School of Mathematics and Physics, University of Tasmania, Australia.
J Theor Biol. 2012 Apr 7;298:16-31. doi: 10.1016/j.jtbi.2011.12.017. Epub 2011 Dec 29.
Recent work has discussed the importance of multiplicative closure for the Markov models used in phylogenetics. For continuous-time Markov chains, a sufficient condition for multiplicative closure of a model class is ensured by demanding that the set of rate-matrices belonging to the model class form a Lie algebra. It is the case that some well-known Markov models do form Lie algebras and we refer to such models as "Lie Markov models". However it is also the case that some other well-known Markov models unequivocally do not form Lie algebras (GTR being the most conspicuous example). In this paper, we will discuss how to generate Lie Markov models by demanding that the models have certain symmetries under nucleotide permutations. We show that the Lie Markov models include, and hence provide a unifying concept for, "group-based" and "equivariant" models. For each of two and four character states, the full list of Lie Markov models with maximal symmetry is presented and shown to include interesting examples that are neither group-based nor equivariant. We also argue that our scheme is pleasing in the context of applied phylogenetics, as, for a given symmetry of nucleotide substitution, it provides a natural hierarchy of models with increasing number of parameters. We also note that our methods are applicable to any application of continuous-time Markov chains beyond the initial motivations we take from phylogenetics.
最近的工作讨论了用于系统发育学的马尔可夫模型的乘法封闭性的重要性。对于连续时间马尔可夫链，通过要求属于模型类的速率矩阵集形成李代数，确保了模型类的乘法封闭性的充分条件。一些著名的马尔可夫模型确实形成李代数，我们将此类模型称为“李马尔可夫模型”。然而，也有一些其他著名的马尔可夫模型明确地不形成李代数（最明显的例子是 GTR）。在本文中，我们将讨论如何通过要求模型在核苷酸置换下具有某些对称性来生成李马尔可夫模型。我们表明，李马尔可夫模型包括，并且因此为“基于群组”和“等变”模型提供了一个统一的概念。对于两个和四个字符状态，给出了具有最大对称性的李马尔可夫模型的完整列表，并显示了包括既不是基于群组也不是等变的有趣示例。我们还认为，我们的方案在应用系统发育学方面令人满意，因为对于核苷酸替代的给定对称性，它提供了具有越来越多参数的模型的自然层次结构。我们还注意到，我们的方法适用于连续时间马尔可夫链的任何应用，而不仅仅是我们从系统发育学中获得的最初动机。
