Abstract

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 ΔV and a local environment, e.g. aqueous solution. For the molecule number N ≪ 1 per ΔV, there was only one molecule at a time inside ΔV or no molecule. The mean rate k of re-entries defined by k = N / τdif was independent of the geometry of ΔV but depended on the size of ΔV and the diffusive properties τdif. The length distribution ℓ of single-molecule trajectories inside ΔV and the measured photon count rates I obeyed power laws with anomalous exponent κ =−1.32 ≈ −4/3.

© 2010 OSA

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  34. A. C. Beveridge, J. H. Jett, R. A. Keller, L. R. Pratt, and T. M. Yoshida, “Reduction of diffusion broadening in flow by analysis of time-gated single-molecule data,” Analyst, DOI: 10.1039/b926956h (2010).
  35. Z. Földes-Papp, S.-C. J. Liao, T. You, E. Terpetschnig, and B. Barbieri, “Confocal fluctuation spectroscopy and imaging,” Curr. Pharm. Biotechnol. 11 (6), in press (2010).
    [Crossref] [PubMed]

2010 (4)

Y. Meroz, I. M. Sokolov, and J. Klafter, “Subdiffusion of mixed origins: when ergodicity and nonergodicity coexist,” Phys. Rev. E 81, 010101 (2010).
[Crossref]

G. Baumann, R. F. Place, and Z. Földes-Papp, “Meaningful interpretation of subdiffusive measurements in living cells (crowded environment) by fluorescence fluctuation microscopy,” Curr. Pharm. Biotechnol. 11, 527–543 (2010).
[Crossref] [PubMed]

L. Luchowski, Z. Gryczynski, Z. Földes-Papp, A. Chang, J. Borejdo, P. Sarkar, and I. Gryczynski, “Polarized fluorescent nanospheres,” Opt. Express 18, 4289–4299 (2010).
[Crossref] [PubMed]

Z. Földes-Papp, S.-C. J. Liao, T. You, E. Terpetschnig, and B. Barbieri, “Confocal fluctuation spectroscopy and imaging,” Curr. Pharm. Biotechnol. 11 (6), in press (2010).
[Crossref] [PubMed]

2009 (2)

Z. Földes-Papp, S.-C. J. Liao, T. You, and B. Barbieri, “Reducing background contributions in fluorescence fluctuation time-traces for single-molecule measurements in solution,” Curr. Pharm. Biotechnol. 10, 532–542 (2009).
[Crossref] [PubMed]

J. Szymanski and M. Weiss, “Elucidating the origin of anomalous diffusion in crowded fluids,” Phys. Rev. Lett. 103, 038102 (2009).
[Crossref] [PubMed]

2008 (3)

A. Lubelski, I. M. Sokolov, and J. Klafter, “Nonergodicity mimics inhomogeneity in single particle tracking,” Phys. Rev. Lett. 100, 0250602 (2008).
[Crossref]

Y. He, S. Burov, R. Metzler, and E. Barkai, “Random time-scale invariant diffusion and transport coefficients,” Phys.Rev. E 101, 058101 (2008).

I. V. Gopich, “Concentration effectss in“single-molecule” spectroscopy,” J. Phys.Chem. B 112, 6214–6220 (2008).
[Crossref]

2007 (2)

Z. Földes-Papp, “Fluorescence fluctuation spectroscopic approaches to the study of a single molecule diffusing in solution and a live cell without systemic drift or convection: a theoretical study,” Curr. Pharm. Biotechnol. 8, 261–273 (2007).
[Crossref] [PubMed]

S. W. Hell, “Far-field optical nanoscopy,” Science 316, 1153–1158 (2007).
[Crossref] [PubMed]

2006 (4)

I. Golding and E. C. Cox, “Physical nature of bacterial cytoplasm,” Phys. Rev. Lett. 96, 098102 (2006).
[Crossref] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–796 (2006).
[Crossref] [PubMed]

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91, 4258–4272 (2006).
[Crossref] [PubMed]

R. Niesner and K.-H. Gericke, “Quantitative determination of the single-molecule detection regime in fluorescence fluctuation microscopy by means of photon counting histogram analysis,” J. Chem. Phys. 124, 134704 (2006).
[Crossref] [PubMed]

