Generating Markov evolutionary matrices for a given branch length

Under a markovian evolutionary process, the expected number of substitutions per site (branch length) that occur when a sequence evolves from another via a transition matrix P can be approximated by −1/4 log(det P). In continuous-time models, it is easy to simulate the process for any given branch length. For discrete-time models, it is not so trivial. In this paper we solve this problem for the most well-known discrete-time models JC69*, K80*, K81*, SSM, and GMM and we provide concise algorithms to generate stochastic matrices of given determinant. These models have the advantage to be nonhomogeneous evolutionary processes.