Using the tangle: A consistent construction of phylogenetic distance matrices for quartets

Abstract

Distance based algorithms are a common technique in the construction of phylogenetic trees from taxonomic sequence data. The first step in the implementation of these algorithms is the calculation of a pairwise distance matrix to give a measure of the evolutionary change between any pair of the extant taxa. A standard technique is to use the log det formula to construct pairwise distances from aligned sequence data. We review a distance measure valid for the most general models, and show how the log det formula can be used as an estimator thereof. We then show that the foundation upon which the log det formula is constructed can be generalized to produce a previously unknown estimator which improves the consistency of the distance matrices constructed from the log det formula. This distance estimator provides a consistent technique for constructing quartets from phylogenetic sequence data under the assumption of the most general Markov model of sequence evolution. * Australian Postgraduate Award † Alexander von Humboldt Fellow

Adventures in invariant theory

“…In data sets where the number of species K is large, where the pairwise nature of logDet can lead to significant loss of evolutionary information, they may also provide alternative or supplementary information to help with inference. In view of the pevious discussion of quantum entanglement, it turns out that for the case of binary characters (D = 2), and three-fold arrays (K = 3) or tripartite marginalisations of higher arity arrays, the Cayley hyperdeterminant (degree n = 4) is precisely such a candidate [27], and we have identified analogous low-degree 'tangles' for D = 3 and 4 [28]. For four taxa, K = 4, and four characters, D = 4, we have found a remarkable, symmetrical set of three degree-five, n = 5, Markov invariants dubbed the 'squangles' (stochastic quartet tangles) [26,29,11].…”

Systematics and symmetry in molecular phylogenetic modelling: perspectives from physics

“…As a first example we look at the famous tangle quantity, quartic in three qubit wavefunctions whose values are known to distinguish between the different classes of entanged states in tripartite systems [62]. The phylogenetics equivalent [63] is for alignments P ijk of binary traits on three species, L = 3 and coding these as {1, 2} , the tangle is the degree four homogeneous polynomial τ (P ) = (P 111 ) 2 (P 222 ) 2 + (P 112 ) 2 (P 221 ) 2 + (P 121 ) 2 (P 212 ) 2 + (P 211 ) 2 (P 122 ) 2 + 4P 111 P 122 P 212 P 221 + 4P 112 P 121 P 211 P 222 − 2P 111 P 112 P 221 P 222 − 2P 111 P 121 P 212 P 222 − 2P 111 P 122 P 211 P 222 − 2P 112 P 121 P 212 P 221 − 2P 112 P 122 P 221 P 211 − 2P 121 P 122 P 212 P 211 known in mathematics as the Cayley hyperdeterminant function. This has the special property of being invariant under transformations of the type Eq (31), up to scaling by the product of determinants 54 , in particular…”

The hyperdeterminant and triangulations of the 4-cube

“…Two of the three maximal cells have f -vector (10,23,21,8) and intersect in a square pyramid; the other cell has f -vector (9,18,15,6) and intersects each of the first two in a triangular prism. The intersection of all three cells is a square.…”

“…Hyperdeterminants have numerous applications ranging from quantum information theory [14,15] to computational biology [1,21] and numerical analysis [7,22]. Of particular interest is the binary case, when r 1 = r 2 = · · · = r n = 2.…”