Burst duration statistics in free diffusion single-molecule fluorescence experiments

In single-molecule fluorescence experiments with freely diffusing molecules, short bursts of photons are typically selected to isolate the most informative segments of the photon stream for subsequent analysis. Each burst has a characteristic duration that, in principle, contains useful information; however, burst duration statistics are rarely analyzed because a suitable theoretical framework has been lacking. We develop a general theory for the burst duration distribution arising from photon bursts generated by a diffusing molecule. By modeling the statistics of arrival times within a burst, we derive an analytic expression for the Laplace transform of the burst duration distribution that fully accounts for experimental burst selection criteria. This expression shows excellent agreement with simulations across a wide range of diffusion coefficients and molecular brightness levels. We further show that a simple three-parameter effective residence-time model can reproduce the full theoretical distribution when fitted. Among its parameters, the residence time has genuine physical significance, scaling inversely with the diffusion coefficient, whereas two other parameters primarily reflect burst selection thresholds. This effective residence time, therefore, provides a reliable marker of molecular mobility and offers a straightforward tool for distinguishing fast and slow molecular species directly from burst duration statistics.