Generating Markov evolutionary matrices for a given branch length

Abstract

Under a markovian evolutionary process, the expected number of substitutions per site (also called branch length) that have occurred when a sequence has evolved from another according to a transition matrix $P$ can be approximated by $-1/4log det P.$ When the Markov process is assumed to be continuous in time, i.e. $P=\exp Qt$ it is easy to simulate this evolutionary process for a given branch length (this amounts to requiring $Q$ of a certain trace). For the more general case (what we call discrete-time models), it is not trivial to generate a substitution matrix $P$ of given determinant (i.e. corresponding to a process of given branch length). In this paper we solve this problem for the most well-known discrete-time models JC*, K80*, K81*, SSM and GMM. These models lie in the class of nonhomogeneous evolutionary models. For any of these models we provide concise algorithms to generate matrices $P$ of given determinant. Moreover, in the first four models, our results prove that any of these matrices can be generated in this way. Our techniques are mainly based on algebraic tools.