## 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 |

Optics Express, Vol. 18, Issue 17, pp. 17883-17896 (2010)

http://dx.doi.org/10.1364/OE.18.017883

Acrobat PDF (893 KB)

### 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* / *τ _{dif}
* was independent of the geometry of ΔV but depended on the size of ΔV and the diffusive properties

*τ*. The length distribution ℓ of single-molecule trajectories inside ΔV and the measured photon count rates

_{dif}*I*obeyed power laws with anomalous exponent

*κ*=−1.32 ≈ −4/3.

© 2010 Optical Society of America

## 1. Introduction

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

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

3. 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]

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]

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

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

6. 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]

*d*= 1.22

_{x–y}*λ*/(2NA) in the x–y plane; the resolution in the z-direction is given by

*d*= 2 n

_{z}*λ*/(NA)

^{2}.

*d*is the distance between two objects just resolved,

*λ*is the wavelength of the exciting laser beam, NA is the numerical aperture of the microscope objective and n is the refraction index of the mounting medium. For example, the resolution limit of fluorescent nanospheres that are 24-nm or 100-nm in diameter and measured at the excitation wavelength of 635 nm or 470 nm with a NA of 1.3 in water with n = 1.3, is 298 nm or 221 nm in the focal plane and 977 nm or 723 nm in z-direction. The projected image of a point source is not a point. It is a diffraction pattern called Airy disk, which is specific to the optics used. The Airy disk has a central maximum and weak concentric side-maxima. The diffraction patterns can be fitted to an Airy disk function or approximated by a two-dimensional Gaussian function. The coordinates of the particle position in the observation volume are the center of mass of each diffraction pattern. Depending on the signal-to-noise ratio, the power of the excitation laser and the physicochemical properties of the fluorophores, e.g. photochemical stability, molecule positions in the focal plane can practically be determined in the range of about 200 nm to 100 nm in solution-phase experiments at the single-molecule and single-particle level, respectively. Localizations with accuracy higher than the diffraction limit of the optics are recorded by additional physical efforts called super-resolution far-field microscopy techniques. Super-resolution in the x-y plane is achieved with stimulated-emission-depletion (STED) microscopy [7

7. 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]

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

9. 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]

10. 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]

11. 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*^{2}(

*t*)〉 = 6

*Dt*is linear with time

*t. D*is the Einstein-Stokes diffusion coefficient. According to the Polya theorem of random Brownian walks, one- and two-dimensional random walks are trivial cases because the single particle and single molecule, respectively, ultimately returns to its starting position in the infinite limit of data collection time (

*t*→ ∞) but its return probability becomes less than one in three or more dimensions [13

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

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

## 2. Numerical simulation

17. 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]

*µ*s. As such, the base diffusion coefficient was, for example, measured with

**= (0,0,1) so that a set of orthogonal vectors**

*ν**𝒮*is generated. Mathematically this means we use the basic set of orthogonal unit vectors in a Cartesian coordinate system as the basis of our calculations

*ℛ*which selects the direction with equal probability randomly from our basis set

_{k}*𝒮**, we create the Brownian track

*ℬ*(

_{n}

*r*_{0},

**) by a sum of independent vectors [18]**

*r*## 3. Results and discussion

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

*S*=

*S*

_{0}·

*δ*(

**)·**

*r**δ*(

*t*)

*a*as lattice spacing. There is no substantial difference of the Monte Carlo simulation and the solution of the diffusion equation with a source term [24

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

*τ*is the width of the time interval.

*τ*is restricted in the real experiment to a lower limit, for example, of 1ms time resolution. In the numerical experiments, the lower limit is, for example, 2.1

*µ*s. We assume that the binary process

*η*(

*t*) is an independent random process without molecular memory. In this context, no molecular memory means there is no hydrodynamic flow or other external forces [27

27. 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]

27. 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]

*λ*= 0,1,2, … and

*k*the mean rate of fluctuations (transitions) in the binary process of Eqs. (1) to (7). A simple model based on rate equations for this process assumes that the probability

*P*(

*λ, τ*) to find

*λ*Brownian tracks either inside or outside the detection volume is defined by the differential-difference equation

*k*defines the moments of the distribution. The solution of this differential-difference equation is given by Eq. (8) and can be explicitly derived [28

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

*k*by

*k*=

*N*/

*τ*, where

_{dif}*τ*is the diffusion time per ΔV [27

_{dif}27. 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]

*k*by simulation (for example, in Fig. 3).

**8**, 261–273 (2007). [CrossRef] [PubMed]

*k*, which is mathematically defined as the time coefficient of the mean value and the variance of the reentry probabilities [27

**8**, 261–273 (2007). [CrossRef] [PubMed]

**8**, 261–273 (2007). [CrossRef] [PubMed]

^{−1}.

