Mareike Fischer

Revisiting an Equivalence Between Maximum Parsimony and Maximum Likelihood Methods in Phylogenetics

Tuffley and Steel (Bull. Math. Biol. 59:581–607, 1997) proved that maximum likelihood and maximum parsimony methods in phylogenetics are equivalent for sequences of characters under a simple symmetric model of substitution with no common mechanism. This result has been widely cited ever since. We show that small changes to the model assumptions suffice to make the two methods inequivalent. In particular, we analyze the case of bounded substitution probabilities as well as the molecular clock assumption. We show that in these cases, even under no common mechanism, maximum parsimony and maximum likelihood might make conflicting choices. We also show that if there is an upper bound on the substitution probabilities which is ‘sufficiently small’, every maximum likelihood tree is also a maximum parsimony tree (but not vice versa).

On the Complexity of Computing MP Distance Between Binary Phylogenetic Trees

Within the field of phylogenetics there is great interest in distance measures to quantify the dissimilarity of two trees. Recently, a new distance measure has been proposed: the Maximum Parsimony (MP) distance. This is based on the difference of the parsimony scores of a single character on both trees under consideration, and the goal is to find the character which maximizes this difference. Here we show that computation of MP distance on two binary phylogenetic trees is NP-hard. This is a highly nontrivial extension of an earlier NP-hardness proof for two multifurcating phylogenetic trees, and it is particularly relevant given the prominence of binary trees in the phylogenetics literature. As a corollary to the main hardness result we show that computation of MP distance is also hard on binary trees if the number of states available is bounded. In fact, via a different reduction we show that it is hard even if only two states are available. Finally, as a first response to this hardness we give a simple Integer Linear Program (ILP) formulation which is capable of computing the MP distance exactly for small trees (and for larger trees when only a small number of character states are available) and which is used to computationally verify several auxiliary results required by the hardness proofs.

On the Maximum Parsimony Distance Between Phylogenetic Trees

Within the field of phylogenetics there is great interest in distance measures to quantify the dissimilarity of two trees. Here, based on an idea of Bruen and Bryant, we propose and analyze a new distance measure: theMaximum Parsimony (MP) distance. This is based on the difference of the parsimony scores of a single character on both trees under consideration, and the goal is to find the character which maximizes this difference. In this article we show that this new distance is a metric and provides a lower bound to the well-known Subtree Prune and Regraft (SPR) distance. We also show that to compute the MP distance it is sufficient to consider only characters that are convex on one of the trees, and prove several additional structural properties of the distance. On the complexity side, we prove that calculating the MP distance is in general NP-hard, and identify an interesting island of tractability in which the distance can be calculated in polynomial time.

On computing the maximum parsimony score of a phylogenetic network

Phylogenetic networks are used to display the relationship among different species whose evolution is not treelike, which is the case, for instance, in the presence of hybridization events or horizontal gene transfers. Tree inference methods such as maximum parsimony need to be modified in order to be applicable to networks. In this paper, we discuss two different definitions of maximum parsimony on networks, “hardwired” and “softwired,” and examine the complexity of computing them given a network topology and a character. By exploiting a link with the problem Multiterminal Cut, we show that computing the hardwired parsimony score for 2-state characters is polynomial-time solvable, while for characters with more states this problem becomes NP-hard but is still approximable and fixed parameter tractable in the parsimony score. On the other hand we show that, for the softwired definition, obtaining even weak approximation guarantees is already difficult for binary characters and restricted network topologies, and fixed-parameter tractable algorithms in the parsimony score are unlikely. On the positive side we show that computing the softwired parsimony score is fixed-parameter tractable in the level of the network, a natural parameter describing how tangled reticulate activity is in the network. Finally, we show that both the hardwired and the softwired parsimony scores can be computed efficiently using integer linear programming. The software has been made freely available

Maximum parsimony on subsets of taxa

In this paper we investigate mathematical questions concerning the reliability (reconstruction accuracy) of Fitchs maximum parsimony algorithm for reconstructing the ancestral state given a phylogenetic tree and a character. In particular, we consider the question whether the maximum parsimony method applied to a subset of taxa can reconstruct the ancestral state of the root more accurately than when applied to all taxa, and we give an example showing that this indeed is possible. A surprising feature of our example is that ignoring a taxon closer to the root improves the reliability of the method. On the other hand, in the case of the two-state symmetric substitution model, we answer affirmatively a conjecture of Li, Steel and Zhang which states that under a molecular clock the probability that the state at a single taxon is a correct guess of the ancestral state is a lower bound on the reconstruction accuracy of Fitchs method applied to all taxa.

