The Binary Perfect Phylogeny with Persistent characters

The binary perfect phylogeny model is too restrictive to model biological events such as back mutations. In this paper we consider a natural generalization of the model that allows a special type of back mutation. We investigate the problem of reconstructing a near perfect phylogeny over a binary set of characters where characters are persistent: characters can be gained and lost at most once. Based on this notion, we define the problem of the Persistent Perfect Phylogeny (referred as P-PP). We restate the P-PP problem as a special case of the Incomplete Directed Perfect Phylogeny, called Incomplete Perfect Phylogeny with Persistent Completion, (refereed as IP-PP), where the instance is an incomplete binary matrix M having some missing entries, denoted by symbol ?, that must be determined (or completed) as 0 or 1 so that M admits a binary perfect phylogeny. We show that the IP-PP problem can be reduced to a problem over an edge colored graph since the completion of each column of the input matrix can be represented by a graph operation. Based on this graph formulation, we develop an exact algorithm for solving the P-PP problem that is exponential in the number of characters and polynomial in the number of species.Comment: 13 pages, 3 figure

Polynomial supertree methods in phylogenomics: algorithms, simulations and software

One of the objectives in modern biology, especially phylogenetics, is to build larger clades of the Tree of Life. Large-scale phylogenetic analysis involves several serious challenges. The aim of this thesis is to contribute to some of the open problems in this context. In computational phylogenetics, supertree methods provide a way to reconstruct larger clades of the Tree of Life. We present a novel polynomial time approach for the computation of supertrees called FlipCut supertree. Our method combines the computation of minimum cuts from graph-based methods with a matrix representation method, namely Minimum Flip Supertrees. Here, the input trees are encoded in a 0/1/?-matrix. We present a heuristic to search for a minimum set of 0/1-flips such that the resulting matrix admits a directed perfect phylogeny. In contrast to other polynomial time approaches, our results can be interpreted in the sense that we try to minimize a global objective function, namely the number of flips in the input matrix. We extend our approach by using edge weights to weight the columns of the 0/1/?-matrix. In order to compare our new FlipCut supertree method with other recent polynomial supertree methods and matrix representation methods, we present a large scale simulation study using two different data sets. Our findings illustrate the trade-off between accuracy and running time in supertree construction, as well as the pros and cons of different supertree approaches. Furthermore, we present EPoS, a modular software framework for phylogenetic analysis and visualization. It fills the gap between command line-based algorithmic packages and visual tools without sufficient support for computational methods. By combining a powerful graphical user interface with a plugin system that allows simple integration of new algorithms, visualizations and data structures, we created a framework that is easy to use, to extend and that covers all important steps of a phylogenetic analysis

Explaining Evolution via Constrained Persistent Perfect Phylogeny

BACKGROUND: The perfect phylogeny is an often used model in phylogenetics since it provides an efficient basic procedure for representing the evolution of genomic binary characters in several frameworks, such as for example in haplotype inference. The model, which is conceptually the simplest, is based on the infinite sites assumption, that is no character can mutate more than once in the whole tree. A main open problem regarding the model is finding generalizations that retain the computational tractability of the original model but are more flexible in modeling biological data when the infinite site assumption is violated because of e.g. back mutations. A special case of back mutations that has been considered in the study of the evolution of protein domains (where a domain is acquired and then lost) is persistency, that is the fact that a character is allowed to return back to the ancestral state. In this model characters can be gained and lost at most once. In this paper we consider the computational problem of explaining binary data by the Persistent Perfect Phylogeny model (referred as PPP) and for this purpose we investigate the problem of reconstructing an evolution where some constraints are imposed on the paths of the tree. RESULTS: We define a natural generalization of the PPP problem obtained by requiring that for some pairs (character, species), neither the species nor any of its ancestors can have the character. In other words, some characters cannot be persistent for some species. This new problem is called Constrained PPP (CPPP). Based on a graph formulation of the CPPP problem, we are able to provide a polynomial time solution for the CPPP problem for matrices whose conflict graph has no edges. Using this result, we develop a parameterized algorithm for solving the CPPP problem where the parameter is the number of characters. CONCLUSIONS: A preliminary experimental analysis shows that the constrained persistent perfect phylogeny model allows to explain efficiently data that do not conform with the classical perfect phylogeny model

Lonesum (0,1)-matrices and poly-Bernoulli numbers of negative index

This thesis shows that the number of (0,1)-matrices with n rows and k columns uniquely reconstructible from their row and column sums are the poly-Bernoulli numbers of negative index, B[subscript n superscript ( -k)] . Two proofs of this main theorem are presented giving a combinatorial bijection between two poly-Bernoulli formula found in the literature. Next, some connections to Fermat are proved showing that for a positive integer n and prime number p B[subscript n superscript ( -p) congruent 2 superscript n (mod p),] and that for all positive integers (x, y, z, n) greater than two there exist no solutions to the equation: B[subscript x superscript ( -n)] + B[subscript y superscript ( -n)] = B[subscript z superscript ( -n)]. In addition directed graphs with sum reconstructible adjacency matrices are surveyed, and enumerations of similar (0,1)-matrix sets are given as an appendix