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] are widely used methods. Their detection capability of slowly and non-exponentially relaxing dynamics makes them ideal for the study of statistical mechanics [2, 3]. 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]. 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]. 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, 6].

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 d_x–y = 1.22λ/(2NA) in the x–y plane; the resolution in the z-direction is given by d_z = 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, 8], saturated structured illumination microscopy (SSIM) [9], 3D stochastic optical reconstruction microscopy (STORM) [10], and photo-activated localization microscopy (PALM) [11].

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]. 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]. This return probability for three dimensions is 0.6594… [14,15].

In most of the papers on random walks related to fluorescence microscopy the simplification of a one or a two dimensional walk is made [1–5, 15]. Here, we will carry out simulations in three dimensions to represent the real experimental setup in single-molecule detection. In this paper, we therefore consider the case of three-dimensional Brownian walks of single molecules or particles in solution-phase, e.g. without immobilization on surfaces like cover slips or hydrodynamic/electrokinetic focusing. We also introduce a 3D observation volume ΔV of cylindrical shape and size of about 0.21 fL for a red excitation or 0.14 fL for a blue excitation (1 fL = 10⁻¹⁵ L). The observation volume itself does not cause a physical restriction for the particles but it partitions the space into two parts, inside and outside the observation volume. In single-molecule detection techniques like one-photon or two-photon fluorescence fluctuation microscopy, the observation volume is cylindrically shaped with a size of about 0.05 to 2 fL. The observation volume is the focus of the excitation laser beam itself. The single molecules or particles move in and out the observation volume at random. Thereby, they get excited and emit fluorescence light. The molecules generate photon showers. The photon showers differ from the mean fluorescence intensity that is obtained by averaging over the whole measurement time T in the experiment. These deviations are due to the binary on-off process of molecule number fluctuations of single molecules. In the real experiment, no information can be obtained about the single molecules as long as the molecule dwells outside the observation volume. With the average molecule number N = 0.0055 for ΔV = 0.21 fL measured with 24-nm nanospheres at 635 nm laser excitation (red excitation) and N = 0.0052 for ΔV = 0.14 fL measured with 100-nm nanospheres at 470 nm laser excitation (blue excitation) in the real fluorescence fluctuation time series, we reached the lower single-molecule detection limit in solution. The contribution of two simultaneously fluorescing molecules to the detected signal during measurement vanishes for N ≪ 1, and the single-molecule detection regime comes close to the average N = 0.048 and N = 0.0057 molecules and particles, respectively, per ΔV [16].

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]. 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 $8.0·\frac{{10}^{-12}{m}^2}{s}=\frac{{\left(10\mathrm{nm}\right)}^2}{2.1\mu s}·\frac{1}{6}$ , 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

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

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

where r ₀ is the origin of the track of n-steps represented as continuous function ℬ_n (r ₀,r) for the end point r. The corresponding generating function is H (z,r ₀,r) = Σ^∞ _n=0 zⁿ ℬ_n (r ₀,r), which allows us to define the moments of the walk [18, 19]. Under the condition that the first two moments exists, mean and variance, the central limit theorem assures that the random walk features a mean velocity and a diffusion coefficient characteristic for diffusion processes [20, 21]. In addition, it is well known that in the mean field approximation a Brownian walk is equivalent to a diffusion process [18]. For Brownian walks, it is also known that there is a scaling behavior of the walk related to temporal and spatial coordinates [22]. It follows that r(t) and θ ^−1/2 r(θ · t) have the same distribution. Thus, changing the temporal scale by a factor θ and the spatial scale by a factor θ ^1/2 gives a process indistinguishable from the original; this is called statistical self-affinity of the Brownian motion [22, 23]. We use these kind of tracks to represent the motion of a single molecule in a force free environment. We checked the two first moments of our tracks to assure that we deal with a Brownian particle undergoing a diffusive motion. The mean and the variance exist and the last of these quantities are shown as an example in Fig. 1. If we simulate a walk, then the relation 〈r ²〉~ n must be satisfied in the mean. For the test, we generated different lengths of random walks and average over a total number m = 1000 tracks for each length. From Fig. 1, we clearly detect that the relation for the Brownian walk is linear in our simulations. Consequently, we are dealing with a classical Brownian particle in our calculations. The measured diffusion coefficients of 24-nm and 100-nm nanospheres in aqueous solution (by fluorescence correlation spectroscopy) were in good agreement with the diffusion coefficients for classical Brownian motion given by the Einstein-Stokes relation.

 

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

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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, 25]. 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(r)·δ(t)

is

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, 25]. Both methods do not give different results.