Perfectly Misleading Distances from Ternary Characters

D. Huson and M. Steel showed that for any two binary phylogenetic trees on the same set of n taxa, there exists a sequence of multistate characters that is homoplasy-free only on the first tree but perfectly additive only on the second one. The original construction of such a sequence required n - 1 character states and it remained an open question whether a sequence using fewer character states can always be found. In the present note we will answer this question by showing that three character states suffice to construct such misleading sequences--even if we insist that they conform to an ultrametric (i.e., fit a molecular clock).

Reduction rules for the maximum parsimony distance on phylogenetic trees

In phylogenetics, distances are often used to measure the incongruence between a pair of phylogenetic trees that are reconstructed by different methods or using different regions of genome. Motivated by the maximum parsimony principle in tree inference, we recently introduced the maximum parsimony (MP) distance, which enjoys various attractive properties due to its connection with several other well-known tree distances, such as TBR and SPR. Here we show that computing the MP distance between two trees, a NP-hard problem in general, is fixed parameter tractable in terms of the TBR distance between the tree pair. Our approach is based on two reduction rules--the chain reduction and the subtree reduction--that are widely used in computing TBR and SPR distances. More precisely, we show that reducing chains to length 4 (but not shorter) preserves the MP distance. In addition, we describe a generalization of the subtree reduction which allows the pendant subtrees to be rooted in different places, and show that this still preserves the MP distance. On a slightly different note we also show that Monadic Second Order Logic (MSOL), posited over an auxiliary graph structure known as the display graph (obtained by merging the two trees at their leaves), can be used to obtain an alternative proof that computation of MP distance is fixed parameter tractable in terms of TBR-distance. We conclude with an extended discussion in which we focus on similarities and differences between MP distance and TBR distance and present a number of open problems. One particularly intriguing question, emerging from the MSOL formulation, is whether two trees with bounded MP distance induce display graphs of bounded treewidth.

On the group theoretical background of assigning stepwise mutations onto phylogenies

BackgroundRecently one step mutation matrices were introduced to model the impact of substitutions on arbitrary branches of a phylogenetic tree on an alignment site. This concept works nicely for the four-state nucleotide alphabet and provides an efficient procedure conjectured to compute the minimal number of substitutions needed to transform one alignment site into another. The present paper delivers a proof of the validity of this algorithm. Moreover, we provide several mathematical insights into the generalization of the OSM matrix to multi-state alphabets. The construction of the OSM matrix is only possible if the matrices representing the substitution types acting on the character states and the identity matrix form a commutative group with respect to matrix multiplication. We illustrate this approach by looking at Abelian groups over twenty states and critically discuss their biological usefulness when investigating amino acids.

Comparing the rankings obtained from two biodiversity indices: the Fair Proportion Index and the Shapley Value

The Shapley Value and the Fair Proportion Index of phylogenetic trees have been frequently discussed as prioritization tools in conservation biology. Both indices rank species according to their contribution to total phylogenetic diversity, allowing for a simple conservation criterion. While both indices have their specific advantages and drawbacks, it has recently been shown that both values are closely related. However, as different authors use different definitions of the Shapley Value, the specific degree of relatedness depends on the specific version of the Shapley Value - it ranges from a high correlation index to equality of the indices. In this note, we first give an overview of the different indices. Then we turn our attention to the mere ranking order provided by either of the indices. We compare the rankings obtained from different versions of the Shapley Value for a phylogenetic tree of European amphibians and illustrate their differences. We then undertake further analyses on simulated data and show that even though the chance of two rankings being exactly identical (when obtained from different versions of the Shapley Value) decreases with an increasing number of taxa, the distance between the two rankings converges to zero, i.e., the rankings are becoming more and more alike. Moreover, we introduce our freely available software package FairShapley, which was implemented in Perl and with which all calculations have been performed.

Non-hereditary Maximum Parsimony trees.

In this paper, we investigate a conjecture by Arndt von Haeseler concerning the Maximum Parsimony method for phylogenetic estimation, which was published by the Newton Institute in Cambridge on a list of open phylogenetic problems in 2007. This conjecture deals with the question whether Maximum Parsimony trees are hereditary. The conjecture suggests that a Maximum Parsimony tree for a particular (DNA) alignment necessarily has subtrees of all possible sizes which are most parsimonious for the corresponding subalignments. We answer the conjecture affirmatively for binary alignments on 5 taxa but also show how to construct examples for which Maximum Parsimony trees are not hereditary. Apart from showing that a most parsimonious tree cannot generally be reduced to a most parsimonious tree on fewer taxa, we also show that compatible most parsimonious quartets do not have to provide a most parsimonious supertree. Last, we show that our results can be generalized to Maximum Likelihood for certain nucleotide substitution models.