What can one learn from two-state single-molecule trajectories?

A time trajectory of an observable that fluctuates between two values (say, on and off), stemming from some unknown multisubstate kinetic scheme, is the output of many single-molecule experiments. Here we show that when all successive waiting times along the trajectory are uncorrelated the on and the off waiting time probability density functions contain all the information. By relating the lack of correlation in the trajectory to the topology of kinetic schemes, we can immediately specify those kinetic schemes that are equally consistent with experiment, and cannot be differentiated by any sophisticated analyses of the trajectory. Correlated trajectories, however, contain additional information about the underlying kinetic scheme, and we consider the strategy that one should use to extract it.