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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 17 — Aug. 16, 2010
  • pp: 17883–17896
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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

Gerd Baumann, Ignacy Gryczynski, and Zeno Földes-Papp  »View Author Affiliations


Optics Express, Vol. 18, Issue 17, pp. 17883-17896 (2010)
http://dx.doi.org/10.1364/OE.18.017883


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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 Optical Society of America

1. Introduction

A number of experimental single-molecule tools are available to measure Brownian dynamics in solution and crowded fluids. Among them fluorescence fluctuation microscopy including statistical autocorrelation analysis like fluorescence correlation spectroscopy [1

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

] are widely used methods. Their detection capability of slowly and non-exponentially relaxing dynamics makes them ideal for the study of statistical mechanics [2

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

, 3

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]

]. The physical properties of the associated stochastic process in solution-phase, i.e. the 3D propagator of random Brownian walks, determine the microscopic pattern. Understanding of such microscopic effects requires being able to translate the real measurement signal from single molecules into state trajectories [4

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]

]. Numerical simulations describe then the single molecule events of particles performing random walks (RWs). Single-molecule studies do not always use information from one individual molecule only or interpret every single molecule in the bulk phase [5

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

]. In contrast to bulk measurements, averaging and other statistical analyses are performed off-line. Bridging single-molecule approaches with ensemble averages should yield interesting results [5

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

, 6

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]

].

A single particle is any independent labeled species. A fluorescent monomer like a dye molecule is one particle, but a fluorescent nanosphere containing many rigidly fixed fluorescent dyes in its inner core is also one particle. Hence, single molecules and single particles can be treated in the same manner. A single fluorescent particle, e.g. fluorescent nanosphere molecule, can be approximated as a point source of emitted fluorescence light if it is smaller than the resolution limit given by the Rayleigh criterion of dx–y = 1.22λ/(2NA) in the x–y plane; the resolution in the z-direction is given by dz = 2 n 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

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

], saturated structured illumination microscopy (SSIM) [9

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]

], 3D stochastic optical reconstruction microscopy (STORM) [10

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]

], and photo-activated localization microscopy (PALM) [11

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]

].

A non-interacting single-molecule in solution, e.g. a fluorescent nanosphere particle, spreads according to Fick’s law as a Gaussian packet in terms of statistical properties of the microscopic jumps [12

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

]. Time and ensemble averages of the Brownian walk are identical. This means that ergodicity is not broken and the mean squared displacement 〈r 2(t)〉 = 6Dt 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]

]. This return probability for three dimensions is 0.6594… [14

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

,15

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

].

2. Numerical simulation

The motion of the Brownian particle is generated with a random number generator delivering pseudo-random numbers used for the steps in all three spatial directions. The mathematical basis of the method and how it is generalized to a fractal and continuous time random walk in a straightforward way are first described in ref. [17

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]

]. The scale of the simulation was set such that the spacing between lattice sites was, for example 10 nm and the time-step was 2.1 µs. As such, the base diffusion coefficient was, for example, measured with 8.0·1012m2s=(10nm)22.1μs·16 , consistent with the experimental setup for the 24-nm nanospheres in aqueous solution. As a specific example, we here consider the way how the Brownian walk is generated.

We generate a random Brownian walk by randomly selecting steps in the three coordinate directions. The three coordinate directions are generated by a permutation of the vector ν = (0,0,1) so that a set of orthogonal vectors 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

𝒮={i,j,ki,j,k3^i=1,j=1,k=1}.
(1)

This set of permuted vectors is extended in all directions positive and negative by the following unification of basis sets

𝒮*=𝒮𝒮={i,j,k,i,j,k}.
(2)

Introducing the random function k which selects the direction with equal probability randomly from our basis set 𝒮*, we create the Brownian track n (r 0,r) by a sum of independent vectors [18

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

]

track:=r0+Σk=1nk(𝒮*)=n(r0,r),
(3)

Fig. 1. Mean square displacement for the three-dimensional Brownian walk. For details, see main text.

3. Results and discussion

The simulation results of Fig. 1 show that by using a standard cubic lattice with a finite lattice spacing, we obtain a normal diffusive motion. The space is not restricted to motion only allowed in some directions, i.e. we generate a random process in all directions which is consistent with the well known mean field theory of random walks on lattices [24

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

, 25

25. 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.

]. The classical solution of the 3D spherical diffusion equation using Fick’s law for the current and the delta function source term S = S 0·δ(rδ(t)

pt(r,t)=D·2p(r,t)+S(r,t)
(4)

is

p(r,t)~t32·exp{r2a·t},
(5)

where −3/2 is the non-fractal, self-affine scaling exponent on a Bravais lattice, standard cubic lattice (sc), with 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]

, 25

25. 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.

