Uncorrelated two-state single molecule trajectories from reducible kinetic schemes

Trajectories of on-off events are the output of many single molecule experiments. Usually, one assumes that the underlying mechanism that generates the trajectory can be described by a kinetic scheme, and by analyzing the trajectory aims at deducing this scheme. In a previous work (O. Flomenbom, J. Klafter and A. Szabo, Biophys. J., in press) we showed that when successive events along a trajectory are uncorrelated, all the information in the trajectory is contained in two basic functions, which are the waiting time probability density functions (PDFs) of the on state and off state. The kinetic schemes that lead to such uncorrelated trajectories were termed reducible. The topologies of reducible kinetic schemes were then given. Here, we provide the mathematical steps that were used to find theses topologies.