Exactly Computing the Parsimony Scores on Phylogenetic Networks Using Dynamic Programming

Abstract

Scoring a given phylogenetic network is the first step that is required in searching for the best evolutionary framework for a given dataset. Using the principle of maximum parsimony, we can score phylogenetic networks based on the minimum number of state changes across a subset of edges of the network for each character that are required for a given set of characters to realize the input states at the leaves of the networks. Two such subsets of edges of networks are interesting in light of studying evolutionary histories of datasets: (i) the set of all edges of the network, and (ii) the set of all edges of a spanning tree that minimizes the score. The problems of finding the parsimony scores under these two criteria define slightly different mathematical problems that are both NPhard. In this article, we show that both problems, with scores generalized to adding substitution costs between states on the endpoints of the edges, can be solved exactly using dynamic programming. We show that our algorithms require O(m p k) storage at each vertex (per character), where k is the number of states the character can take, p is the number of reticulate vertices in the network, m = k for the problem with edge set (i), and m = 2 for the problem with edge set (ii). This establishes an O(nm p k 2 ) algorithm for both the problems (n is the number of leaves in the network), which are extensions of Sankoff's algorithm for finding the parsimony scores for phylogenetic trees. We will discuss improvements in the complexities and show that for phylogenetic networks whose underlying undirected graphs have disjoint cycles, the storage at each vertex can be reduced to O(mk), thus making the algorithm polynomial for this class of networks. We will present some properties of the two approaches and guidance on choosing between the criteria, as well as traverse through the network space using either of the definitions. We show that our methodology provides an effective means to study a wide variety of datasets.

Evidence for a Resting State Network Abnormality in Adults Who Stutter

“…Spanning trees are subsets of graphs that cover all nodes without any loop ( Stam, 2014 ; Tewarie et al, 2015 ). The MST is the tree with the lowest total cost ( Graham and Hell, 1985 ); the unique, unweighted, binary graph that is made by the shortest path between all pairs of nodes without the occurrence of a loop. Therefore, the minimum connectivity cost involved in spanning all nodes is recovered by MST analysis ( Tewarie et al, 2015 ).…”

Molecular phylogeny ofIndo‐Pacific carpenter ants (Hymenoptera:Formicidae,Camponotus) reveals waves of dispersal and colonization from diverse source areas

“…We remain open to the hypothesis that hybridization has contributed to the lack of deep phylogenetic resolution and of distinct morphological characters that are useful for taxonomic purposes. Using a manageable subset of our data to develop a parsimony network analysis, it was found that certain species were best explained as the direct results of reticulate events (the joining of lineages through time) in the most parsimonious network (Kannan and Wheeler, ), and we recommend expanding this line of investigation with Camponotus going forward.…”

Phylogenetic network analysis as a parsimony optimization problem

“…Hardwired costs on the other hand ( H ( N , C ) score ) do not depend on display trees, but are the sum of the weights of all edges ( e ) in the network N , where the edge weights ( w ( e )) are the minimum number of character changes between vertex states that bound each edge [ 21 , 22 ]. …”