Published: February 09, 2011 r2011 American Chemical Society 5838 dx.doi.org/10.1021/jp1099877 | J. Phys. Chem. A 2011, 115, 5838–5846 ARTICLE pubs.acs.org/JPCA Single Molecule Diffusion and the Solution of the Spherically Symmetric Residence Time Equation Noam Agmon* The Fritz Haber Research Center, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel ABSTRACT: The residence time of a single dye molecule diffusing within a laser spot is propotional to the total number of photons emitted by it. With this application in mind, we solve the spherically symmetric “residence time equation” (RTE) to obtain the solution for the Laplace transform of the mean residence time (MRT) within a d-dimensional ball, as a function of the initial location of the particle and the observation time. The solutions for initial conditions of potential experimental interest, starting in the center, on the surface or uniformly within the ball, are explicitly presented. Special cases for dimensions 1, 2, and 3 are obtained, which can be Laplace inverted analytically for d = 1 and 3. In addition, the analytic short- and long-time asymptotic behaviors of the MRT are derived and compared with the exact solutions for d = 1, 2, and 3. As a demonstration of the simplification afforded by the RTE, the Appendix obtains the residence time distribution by solving the Feynman-Kac equation, from which the MRT is obtained by differentiation. Single-molecule diffusion experiments could be devised to test the results for the MRT presented in this work. ’ INTRODUCTION When a laser beam is focused onto a tiny volume element (e.g., 1 fl) in a solution containing a very low (subnanomolar) concentra- tion (c) of a fluorophore, fluorescence bursts can be observed. 1-8 These photonic fluctuations are largely due to a single dye molecule that diffuses (di ffusion coefficient D) in and out of the laser focus, until it eventually escapes to large distances from it (Figure 1). Under steady-state illumination, for an idealized scenario in which the laser spot is a three-dimensional ball of radius R, which is uniformly illuminated, dye molecules arrive at its surface with the diffusion-control rate coefficient 9 4πDRc, which determines the waiting time between bursts. Once on the surface, the particle resides in the ball for an average duration R 2 /3D, 2,10 which determines the average burst duration (hence also the average number of photons emitted). 2 A more detailed theoretical discus- sion can be found in section IV of ref 11. Consequently, under steady-state conditions it suffices to con- sider particles starting on the surface of the sphere (those starting within the sphere contribute only a fast initial transient). Their mean residence time (MRT) within a three-dimensional ball (B 3 ) for an infinite observation time (t f ¥), denoted here by Æτ B 3 (¥|R)æ, is thus a fundamental quantity relevant for analyzing fluorescence bursts from a single freely diffusing dye molecule. More generally, the distribution of the number of emitted photons 11,12 is related to the distribution, F τ (¥|R), of the residence time τ (see Appendix). Interestingly, not only a spot of light can be generated but also a “spot of protons”, namely a spatial pH jump. 13 Dyes diffusing through this spot will change their protonation state, and this could be detected spectroscopically. We have previously evaluated these quantities for t f ¥ and an arbitrary starting point r. 14 Setting r = R in eq 3.16 of ref 14 (with the evident change of notations) indeed yields Æτ B 3 (¥|R)æ = R 2 /3D, as suggested earlier by Eigen. 10 Single molecule diffusion experiments can be performed also on surfaces, membranes, 15 or filaments, 16,17 and these depend on the MRT for dimensions d = 2 or 1, respectively. For example, one- dimensional single-molecule motion occurs when motor proteins move along cellular filaments such as myosin on actin or kinesin/ dynein on microtubules. In the so-called “single-motor assay”, the filament is attached to a glass surface and the motor protein is Figure 1. Schematic depiction of the residence time scenario. When the trajectory of the diffusing particle resides within the spherical domain of radius R, it is photoexcited by a continuous illumination source, resulting in photon emission (yellow stars). Thus the total number of photons emitted is proportional to the particle’s residence time within the domain. The red trajectory starts outside the domain, whereas the green one starts inside it. Special Issue: Victoria Buch Memorial Received: October 18, 2010 Revised: December 24, 2010