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 4 log detP. When the Markov process is assumed to be continuous in time, i.e. P = expQt 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 JC69∗, K81∗, K80∗, 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.