2005 (1)

M. G. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. USA 102, 13081–13086 (2005).
[Crossref] [PubMed]

2004 (1)

G. Zumofen, J. Hohlbein, and C. G. Huebner, “Recurrence and photon statistics in fluorescence fluctuation spectroscopy,” Phys. Rev. Lett. 93, 260601 (2004).
[Crossref]

2002 (1)

Z. Földes-Papp, “Theory of measuring the selfsame single fluorescent molecule in solution suited for studying individual molecular interactions by SPSM-FCS,” Pteridines,  13, 73–82 (2002).

2001 (2)

G. Seisenberger, M. U. Ried, T. Endres, H. Buning, M. Hallek, and C. Brauchle, “Real-time single-molecule imaging of the infection pathway of an adeno-associated virus,” Science 294, 1929–1932 (2001).
[Crossref] [PubMed]

Z. Földes-Papp, “Ultrasensitive detection and identification of fluorescen molecules by FCS: impact for immunobiology,” Proc. Natl. Acad. Sci. USA 98, 11509–11514 (2001).
[Crossref] [PubMed]

2000 (1)

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. USA 97, 8206–8210 (2000).
[Crossref] [PubMed]

1999 (1)

Y. Chen, J. D. Muller, P. T. C. So, and E. Gratton, “The photon counting histogram in fluorescence fluctuation spectroscopy,” Biophys. J. 77, 553–567 (1999).
[Crossref] [PubMed]

1990 (1)

J.-P. Bouchaud and A. Georges, “Anomalous diffsuion in disordered media: statistical mechanisms, models and physical applications,” Phys. Rep. 12, 195 (1990).

1979 (1)

H. Risken and H. D. Vollmer, “On the application of truncated generalized Fokker-Planck equations,” Z. Physik B 35, 313 (1979).
[Crossref]

1965 (1)

E. W. Montroll and G. H. Weiss, “Random walks on lattices,” J. Math. Phys. 6, 364 (1965).
[Crossref]

1939 (1)

G. N. Watson, “The triple integrals,” Quat. J. Math. 10, 266 (1939).
[Crossref]

1921 (1)

G. Polya, “Uber eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend der Irrfahrt im Strassennetz,” Math.Ann. 84, 149–160 (1921).
[Crossref]

Barbieri, B.

Z. Földes-Papp, S.-C. J. Liao, T. You, E. Terpetschnig, and B. Barbieri, “Confocal fluctuation spectroscopy and imaging,” Curr. Pharm. Biotechnol. 11 (6), in press (2010).
[Crossref] [PubMed]

Z. Földes-Papp, S.-C. J. Liao, T. You, and B. Barbieri, “Reducing background contributions in fluorescence fluctuation time-traces for single-molecule measurements in solution,” Curr. Pharm. Biotechnol. 10, 532–542 (2009).
[Crossref] [PubMed]

Barkai, E.

Y. He, S. Burov, R. Metzler, and E. Barkai, “Random time-scale invariant diffusion and transport coefficients,” Phys.Rev. E 101, 058101 (2008).

Bates, M.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–796 (2006).
[Crossref] [PubMed]

Baumann, G.

G. Baumann, R. F. Place, and Z. Földes-Papp, “Meaningful interpretation of subdiffusive measurements in living cells (crowded environment) by fluorescence fluctuation microscopy,” Curr. Pharm. Biotechnol. 11, 527–543 (2010).
[Crossref] [PubMed]

Beveridge, A. C.

A. C. Beveridge, J. H. Jett, R. A. Keller, L. R. Pratt, and T. M. Yoshida, “Reduction of diffusion broadening in flow by analysis of time-gated single-molecule data,” Analyst, DOI: 10.1039/b926956h (2010).

Borejdo, J.

Bouchaud, J.-P.

J.-P. Bouchaud and A. Georges, “Anomalous diffsuion in disordered media: statistical mechanisms, models and physical applications,” Phys. Rep. 12, 195 (1990).

Brauchle, C.

