Anomalous behavior in length distributions of 3D random Brownian walks and measured photon count rates within observation volumes of single-molecule trajectories in fluorescence fluctuation microscopy

Based on classical mean-field approximation using the diffusion equation for ergodic normal motion of single 24-nm and 100-nm nanospheres, we simulated and measured molecule number counting in fluorescence fluctuation microscopy. The 3D-measurement set included a single molecule, or an ensemble average of single molecules, an observation volume △V and a local environment, e.g. aqueous solution. For the molecule number N ≪ 1 per κV, there was only one molecule at a time inside △V or no molecule. The mean rate k of re-entries defined by k = N / τ_dif was independent of the geometry of △V but depended on the size of △V and the diffusive properties τ_dif. The length distribution ℓ of single-molecule trajectories inside △V and the measured photon count rates I obeyed power laws with anomalous exponent κ =-1.32 ≈-4/3.