]. Both methods do not give different results.

η(t)={1insideΔV,emittedfluorescencedetected,0outsideΔV,noflorescencedetected.
(6)

In this original article, the track of Brownian particles is examined inside and outside the observation volume ΔV. If the particle is observed inside the detection volume ΔV we record this state as 1 and if the particle is outside the detection volume we record this state as 0. In this way, each step of the random walk of the 3D Brownian track is converted into a binary sequence of events. The reduction from a three dimensional path to a one dimensional sequence of binary states corresponds to the measuring process, which detects a signal or not. An example for a simulated signal and the real fluorescence signal are shown in Fig. 2. The results assume that uncorrelated photons are measured which means they are statistically independent. Because the experimental apparatus requires time intervals of several milliseconds up to seconds and even longer, the photon correlations will be lost for times much longer than the coherence time that is the inverse of the bandwidth of the laser. This phenomenon can be understood by noting that if the simulation/measurement time T is very large, many fluctuations take place, and hence we measure an average value of the fluctuations and not the fluctuation itself. The longer the time interval T, the closer the measured value approaches the mean value. As a consequence, the measured statistics approaches the uncorrelated Poisson distribution. We provide a direct test of single-molecule trajectories in solution-phase by means of fluorescence fluctuation microscopy. Our analysis moves beyond unphysical assumptions of theoretical diffusive measurements in solution-phase by fluorescence fluctuation microscopy.

The extraction of the information from the signal is based on the relation

P(τ)=P(η(t+τ)η(t)=λ),
(7)

where 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, 2.1µs. 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]

]. The theoretical result for this counting statistics of molecule number fluctuations of Eqs. (4) and (5) under ΔV constraint given by Eqs. (6) and (7) is Poisson distributed [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]

]; i.e.

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.
P(τ)=ekτ(kτ)λλ!,
(8)

with the events (mathematically speaking, transitions) 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

dP(λ,τ)dτ=k(P(λ1,τ)P(λ,τ)),
(9)

where 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]

]. Under the conditions N < 1 per size ΔV, we specified k by k = N / τdif, where τdif is the diffusion time per ΔV [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]

]. Here, we first confirm the correctness of this specification for k by simulation (for example, in Fig. 3).

Since the elementary process of Eq. (1) is caused by the 3-dimensional Brownian motion that is not different inside and outside the observation volume ΔV, the single-molecule fluctuation counting statistics Eqs. (8) and (9) did not depend on the geometry of ΔV as we theoretically predicted in ref. [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]

] and observed in Fig. 3. We found exactly the same mean number of reentries k, which is mathematically defined as the time coefficient of the mean value and the variance of the reentry probabilities [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]

], for a spherical observation volume of the same size. However, the counting statistics Eqs. (7) and (8) depends on the diffusive properties of the single molecules represented by the diffusion coefficient and, therefore, the diffusion time per size of ΔV [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]

].

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.

Next, we simulated the experimentally well defined distribution of times between the fluctuation maxima, i.e. the off-time distribution, and compared it with the off-times of the real signal. For molecule number fluctuations that satisfy Poisson statistics (Eq. (8)), the times between photon bursts in Fig. 2 are given by

pt(Δt)=β·exp{β·Δt}.
(10)

The probability density function of off-times Δt (Eq. (10)) has units of time−1. β can have any value between 0 and infinity. The greater β the more sharply the distribution curve pt(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.

When the time intervals between diffusive jumps of Brownian trajectories show fractal characteristics, the mean squared displacement of free diffusion can scale as 〈r 2(t) − r 2(0)〉 ∝ tγ˜ [17

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]

]. This behavior is known as anomalous diffusion but sub-diffusion of mixed origin can coexist as found recently [4

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]

]. Thus, it is important to measure anomalous dynamics on different length scales or timescales and to couple the analysis of how experimental parameters change with predictions from different mechanistic models. With imprecise parameter definitions, this type of analysis is not possible.

Given by the diffusion coefficient we have a one to one representation between the on-time distribution shown in Fig. 5 and the on-length distribution. The on-time is the time the molecule is in the observation volume ΔV. The on-lengths are the length distribution 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 Δtoff 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 Δtoff for a successive on-length l i+1, which is the right-hand length of the time interval satisfying l i+1(Δt) ∈ [ti,ti+Δt) with 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 Δtoff is unity and preserve the magnitudes of the on-lengths at all time points tiT. Strictly speaking, the outcomes of simulation and real measurement, respectively, are mutually exclusive at any Δtoff for all time points tiT. In Fig. 6, subsets of on-lengths are plotted at each Δtoff in multiples of time resolution for time points tiT. The subsets of measured photon count rates of fluctuation maxima as function of Δtoff are extracted from the real time series measurement in the same way and depicted in the inserts of Fig. 6.

