Centre for Research in Mathematics and Data Science, Western Sydney University, Sydney, NSW, Australia.
Institut für Molekulare Evolution, Heinrich-Heine Universität, Düsseldorf, Germany.
Bull Math Biol. 2020 Nov 21;82(12):151. doi: 10.1007/s11538-020-00832-w.
In phylogenetics, it is of interest for rate matrix sets to satisfy closure under matrix multiplication as this makes finding the set of corresponding transition matrices possible without having to compute matrix exponentials. It is also advantageous to have a small number of free parameters as this, in applications, will result in a reduction in computation time. We explore a method of building a rate matrix set from a rooted tree structure by assigning rates to internal tree nodes and states to the leaves, then defining the rate of change between two states as the rate assigned to the most recent common ancestor of those two states. We investigate the properties of these matrix sets from both a linear algebra and a graph theory perspective and show that any rate matrix set generated this way is closed under matrix multiplication. The consequences of setting two rates assigned to internal tree nodes to be equal are then considered. This methodology could be used to develop parameterised models of amino acid substitution which have a small number of parameters but convey biological meaning.
在系统发生学中，感兴趣的是满足矩阵乘法封闭的速率矩阵集，因为这使得可以在不计算矩阵指数的情况下找到相应的转移矩阵集。参数数量少也很有优势，因为在应用中，这将减少计算时间。我们探索了一种通过为内部树节点分配速率并为叶子分配状态来从有根树结构构建速率矩阵集的方法，然后将两个状态之间的变化速率定义为分配给这两个状态最近共同祖先的速率。我们从线性代数和图论的角度研究了这些矩阵集的性质，并表明以这种方式生成的任何速率矩阵集都在矩阵乘法下封闭。然后考虑将两个分配给内部树节点的速率设置为相等的后果。这种方法可以用于开发具有少量参数但传达生物学意义的氨基酸替代参数化模型。
