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 describes the underlying mechanism that generates the trajectory using a kinetic scheme, and by analyzing the trajectory aims at deducing this scheme. In a previous work [O. Flomenbom, J. Klafter, and A. Szabo, submitted (2004)], 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 functions (PDFs) of the on state and of the off state. The kinetic schemes that lead to such uncorrelated trajectories were termed reducible. Here we discuss the reasons that lead to reducible schemes. In particular, the topology of reducible schemes is characterized and proven.