G. Seisenberger, M. U. Ried, T. Endres, H. Buning, M. Hallek, and C. Brauchle, “Real-time single-molecule imaging of the infection pathway of an adeno-associated virus,” Science 294, 1929–1932 (2001).
[Crossref] [PubMed]

Buning, H.

G. Seisenberger, M. U. Ried, T. Endres, H. Buning, M. Hallek, and C. Brauchle, “Real-time single-molecule imaging of the infection pathway of an adeno-associated virus,” Science 294, 1929–1932 (2001).
[Crossref] [PubMed]

Burov, S.

Y. He, S. Burov, R. Metzler, and E. Barkai, “Random time-scale invariant diffusion and transport coefficients,” Phys.Rev. E 101, 058101 (2008).

Chang, A.

Chen, Y.

Y. Chen, J. D. Muller, P. T. C. So, and E. Gratton, “The photon counting histogram in fluorescence fluctuation spectroscopy,” Biophys. J. 77, 553–567 (1999).
[Crossref] [PubMed]

Cox, E. C.

I. Golding and E. C. Cox, “Physical nature of bacterial cytoplasm,” Phys. Rev. Lett. 96, 098102 (2006).
[Crossref] [PubMed]

Dyba, M.

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. USA 97, 8206–8210 (2000).
[Crossref] [PubMed]

Egner, A.

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. USA 97, 8206–8210 (2000).
[Crossref] [PubMed]

Endres, T.

G. Seisenberger, M. U. Ried, T. Endres, H. Buning, M. Hallek, and C. Brauchle, “Real-time single-molecule imaging of the infection pathway of an adeno-associated virus,” Science 294, 1929–1932 (2001).
[Crossref] [PubMed]

Falconer, K. J.

K. J. Falconer, Fractal Geometry (Wiley, Chichester, 2003).
[Crossref]

Földes-Papp, Z.

G. Baumann, R. F. Place, and Z. Földes-Papp, “Meaningful interpretation of subdiffusive measurements in living cells (crowded environment) by fluorescence fluctuation microscopy,” Curr. Pharm. Biotechnol. 11, 527–543 (2010).
[Crossref] [PubMed]

Z. Földes-Papp, S.-C. J. Liao, T. You, E. Terpetschnig, and B. Barbieri, “Confocal fluctuation spectroscopy and imaging,” Curr. Pharm. Biotechnol. 11 (6), in press (2010).
[Crossref] [PubMed]

L. Luchowski, Z. Gryczynski, Z. Földes-Papp, A. Chang, J. Borejdo, P. Sarkar, and I. Gryczynski, “Polarized fluorescent nanospheres,” Opt. Express 18, 4289–4299 (2010).
[Crossref] [PubMed]

Z. Földes-Papp, S.-C. J. Liao, T. You, and B. Barbieri, “Reducing background contributions in fluorescence fluctuation time-traces for single-molecule measurements in solution,” Curr. Pharm. Biotechnol. 10, 532–542 (2009).
[Crossref] [PubMed]

Z. Földes-Papp, “Fluorescence fluctuation spectroscopic approaches to the study of a single molecule diffusing in solution and a live cell without systemic drift or convection: a theoretical study,” Curr. Pharm. Biotechnol. 8, 261–273 (2007).
[Crossref] [PubMed]

Z. Földes-Papp, “Theory of measuring the selfsame single fluorescent molecule in solution suited for studying individual molecular interactions by SPSM-FCS,” Pteridines,  13, 73–82 (2002).

Z. Földes-Papp, “Ultrasensitive detection and identification of fluorescen molecules by FCS: impact for immunobiology,” Proc. Natl. Acad. Sci. USA 98, 11509–11514 (2001).
[Crossref] [PubMed]

Georges, A.

J.-P. Bouchaud and A. Georges, “Anomalous diffsuion in disordered media: statistical mechanisms, models and physical applications,” Phys. Rep. 12, 195 (1990).

Gericke, K.-H.

R. Niesner and K.-H. Gericke, “Quantitative determination of the single-molecule detection regime in fluorescence fluctuation microscopy by means of photon counting histogram analysis,” J. Chem. Phys. 124, 134704 (2006).
[Crossref] [PubMed]

Girirajan, T. P. K.

