Theory of the fluorescence of single molecules undergoing multistate conformational dynamics

The fluorescence decay of an N-state system observed in certain single-mole cule experiments is multiexponential with random amplitudes when the durati on of the experiment, T, is finite. We derive the probability density for t hese amplitudes. When Tis sufficiently large, the probability density for a given amplitude is shown to be a Gaussian centered at the equilibrium prob ability of the corresponding state, The width of the Gaussian is determined by the correlation time describing equilibrium fluctuations of the populat ion 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 t he two-state system derived previously. Illustrative calculations for a thr ee-state system are given which compare the exact. Gaussian, and the equiva lent two-state results.