*β*can have any value between 0 and infinity. The greater

*β*the more sharply the distribution curve

*p*(Δt) slopes for Δt → 0. Thus, it is much more likely to have short times-in-between fluctuation maxima than long ones. In Fig. 4, a one-timescale scenario is clearly seen in the half-logarithmic plot. Furthermore, the off-times in the fast regime, which is defined as the off-times at which the system is equilibrated on the experimental timescale, are in reasonable agreement with the measured off-times of the fluorescent nanospheres in aqueous solution.

_{t}

*r*^{2}(

*t*) −

*r*^{2}(0)〉 ∝

*t*[17

^{γ˜}17. 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]

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]

*l*of single-molecule trajectories inside ΔV giving rise to the measured photon count rates. Let us now consider the observable of the on-length distribution for the 3D-measurement set of Eq. (1). The non-fractal exponent −3/2 is a consequence of the analytic solution given in Eq. (5). In practice, real measurements of the on-time distribution of the on-lengths lead to various experimental difficulties as is evident upon carrying out the procedure of Eq (1). Actually, it is the experimental experiences that are very difficult to realize; we reach the limitations of our experimental fluctuation techniques because of the necessity of small noise measurements. Motivated by the molecule number fluctuations of Eqs. (4) and (5) under ΔV constraint given by Eqs. (6) and (7) the purpose of this analysis is to describe Brownian motion within small observation volumes as observed in many single molecule / single particle experiments in liquid environment. Therefore, we take the on-time distribution that is identical with the experimentally accessible off-time distribution of Eq. (10) with respect to Δt rather than t in our Eq. (6). We then choose the variable Δt that is given by two successive molecular number fluctuations of the signal and apply the rule Δt

_{off}for the respective on-lengths of single-molecule trajectories at any point of the 3D-Brownian molecule track. With the corresponding boundary-value condition at any Δt

_{off}for a successive on-length

*l*

_{i+1}, which is the right-hand length of the time interval satisfying

*l*

_{i+1}(Δt) ∈ [

*t*,

_{i}*t*+Δt) with

_{i}*i*the number of time intervals of equal length, we assure that the sum (or integral) of the mathematical probabilities of variate values

*l*

_{i+1}at any Δt

_{off}is unity and preserve the magnitudes of the on-lengths at all time points

*t*∈

_{i}*T*. Strictly speaking, the outcomes of simulation and real measurement, respectively, are mutually exclusive at any Δt

_{off}for all time points

*t*∈

_{i}*T*. In Fig. 6, subsets of on-lengths are plotted at each Δt

_{off}in multiples of time resolution for time points

*t*∈

_{i}*T*. The subsets of measured photon count rates of fluctuation maxima as function of Δt

_{off}are extracted from the real time series measurement in the same way and depicted in the inserts of Fig. 6.

*δ*V defined below (see Eqs. (11–16)). As a specific example we construct a three-dimensional grid of planes parallel to the

*xy*,

*yz*, and

*xz*planes. Hence, the three-dimensional region ΔV is subdivided into sub-regions which are rectangular parallelepipeds. By the 3-dimensional grid we define a neighborhood

*δ*V for the single-molecule track within ΔV. The diagonal of the parallelepiped

*δ*V is given by the magnitude of the on-length of the track. This is an estimation where the real molecule is located. The origin of the parallelepiped is given by

**that is the position vector of the parallelepiped. Any point**

*ξ**W*within the bounds of ΔV is defined by the components (

*x,y,z*) and by the rectangular coordinates (

*u,v,w*) of the parallelepiped. A one to one map between points in

*xyz*and in

*uvw*is given by the transformation equations

**,**

*i***,**

*j***are the rectangular unit vectors having the direction of the positive**

*k**x, y, z*axes of the Cartesian coordinate system. The position vector

**is taken from the origin 0 to point**

*ξ**W*. The vector function

**(**

*ξ**u,v,w*) is continuous at (

*u*

_{0},

*v*

_{0},

*w*

_{0}). If

*u*varies and

*v, w*are kept constant,

**describes coordinate curves through**

*ξ**W*.

**also describes coordinate curves through**

*ξ**W*if

*v*varies with

*u, w*= const., and if

*w*varies with

*u, v*= const.. Hence, the diagonal

*d*of the parallelepiped

**ξ***δ*V is given by

*e*_{1}as unit vector at point

*W*in the direction of the vector ∂

*/∂*

**ξ***u*, we have ∂

*/∂*

**ξ***u*=

*h*

_{1}

*e*_{1}with

*h*

_{1}= ∣∂

*/∂*

**ξ***u*∣, and similarly ∂

*/∂*

**ξ***v*=

*h*

_{2}

*e*_{2}with

*h*

_{2}= ∣∂

*/∂*

**ξ***v*∣, ∂

*/∂*

**ξ***w*=

*h*

_{3}

*e*_{3}with

*h*

_{3}= ∣∂

*/∂*

**ξ***w*∣.

*e*_{1},

*e*_{2},

*e*_{3}are mutually perpendicular at any point

*W*.