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91, 4258–4272 (2006).
[Crossref] [PubMed]

Golding, I.

I. Golding and E. C. Cox, “Physical nature of bacterial cytoplasm,” Phys. Rev. Lett. 96, 098102 (2006).
[Crossref] [PubMed]

Gopich, I. V.

I. V. Gopich, “Concentration effectss in“single-molecule” spectroscopy,” J. Phys.Chem. B 112, 6214–6220 (2008).
[Crossref]

Gratton, E.

Y. Chen, J. D. Muller, P. T. C. So, and E. Gratton, “The photon counting histogram in fluorescence fluctuation spectroscopy,” Biophys. J. 77, 553–567 (1999).
[Crossref] [PubMed]

Gryczynski, I.

Gryczynski, Z.

Gustafsson, M. G.

M. G. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. USA 102, 13081–13086 (2005).
[Crossref] [PubMed]

Hallek, M.

G. Seisenberger, M. U. Ried, T. Endres, H. Buning, M. Hallek, and C. Brauchle, “Real-time single-molecule imaging of the infection pathway of an adeno-associated virus,” Science 294, 1929–1932 (2001).
[Crossref] [PubMed]

He, Y.

Y. He, S. Burov, R. Metzler, and E. Barkai, “Random time-scale invariant diffusion and transport coefficients,” Phys.Rev. E 101, 058101 (2008).

Hell, S. W.

S. W. Hell, “Far-field optical nanoscopy,” Science 316, 1153–1158 (2007).
[Crossref] [PubMed]

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. USA 97, 8206–8210 (2000).
[Crossref] [PubMed]

Hess, S. T.

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91, 4258–4272 (2006).
[Crossref] [PubMed]

Hohlbein, J.

G. Zumofen, J. Hohlbein, and C. G. Huebner, “Recurrence and photon statistics in fluorescence fluctuation spectroscopy,” Phys. Rev. Lett. 93, 260601 (2004).
[Crossref]

Huebner, C. G.

G. Zumofen, J. Hohlbein, and C. G. Huebner, “Recurrence and photon statistics in fluorescence fluctuation spectroscopy,” Phys. Rev. Lett. 93, 260601 (2004).
[Crossref]

Hughes, B. D.

B. D. Hughes, Random Walks and Random Environments (Clarendon Press, Oxford, 1995).

Jakobs, S.

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. USA 97, 8206–8210 (2000).
[Crossref] [PubMed]

Jett, J. H.

A. C. Beveridge, J. H. Jett, R. A. Keller, L. R. Pratt, and T. M. Yoshida, “Reduction of diffusion broadening in flow by analysis of time-gated single-molecule data,” Analyst, DOI: 10.1039/b926956h (2010).

Keller, R. A.

A. C. Beveridge, J. H. Jett, R. A. Keller, L. R. Pratt, and T. M. Yoshida, “Reduction of diffusion broadening in flow by analysis of time-gated single-molecule data,” Analyst, DOI: 10.1039/b926956h (2010).

Klafter, J.

Y. Meroz, I. M. Sokolov, and J. Klafter, “Subdiffusion of mixed origins: when ergodicity and nonergodicity coexist,” Phys. Rev. E 81, 010101 (2010).
[Crossref]

A. Lubelski, I. M. Sokolov, and J. Klafter, “Nonergodicity mimics inhomogeneity in single particle tracking,” Phys. Rev. Lett. 100, 0250602 (2008).
[Crossref]

Klar, T. A.

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. USA 97, 8206–8210 (2000).
[Crossref] [PubMed]

Levy, P.

P. Levy, Processus stochastiques et mouvement Brownien (Gauthier-Villars, Paris, 1965).

Liao, S.-C. J.

Z. Földes-Papp, S.-C. J. Liao, T. You, E. Terpetschnig, and B. Barbieri, “Confocal fluctuation spectroscopy and imaging,” Curr. Pharm. Biotechnol. 11 (6), in press (2010).
[Crossref] [PubMed]

Z. Földes-Papp, S.-C. J. Liao, T. You, and B. Barbieri, “Reducing background contributions in fluorescence fluctuation time-traces for single-molecule measurements in solution,” Curr. Pharm. Biotechnol. 10, 532–542 (2009).
[Crossref] [PubMed]

Lubelski, A.

