Theory of the Fluorescence of Single Molecules Undergoing Multistate Conformational Dynamics

The fluorescence decay of an N -state system observed in certain single-molecule experiments is multiexponential with random amplitudes when the duration of the experiment, T, is finite. We derive the probability density for these amplitudes. When T is sufficiently large, the probability density for a given amplitude is shown to be a Gaussian centered at the equilibrium probability of the corresponding state. The width of the Gaussian is determined by the correlation time describing equilibrium fluctuations of the population of this state. To get an expression applicable at moderately large T we approximately map the N -state problem to a two-state problem. This allows us to make use of an analytical expression for the probability density of the two-state system derived previously. Illustrative calculations for a three-state system are given which compare the exact, Gaussian, and the equivalent two-state results.