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. 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.

Our analysis embodied in Eqs. (11)–(16) is motivated by the wide range of applicability of box-counting. For example, it can be applied to a distribution of points as easily as it can be applied to a continuous curve. The topography is overlaid with a grid of boxes; grids of different size boxes are used. However, the box counting here is different from the classical one in which the box size is varied. We use instead a single sized box ΔV including different sublengths corresponding to a measure δV 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 δV for the single-molecule track within ΔV. The diagonal of the parallelepiped δV 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 ξ that is the position vector of the parallelepiped. Any point 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

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) ~ ℓκ 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) ~ ℓκ 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).
x=f(u,v,w),y=g(u,v,w),z=h(u,v,w);
(11)
ξ=xi+yj+zk=f(u,v,w)i+g(u,v,w)j+h(u,v,w)k,
(12)

where i, j, k are the rectangular unit vectors having the direction of the positive x, y, z axes of the Cartesian coordinate system. The position vector ξ is taken from the origin 0 to point 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, ξ describes coordinate curves through W. ξ also describes coordinate curves through W if v varies with u, w = const., and if w varies with u, v = const.. Hence, the diagonal dξ of the parallelepiped δV is given by

dξ=ξuv,w=constdu+ξvu,w=constdv+ξwu,v=constdw.
(13)

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

dξ=h1du+h2dv+h3dw.
(14)

The arc length dl is given by

dl2=dξ·dξ=h12du2+h22dv2+h32dw2,
(15)

where 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 δV = ∣∂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(ξ) of finding the single particle at any position within the bounds of ΔV is

P(ξ)=P(h1u(l),h2v(l),h3w(l)),
(16)

The problem we here addressed is irregularity by 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]

]. Several examples of power-law ‘fragmentation’ were given in ref. [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]

] but we first found that normal diffusion can show anomalous behavior (Fig. 6) which is only characterized for fractal motion (anomalous diffusion) in, for example, crowded environment. The origin and nature of the anomalous behavior are due to the fact that the single-molecule trajectories differ by jumps of regions of size 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]

] is the fact that once an event on the scale of size 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.

The currently well accepted approach to measure a single molecule as it flows through a well-defined probe/observation volume is often not true single molecule [34

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).

]. Although there is only one analyte molecule in the observation volume during the measurement, poor signal-to-noise requires that bursts from many analyte species must be averaged in order to achieve a reasonable signal-to-noise ratio. This makes it difficult to distinguish between rare confirmers with a strong signal that occasionally pass through the observation volume (or confirmers in dynamic equilibrium) from a mixture of stable confirmers (Richard. A. Keller, personal communication, Los Alamos); molecules that travel fast get there first and have less time to diffuse and the diffusion width is small [34

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).

]. In very dilute solutions without flow, with very high probability the first molecule to enter the observation volume is the molecule that just left [35

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]

]. The reentry time depends on the size of the observation volume, the diffusion coefficient and the molar bulk concentration of other molecules of the same kind that are not the original molecule [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]

].

4. Conclusions

In this paper, we have studied by a Monte Carlo simulation method the real 3D-measurement set in fluorescence fluctuation microscopy where no information can be obtained about the single molecules as long as the molecule dwells outside the 3D observation volume ΔV. At the single-molecule level, there is only one molecule at a time inside ΔV or no molecule. The dynamics of the molecule remains hidden unless it is the dynamics of the molecule itself that causes the change in the molecule number fluctuations across ΔV. Photon trajectories for each molecule can also be obtained. The Poisson single-molecule fluctuation counting statistics for the molecule number N < 1 per ΔV depends on the diffusive properties of the single molecules represented by the diffusion coefficient and, therefore, the diffusion time τdif per size of ΔV. Most important in our findings is that the mean rate of re-entries k defined by k = N / τdif is independent of the geometry of the observation volume ΔV but depends on its size and the diffusive properties τdif of the single molecules. Besides the well-known non-fractal exponent −3/2 due to the self-affine scaling of classical Brownian motion, length distribution within the bounds ΔV of single-molecule trajectories, i.e. the so-called on-length distribution, and the measured photon count rates I obey the power laws P(ℓ) ~ ℓκ and P(I) ~ Iκ with the anomalous exponent κ =−1.32 ≈ −4/3 that is the box counting dimension conjectured but not proved by B.B. Mandelbrot. The observed power-law behavior is linked to the molecular level because the on-length distribution is not perfectly isotropic in equilibrium. The power law provides an expression for the violation of perfect isotropy.