A. Lubelski, I. M. Sokolov, and J. Klafter, “Nonergodicity mimics inhomogeneity in single particle tracking,” Phys. Rev. Lett. 100, 0250602 (2008).
[Crossref]

Luchowski, L.

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983), pp.237–243 and pp. 326–334.

Mason, M. D.

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91, 4258–4272 (2006).
[Crossref] [PubMed]

Mazo, M. R.

M. R. Mazo, Brownian motion (Oxford Univ. Press, Oxford, 2009).

Meroz, Y.

Y. Meroz, I. M. Sokolov, and J. Klafter, “Subdiffusion of mixed origins: when ergodicity and nonergodicity coexist,” Phys. Rev. E 81, 010101 (2010).
[Crossref]

Metzler, R.

Y. He, S. Burov, R. Metzler, and E. Barkai, “Random time-scale invariant diffusion and transport coefficients,” Phys.Rev. E 101, 058101 (2008).

Montroll, E. W.

E. W. Montroll and G. H. Weiss, “Random walks on lattices,” J. Math. Phys. 6, 364 (1965).
[Crossref]

E. W. Montroll and M. F. Schlesinger, in Studies in statistical mechanics, edited by J. L. Lebowitz and E. W. Montroll, (Elsevier, New York, 1984), vol.11.

Muller, J. D.

Y. Chen, J. D. Muller, P. T. C. So, and E. Gratton, “The photon counting histogram in fluorescence fluctuation spectroscopy,” Biophys. J. 77, 553–567 (1999).
[Crossref] [PubMed]

Niesner, R.

R. Niesner and K.-H. Gericke, “Quantitative determination of the single-molecule detection regime in fluorescence fluctuation microscopy by means of photon counting histogram analysis,” J. Chem. Phys. 124, 134704 (2006).
[Crossref] [PubMed]

Place, R. F.

G. Baumann, R. F. Place, and Z. Földes-Papp, “Meaningful interpretation of subdiffusive measurements in living cells (crowded environment) by fluorescence fluctuation microscopy,” Curr. Pharm. Biotechnol. 11, 527–543 (2010).
[Crossref] [PubMed]

Polya, G.

G. Polya, “Uber eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend der Irrfahrt im Strassennetz,” Math.Ann. 84, 149–160 (1921).
[Crossref]

Pratt, L. R.

A. C. Beveridge, J. H. Jett, R. A. Keller, L. R. Pratt, and T. M. Yoshida, “Reduction of diffusion broadening in flow by analysis of time-gated single-molecule data,” Analyst, DOI: 10.1039/b926956h (2010).

Ried, M. U.

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Figures (6)

Fig. 1.
Fig. 1.

Mean square displacement for the three-dimensional Brownian walk. For details, see main text.

Fig. 2.
Fig. 2.

Digital single-molecule detection in solution. In laser-induced fluorescence fluctuation detection at the single-molecule level, the detected fluorescence becomes digital since the time-averaged molecule number in the tiny observation volume ΔV is much smaller then unity. The molecules are only detected when they pass through the observation volume, i.e. the focused laser beam [1, 4, 6]. Upper panel: The simulation signal. Lower panel: Real binary signal measured in aqueous solution of sonicated 24-nm fluorescent nanospheres from an average number of molecules N = 0.0055 or 43 picomolar for ΔV = 0.21 fL. We observed 2541 fluctuations above the background of 3000 photon counts per second for T = 300 s. A pulsed diode laser at wavelength of 635 nm was used and operated at 20 MHz repetition rate at 20 µW laser power intensity after objective. Experimental measurement details are described elsewhere [26]. The measurement analysis was performed with the developed ISS Fluctuation Analyzer TZ software package.

Fig. 3.
Fig. 3.

