Reentries of a single molecule in the confocal, femtoliter-sized probe region (about 10-16 L and less) are significant because during measurement times they give rise to fluctuation phenomena such as molecule number fluctuations at the single-molecule level in solution without immobilization or hydrodynamic focusing. These fluctuations are the fundamental physical process on which, for example, fluorescence correlation spectroscopy and two-color fluorescence cross-correlation spectroscopy are based. The reentries of just one molecule in the confocal probe region are theoretically examined in this original article using a hidden, continuous-time Markov model. The system is not set up to have systemic drift or convection. It is found that the reentries obey certain conditions and analytical expressions for the reentry probabilities are obtained first. In particular, the time constant of the mean value and the variance of the reentry probabilities are obtained. The fractions of non-meaningful reentries and meaningful reentries are found for these experimental situations. Therewith, the concentration dependence of the meaningful time that one can study bimolecular reactions of the selfsame molecule in the confocal probe region is derived for the first time. The meaningful time in the probe volume is proportional to the diffusion time of the selfsame molecule and related inversely to the size of the given confocal probe volume. For small molecules, i.e. small diffusion times at a given size of the confocal probe region, one needs lower concentrations of molecules of the same kind in the bulk phase, whereas large molecules can be studied at higher concentrations. The selfsame molecule scenario is compared with the molecular scenario that a second molecule enters the probe volume at random as a function of the meaningful time. The analytical solutions of the physical reentry model (mechanism) hold for the one-, two- (membrane), or three- (solution, live cell) dimensional Brownian motion.
Keywords: Single molecule, re-entries, meaningful time for measuring a single molecule diffusing in solution or a live cell without systemic drift or convection, quantitative relationships, theory, fluorescence fluctuation spectroscopic approaches, fluorescence correlation spectroscopy, two-color fluorescence cross-correlation spectroscopy, molecule number fluctuations