regrafting (SPR) and Tree-Bisection-Reconnection (TBR) neighborhoods, we introduce the concept of Progressive Neighborhood (PN) which consists in constraining progressively the size of the neighborhood as the search advances. We empirically show that applied to the Maximum Parsimony problem, this progressive neighborhood turns out to be more efficient and robust than the classic neighborhoods using a descent algorithm. Indeed, it allows to find better solutions with a smaller number of iterations or trees evaluated.

Mapping Edge Sets to Splits in Trees: the Path Index and Parsimony

Associated to any finite tree, there are simple vector spaces over $$ {\user2{\mathbb{Z}}}_{2} $$ and linear transformations between them that relate collections of edge-disjoint paths, sets of leaves of even cardinality, and bipartitions of the leaf set. In this paper, we use this connection to introduce and study an apparently new integer-valued invariant, the ‘path index’ of a tree. In the case of trivalent (or ‘binary’) trees, this index has an interesting recursive description that allows its easy calculation implying in particular that the path index of such a tree never exceeds one quarter of the number of its leaves and coincides with that number exclusively for the (unique!) trivalent tree with four leaves. We then show how our algebraic perspective has some other uses—for example, it relates to Hadamard conjugation first described by Mike Hendy, and it provides a way to study a combinatorial optimization problem considered in phylogenetics called the small maximum parsimony problem.

PRec-I-DCM3: a parallel framework for fast and accurate large-scale phylogeny reconstruction

Accurate reconstruction of phylogenetic trees often involves solving hard optimisation problems, particularly the Maximum Parsimony (MP) and Maximum Likelihood (ML) problems. Various heuristics yield good results for these problems within reasonable time only on small datasets. This is a major impediment for large-scale phylogeny reconstruction. Roshan et al. introduced Rec-I-DCM3, an efficient and accurate meta-method for solving the MP problem on large datasets of up to 14,000 taxa. We improve the performance of Rec-I-DCM3 via parallelisation. The experiments demonstrate that our parallel method, PRec-I-DCM3, achieves significant improvements, both in speed and accuracy, over its sequential counterpart.

Landscapes on spaces of trees

Combinatorial optimization problems defined on sets of phylogenetic trees are an important issue in computational biology, for instance the problem of reconstructing a phylogeny using maximum likelihood or parsimony approaches. The collection of possible phylogenetic trees is arranged as a so-called Robinson graph by means of the nearest neighborhood interchange move. The coherent algebra and spectra of Robinson graphs are discussed in some detail as their knowledge is important for an understanding of the landscape structure. We consider simple model landscapes as well as landscapes arising from the maximum parsimony problem, focusing on two complementary measures of ruggedness: the amplitude spectrum arising from projecting the cost functions onto the eigenspaces of the underlying graph and the topology of local minima and their connecting saddle points.

Proof of the populous path algorithm for missing mutations in parsimony trees

In the maximum parsimony problem, one produces an ancestral branching arrangement (phylogenetic tree, dendrogram) and ancestral messenger RNA sequences which account for contemporary amino acid sequences with the fewest number of nucleotide replacements consistent with the genetic code. This parsimony approach is not likely to capture the full extent of mutational change on branches of the tree which are represented by relatively few contemporary species, either due to inadequate data or extinction. A computer algorithm introduced by Goodman, Moore, Barnabas & Matsuda (1974) seeks to equalize this disparity by propagating mutational information from pairs of points in the dendrogram with many intervening links to pairs of points with few intervening links. Two pairs of points are assumed to be comparable if the number of aligned, non-matching nucleotides for one pair of points equals that for the other pair. This paper demonstrates mathematically that the Goodman et al. (1974) algorithm adds the minimum possible new mutations to the tree, consistent with such a propagation procedure.