  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.