Acknowledgments

We would like to thank Karol (Zygmunt) Gryczynski for providing his measurement facility of fluorescence fluctuation microscopy at the Center for Commercialization of Fluorescence Technologies (CCFT), University of North Texas Health Science Center, TX 76107, USA. We also thank Rafal Luchowski at CCFT for performing measurements during the visiting professorship of Zeno Foldes-Papp at the CCFT and Department of Molecular Biology and Immunology, University of North Texas Health Science Center, TX 76107, USA. The ISS Fluctuation Analyzer TZ software package was developed by Zeno Foldes-Papp and Tiefeng You at ISS in Champaign, IL 61822, USA.

We acknowledge the interesting contributions of Richard A. Keller, Los Alamos, National Laboratory, USA, to the concepts of measuring a true single molecule under flow conditions; the properties of a single molecule in dynamic equilibrium can be studied by increasing the observation time so that many cycles of reentries can be obtained for each species. The approach to take advantage of reentries to extend the measurement time of the same single molecule will increase the accuracy of kinetic measurements of molecules in dynamic equilibrium.

We thank Enrico Gratton, University of California Irvine, Laboratory of Fluorescence Dynamics, USA, for his comments in the preparation of the final manuscript.

Zeno Foldes-Papp, who is the principal investigator, acknowledges financial support in part from his Austrian FWF Science Fund collaborative research project P20454-N13, the Center for Commercialization of Fluorescence Technologies (CCFT), the University of North Texas Health Science Center, and from the German University in Cairo, the University of Ulm (Germany) as well as the bwGRiD Cluster Ulm that is part of the high performance computing facilities of the Federal State of Baden-Wuerttemberg (Germany), where most of the very time-consuming and expensive numercial calculations were executed. Zeno Foldes-Papp has visiting professorships at the CCFT and Department of Molecular Biology and Immunology, University of North Texas Health Science Center, TX 76107, USA, at ISS in Champaign, IL 61822, USA and at the Mathematics Department of the German University in Cairo.

References and links

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]

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]

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]

12.

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

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]

15.

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

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. 124, 134704 (2006). [CrossRef] [PubMed]

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]

18.

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

19.

F. Spitzer, Principles of random walk (Springer, New York, 2001).

20.

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

21.

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

22.

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

23.

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

24.

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

25.

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.

26.

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]

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]

28.

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

29.

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

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]

31.

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).

32.

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

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. 10, 532–542 (2009). [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]

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

  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 Stat. Nonlin. Soft Matter Phys. 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]
  6. 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]
  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. U.S.A. 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. U.S.A. 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]
  12. Y. He, S. Burov, R. Metzler, and E. Barkai, “Random time-scale invariant diffusion and transport coefficients,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 101, 058101 (2008).
  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,” Q. J. Math. 10, 266 (1939). [CrossRef]
  15. B. D. Hughes, Random Walks and Random Environments (Clarendon Press, Oxford, 1995).
  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. 124, 134704 (2006). [CrossRef] [PubMed]
  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]
  18. M. R. Mazo, Brownian motion (Oxford Univ. Press, Oxford, 2009).
  19. F. Spitzer, Principles of random walk (Springer, New York, 2001).
  20. J.-P. Bouchaud, and A. Georges, “Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications,” Phys. Rep. 12, 195 (1990).
  21. P. Levy, Processus stochastiques et mouvement Brownien (Gauthier-Villars, Paris, 1965).
  22. K. J. Falconer, Fractal Geometry (Wiley, Chichester, 2003). [CrossRef]
  23. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983), pp.237–243 and pp. 326–334.
  24. E. W. Montroll, and G. H. Weiss, “Random walks on lattices,” J. Math. Phys. 6, 364 (1965). [CrossRef]
  25. 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.
  26. 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]
  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]
  28. H. Risken, and H. D. Vollmer, “On the application of truncated generalized Fokker-Planck equations,” Z. Physik B 35, 313 (1979). [CrossRef]
  29. I. V. Gopich, “Concentration effects in “single-molecule” spectroscopy,” J. Phys. Chem. B 112, 6214–6220 (2008). [CrossRef]
  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]
  31. 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).
  32. G. Zumofen, J. Hohlbein, and C. G. Huebner, “Recurrence and photon statistics in fluorescence fluctuation spectroscopy,” Phys. Rev. Lett. 93, 260601 (2004). [CrossRef]
  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. 10, 532–542 (2009). [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 (Lond.) (2010), doi:10.1039/b926956h.
  35. Z. Földes-Papp, S.-C. J. Liao, T. You, E. Terpetschnig, and B. Barbieri, “Confocal fluctuation spectroscopy and imaging,” Curr. Pharm. Biotechnol.in press. [CrossRef] [PubMed]

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