Generating Markov evolutionary matrices for a given branch length
  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. 
Marta Casanellas and Anna Kedzierska
q-bio.PE