*h*

_{1},

*h*

_{2},

*h*

_{3}are scale factors. Eq. (13) then reads

*dl*is given by

*dl*

^{2}is the squared on-length of the diagonal in the metric estimator for the single-molecule track. The volume of the parallelepiped within ΔV is

*δ*V = ∣∂

*/∂*

**ξ***u*,∂

*/∂*

**ξ***v*,∂

*/∂*

**ξ***w*∣dudvdw =

*h*

_{1}

*u*(

*l*)

*h*

_{2}

*v*(

*l*)

*h*

_{3}

*w*(

*l*), where ∣,,∣ is the vector triple product. The probability

*P*(

**) of finding the single particle at any position within the bounds of ΔV is**

*ξ***is the stochastic location of the track estimator for the track at location**

*ξ***. The track**

*r***is included in**

*r**δ*V. Note that

*δ*V is not necessarily completely covered by ΔV, however it is guaranteed by this construction that the single-molecule track

**is completely inside ΔV. That is ΔV ≤ ∑**

*r*_{i}

*δ*V

_{i}where

*i*is the total number of on-tracks. If

*l*is sufficiently small, then the approximated location

**is approaching the track position**

*ξ***. Like Eq. (15), Eq. (16) is exact, but formal, in that it merely defines the properties of**

*r**dl*,

*P*(

**) and**

*ξ**P*(

*dl*) in terms of those of

*δ*V. The utility of these equations was to model

*P*in some simple way, as a stochastic process with specified statistical properties like

*P*(ℓ) ~ ℓ

*and*

^{κ}*P*(

*I*) ~

*I*as depicted in Fig. 6.

^{κ}*κ*is not due to the self-affinity of normal Brownian motion; the anomalous exponent of normal Brownian motion is based on the Poisson process given by Δt

_{off}. The anomalous exponent of −4/3 is found by our simulations and by the real measurements performed as depicted in Fig. 6. As shown by Eqs. (11–16), there is a link between the classical analytical solution of the Brownian motion represented by our simulations and the box counting approach. The magnitude of

*κ*was given in a conjecture based on box counting by B.B. Mandelbrot [23]. It also yields the result that the commonly accepted theoretical assumption

*P*(

**) = 1/ΔV = constant for**

*r***∈ ΔV in the analysis of experimental Photon Counting Histograms (PCH) for single-particle tracking [30**

*r*30. 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]

*P*(

**) for the track at location**

*ξ***,**

*r**P*(

**) depends on the position of the single molecule within ΔV given by Eq. (16), so does**

*ξ**P*(

**). In molecular system studied at a many-molecule level there are many molecules presents even at low concentrations and**

*r**P*(

**) might be a constant. We derived an explicit expression for**

*r**P*(

**) within the observation volume ΔV in cylindrical coordinates of radial diffusion in 3D space corresponding to a Fokker-Planck equation for 3D single-molecule track [31].**

*r**chance*which occurred when the form of normal single-molecule motion in a system without immobilization on surfaces or hydrodynamic/electrokinetic flow is constrained by an observation/detection volume. Irregularity by

*chance*also included quantitative information derived from small average molecular numbers

*N*≪ 1 per ΔV in the true single-molecule detection regime of real measurements as demonstrated (e.g., contribution of two simultaneously fluorescing molecules to the detected signal, bursts of photons emitted by chemical aggregates of two or more particles) and thus requires the quantification of a single molecule at a number of ‘cycles’. That is the main difference of our ansatz to the paper of Zumofen et al., 2004 [32

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

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

*l*

_{1}on a time

*t*(

*l*

_{1}) that is much shorter than the time needed to jump a region of size

*l*

_{2}>

*l*

_{1}. Thus, the Brownian walks of short on-lengths occur on a time scale such that long on-lengths are effectively frozen. This feature is central for understanding that the fractal power law allows completely equilibrated and non-equilibrated modes of Brownian walks to coexist at some time. An additional important consequence of the violation of perfect isotropy [13

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

*ξ*has taken place, the details of motion on scales

*l*<

*ξ*start developing between nearby jumps, while the general pattern formed by the dynamics on scale

*ξ*hardly change. Single-molecule spectroscopy and imaging have revealed that the behavior of macroscopic systems can be influenced by events that occur in microscopic non-equilibrium processes.