Fluctuation number distribution η(t) taken from the binary reduction of the Brownian track. The graph is generated from 140 tracks of 20000 steps. The 3D representation shows the frequency distribution depending on the time lag τ and λ for the 100-nm nanospheres measured with a cylindrically-shaped observation volume of ΔV = 0.14 fL. We confirmed by simulation the measured mean rate of re-entries (transitions) k = N / τdif = 0.0052 / 2.79·10−3 s = 1.86 s −1, where τdif is the measured diffusion time, for example, of the 100-nm nanospheres in aqueous solution.

Fig. 4.
Fig. 4.

Off-time distribution at the boundary of the measuring volume. The off-time is defined in the main text. Upper panel: the graph is based on n = 29824 Brownian tracks for an observation volume of 0.21 fL (red excitation volume). n represents the time evolution of Δt. Lower panel: the graph is from the measurement of 24-nm nanospheres (red excitation) at 43 pmolar. There is only one physical Poisson process in the signal and therefore the half-logarithmic plots have to be fitted to one line instead of two lines; the data are noisy. The simulated and measured β values were in good agreement.

Fig. 5.
Fig. 5.

Simulated on-time distribution at the boundary of the measuring volume 0.14 fL. The on-time is defined in the main text. n represents the time evolution. The scaling exponent is the well known non-fractal exponent−3/2 due to the self-affine scaling of normal Brownian motion by a factor θ [23]. The same result was obtained for the 0.21 fL observation volume.

Fig. 6.
Fig. 6.

Measured photon count rate (insert) for 24-nm nanospheres in aqueous solution as function of the on-lengths of single-molecule trajectories. PDF: probability density function. Δtoff is given in multiples of time resolution and runs from 1 (upper panel) to 2 (lower panel) plotted in the real measurements (inserts) and the simulations. In the measurement, Δtoff = 1 ms occurred with a frequency of 386 and Δtoff = 2 ms with a frequency of 419. For Δtoff ≥ 3, the measured number of successive fluctuations was small yielding very noisy statistics. P(ℓ) ~ ℓ κ with κ = −1.32 ≈ −4/3 was found. The photon count rate I (insert) obeyed the same power law as the on-length. κ = - 4/3 is the conjectured but not proved value of the box counting dimension given by B.B. Mandelbrot [23]. In the main text, we show how the power-law relations P(ℓ) ~ ℓ κ and P(I) ~ Iκ can be related to the fractal relations with the fractal or anomalous dimension κ. Number of tracks used in the simulation n = 8000 of a total number of random steps of T = 30000. The same κ values of the box counting dimension were found for 100-nm nanospheres (data not shown).

Equations (16)

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𝒮 = { i , j , k i , j , k 3 ^ i = 1 , j = 1 , k = 1 } .
𝒮 * = 𝒮 𝒮 = { i , j , k , i , j , k } .
track : = r 0 + Σ k = 1 n k ( 𝒮 * ) = n ( r 0 , r ) ,
p t ( r , t ) = D · 2 p ( r , t ) + S ( r , t )
p ( r , t ) ~ t 3 2 · exp { r 2 a · t } ,
η ( t ) = { 1 inside Δ V , emitted fluorescence detected , 0 outside Δ V , no florescence detected .
P ( τ ) = P ( η ( t + τ ) η ( t ) = λ ) ,
P ( τ ) = e k τ ( k τ ) λ λ ! ,
d P ( λ , τ ) d τ = k ( P ( λ 1 , τ ) P ( λ , τ ) ) ,
p t ( Δ t ) = β · exp { β · Δ t } .
x = f ( u , v , w ) , y = g ( u , v , w ) , z = h ( u , v , w ) ;
ξ = x i + y j + z k = f ( u , v , w ) i + g ( u , v , w ) j + h ( u , v , w ) k ,
d ξ = ξ u v , w = const d u + ξ v u , w = const d v + ξ w u , v = const d w .
d ξ = h 1 d u + h 2 d v + h 3 d w .
dl 2 = d ξ · d ξ = h 1 2 d u 2 + h 2 2 d v 2 + h 3 2 d w 2 ,
P ( ξ ) = P ( h 1 u ( l ) , h 2 v ( l ) , h 3 w ( l ) ) ,

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