In florescence fluctuation microscopy measurements, the point spread function (PSF) is used to define an observation volume ΔV which is governed by the 1/e ² decay of the laser intensity distribution. In our model, we assume that ΔV is a finite, well defined volume. Strictly speaking, the PSF extends to infinity, however outside the 1/e ² decay regime the intensity does not contribute to the measurement signal above the background noise. Consequently, there is a simple in and out jump of the particle or an on and off of the signal constrained by ΔV. This represents the real molecule number fluctuation for the measurements. Our goal here is not to discuss the PSF in more detail because the extension of this probability to infinity is merely formal in a real experimental situation with background noise. The molecule number fluctuations inside ΔV between 1 molecule and 0 molecule are physically related to the time-dependent response gathered as a time series measurement. To describe the dynamics of the molecule itself, we introduce the scalar function η(t) for the real experimental situation of measuring a single molecule at a time

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

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]. 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]; 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.

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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

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]. Under the conditions N < 1 per size ΔV, we specified k by k = N / τ_dif, where τ_dif is the diffusion time per ΔV [27]. 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] 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], 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].

 

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 / τ_dif = 0.0052 / 2.79·10⁻³ s = 1.86 s ⁻¹, where τ_dif is the measured diffusion time, for example, of the 100-nm nanospheres in aqueous solution.

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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

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 p_t(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)〉 ∝ t^{$\tilde{\gamma }$} [17]. This behavior is known as anomalous diffusion but sub-diffusion of mixed origin can coexist as found recently [4]. 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 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_i+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 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.

 

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.

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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.

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Figure 6 proves that the on-lengths of single-molecule tracks r(n) represent the source of the real photon count rates measured. The simulated single-molecule lengths and the real photon count rates are not separate. Small on-lengths of the single-molecule track r(n) within the observation volume ΔV correspond to small photon count rates I _i+1(Δt) ∈ [t_i,t_i+Δt) at a certain Δt for time points t_i∈ T of the fluorescence fluctuation photon stream. Large on-lengths of the single molecules are related to higher photon counts rates. The accuracy of the fluorescence fluctuation measurements with an average molecule number N = 0.0055 for ΔV = 0.21 fL has not allowed us to map the large photon count rates of the measurement. Even at very low concentrations there are bursts of photons emitted by chemical aggregates of two or more particles, but the majority of bursts originate from single particles; therefore, the distribution of photons obtained from such trajectories can be approximated by a single-molecule one [29]. Thus, the single-molecule description is valid when photon counts are not too high. The magnitude of the anomalous exponent κ in Fig. 6 is determined by the conjectured box counting dimension [23]. We show below that the exponent associated with this fractal or anomalous power-law scaling is mathematically sound and that the model parameters can be understood in terms of the molecule number fluctuations of Eqs. (4) and (5) under ΔV constraint given by Eqs. (6) and (7) of the real sample. For this purpose, we first prove why it is appropriate to refer to the power-law exponent κ in Fig. 6 as a fractal or anomalous dimension by using a box-counting method.

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. Δt_off 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, Δt_off = 1 ms occurred with a frequency of 386 and Δt_off = 2 ms with a frequency of 419. For Δt_off ≥ 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) ~ I^κ 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).

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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

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

The arc length dl is given by

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

where ξ is the stochastic location of the track estimator for the track at location r. The track r is included in δV. Note that δV is not necessarily completely covered by ΔV, however it is guaranteed by this construction that the single-molecule track r is completely inside ΔV. That is ΔV ≤ ∑_i δV_i where i is the total number of on-tracks. If l is sufficiently small, then the approximated location ξ is approaching the track position r. Like Eq. (15), Eq. (16) is exact, but formal, in that it merely defines the properties of dl, P(ξ) and P(dl) in terms of those of δV. 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^κ and P(I) ~ I^κ as depicted in Fig. 6. 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(r) = 1/ΔV = constant for r ∈ ΔV in the analysis of experimental Photon Counting Histograms (PCH) for single-particle tracking [30] is not valid. As we prove by our stochastic track estimator P(ξ) for the track at location r, P(ξ) depends on the position of the single molecule within ΔV given by Eq. (16), so does P(r). In molecular system studied at a many-molecule level there are many molecules presents even at low concentrations and P(r) might be a constant. We derived an explicit expression for P(r) 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].

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]. Several examples of power-law ‘fragmentation’ were given in ref. [32] 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] 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.

In the present study, the temporal quantification of the elementary events of Brownian motion (normal diffusion) that tend to progress towards a state of equilibrium is used to predict macroscopic behavior of a system as it approaches equilibrium. Major improvements in the sensitivity of detection permitted the emergence of fluorescence fluctuation microscopy as a strong contender to other single-molecule detection formats [33]. Bridging single-molecule approaches with ensemble averages has been instrumental in establishing that true single-molecule quantification can be rendered accurate when proper algorithms for simulation and experimental validation are used. The physical existence of irregular patterns in real and simulated data sets for small molecule numbers per observation volume challenges us to briefly discus below related subtleties. In such cases, our approach may be adopted for other single-molecule investigations.

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]. 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]. 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]. 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].

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 τ_dif per size of ΔV. Most important in our findings is that the mean rate of re-entries k defined by k = N / τ_dif is independent of the geometry of the observation volume ΔV but depends on its size and the diffusive properties τ_dif 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) ~ ℓ^κ and P(I) ~ I^κ 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.