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]

**8**, 261–273 (2007). [CrossRef] [PubMed]

## 4. Conclusions

## Acknowledgments

## References and links

1. | J. Szymanski and M. Weiss, “Elucidating the origin of anomalous diffusion in crowded fluids,” Phys. Rev. Lett. |

2. | I. Golding and E. C. Cox, “Physical nature of bacterial cytoplasm,” Phys. Rev. Lett. |

3. | 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 |

4. | Y. Meroz, I. M. Sokolov, and J. Klafter, “Subdiffusion of mixed origins: when ergodicity and nonergodicity coexist,” Phys. Rev. E |

5. | A. Lubelski, I. M. Sokolov, and J. Klafter, “Nonergodicity mimics inhomogeneity in single particle tracking,” Phys. Rev. Lett. |

6. | Z. Földes-Papp, “Ultrasensitive detection and identification of fluorescen molecules by FCS: impact for immunobiology,” Proc. Natl. Acad. Sci. USA |

7. | 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 |

8. | S. W. Hell, “Far-field optical nanoscopy,” Science |

9. | M. G. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. USA |

10. | M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods |

11. | S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. |

12. | Y. He, S. Burov, R. Metzler, and E. Barkai, “Random time-scale invariant diffusion and transport coefficients,” Phys.Rev. E |

13. | G. Polya, “Uber eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend der Irrfahrt im Strassennetz,” Math.Ann. |

14. | G. N. Watson, “The triple integrals,” Quat. J. Math. |

15. | B. D. Hughes, |

16. | 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. |

17. | 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. |

18. | M. R. Mazo, |

19. | F. Spitzer, |

20. | J.-P. Bouchaud and A. Georges, “Anomalous diffsuion in disordered media: statistical mechanisms, models and physical applications,” Phys. Rep. |

21. | P. Levy, |

22. | K. J. Falconer, |

23. | B. B. Mandelbrot, |

24. | E. W. Montroll and G. H. Weiss, “Random walks on lattices,” J. Math. Phys. |

25. | E. W. Montroll and M. F. Schlesinger, in |

26. | L. Luchowski, Z. Gryczynski, Z. Földes-Papp, A. Chang, J. Borejdo, P. Sarkar, and I. Gryczynski, “Polarized fluorescent nanospheres,” Opt. Express |

27. | 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. |

28. | H. Risken and H. D. Vollmer, “On the application of truncated generalized Fokker-Planck equations,” Z. Physik B |

29. | I. V. Gopich, “Concentration effectss in“single-molecule” spectroscopy,” J. Phys.Chem. B |

30. | Y. Chen, J. D. Muller, P. T. C. So, and E. Gratton, “The photon counting histogram in fluorescence fluctuation spectroscopy,” Biophys. J. |

31. | Z. Földes-Papp, “Theory of measuring the selfsame single fluorescent molecule in solution suited for studying individual molecular interactions by SPSM-FCS,” Pteridines , |

32. | G. Zumofen, J. Hohlbein, and C. G. Huebner, “Recurrence and photon statistics in fluorescence fluctuation spectroscopy,” Phys. Rev. Lett. |

33. | 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. |

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. |

**OCIS Codes**

(170.6280) Medical optics and biotechnology : Spectroscopy, fluorescence and luminescence

(180.1790) Microscopy : Confocal microscopy

(180.2520) Microscopy : Fluorescence microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: April 27, 2010

Revised Manuscript: July 6, 2010

Manuscript Accepted: July 21, 2010

Published: August 4, 2010

**Virtual Issues**

Vol. 5, Iss. 13 *Virtual Journal for Biomedical Optics*

**Citation**

Gerd Baumann, Ignacy Gryczynski, and Zeno Földes-Papp, "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," Opt. Express **18**, 17883-17896 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-17883

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### References

- J. Szymanski, and M. Weiss, “Elucidating the origin of anomalous diffusion in crowded fluids,” Phys. Rev. Lett. 103, 038102 (2009). [CrossRef] [PubMed]
- I. Golding, and E. C. Cox, “Physical nature of bacterial cytoplasm,” Phys. Rev. Lett. 96, 098102 (2006). [CrossRef] [PubMed]
- 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]
- Y. Meroz, I. M. Sokolov, and J. Klafter, “Subdiffusion of mixed origins: when ergodicity and nonergodicity coexist,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 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]
- Z. Földes-Papp, “Ultrasensitive detection and identification of fluorescent molecules by FCS: impact for immunobiology,” Proc. Natl. Acad. Sci. U.S.A. 98, 11509–11514 (2001). [CrossRef] [PubMed]
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