Systematics and symmetry in molecular phylogenetic modelling: perspectives from physics

Phylogenetics is the suite of mathematical and computational methods which provide biologists with the means to infer past evolutionary relationships between observed species. The aim of this review is to present and analyze the probabilistic models of mathematical phylogenetics which have been intensively used in recent years, as the foundations on which the practical implementations are based. We outline the development of theoretical phylogenetics, from the earliest studies based on morphological characters, through to the use of molecular data in a wide variety of forms. We bring the lens of mathematical physics to bear on the formulation of theoretical models, focussing on the applicability of many methods from the toolkit of that tradition—techniques of groups and representations to guide model specification and to exploit the multilinear setting of the models in the presence of underlying symmetries; extensions to coalgebraic properties of the generators associated to rate matrices underlying the models, and possibilities to marry these with the graphical structures (trees and networks) which form the search space for inferring evolutionary trees.

Particular aspects which we wish to present to a readership accustomed to thinking from physics, include relating model classes to structural data on relevant matrix Lie algebras, as well as using manipulations with group characters (especially the operation of plethysm, for computing tensor powers) to enumerate various natural polynomial invariants, which can be enormously helpful in tying down robust, low-parameter quantities for use in inference (some of which have only come to light through our perspective). Above all, we wish to emphasize the many features of multipartite entanglement which are shared between descriptions of quantum states on the physics side, and the multi-way tensor probability arrays arising in phylogenetics. In some instances, well-known objects such as the Cayley hyperdeterminant (the 'tangle') can be directly imported into the formalism—in this case, for models with binary character traits, and for providing information about triplets of taxa. In other cases new objects appear, such as the remarkable 'squangle' invariants for quartet tree discrimination, which for DNA data are of quintic degree, with their own unique interpretation in the phylogenetic modelling context. All this hints strongly at the natural and universal presence of entanglement as a phenomenon which reaches across disciplines. We hope that this broad perspective may in turn furnish new insights of use in physics.