Fluorescence Molecule Counting for Single-Molecule Studies in Crowded Environment of Living Cells without and with Broken Ergodicity

Abstract

We present a new approach to distinguish between non-ergodic and ergodic behavior. Performing ensemble averaging in a subpopulation of individual molecules leads to a mean value that can be similar to the mean value obtained in an ergodic system. The averaging is carried out by minimizing the variation between the sum of the temporal averaged mean square deviation of the simulated data with respect to the logarithmic scaling behavior of the subpopulation. For this reason, we first introduce a kind of Continuous Time Random Walks (CTRW), which we call Limited Continuous Time Random Walks (LCTRW) on fractal support. The random waiting time distributions are sampled at points which fulfill the condition N < 1, where N is the Poisson probability of finding a single molecule in the femtoliter-sized observation volume ΔV at the single-molecule level. Given a subpopulation of different single molecules of the same kind, the ratio T/ Tm between the measurement time T and the meaningful time Tm, which is the time for observing just one and the same single molecule, is the experimentally accessible quantity that allows to compare different molecule numbers in the subpopulation. In addition, the mean square displacement traveled by the molecule during the time t is determined by an upper limit of the geometric dimension of the living cell or its nucleus.

Anomalous Subdiffusive Measurements by Fluorescence Correlations Spectroscopy and Simulations of Translational Diffusive Behavior in Live Cells

“…However, those measurements and theory only considered a single "dynamic" exponent [30][31][32][33][34]. For the first time, we have developed the theoretical formulation and an FCS-based method in order to separate the spatial subdiffusion from temporal subdiffusion [1,2]. We have observed that FCS is superior at calculating spatially and temporally variable diffusion coefficients and our stimulated and experimental data support this.…”

“…As opposed to normal translational diffusion, in which the movement of molecules is not correlated with their previous position, anomalous translational diffusion molecules are spatially and temporally correlated [1][2][3]. This spatio-temporal correlation reflects a fundamentally different behavior compared with the one in dilute or very dilute solution which, for example, affects the spread of molecules within live cells.…”

“…to processes governed by steps or waiting times with heavy-tailed distributions are used to describe heterogeneities in live cells [1,2]. At the single-molecule level, chemical and physical processes are stochastic, meaning that a reaction takes a variable time to complete.…”

Fractional adsorption diffusion

“…The origin of such kind of diffusion can be either a crowded environment as in cells [9], [2] or the result of a continuous time random walks [24]. However, the spatial diffusion is still formulated using ordinary differentiation.…”

“…The subject became recently again of interest when the ideas of Delahay were used in Monte Carol simulations by Seri-Levy and Avnir [25] and generalized to a fractional formulation by Giona and Giustiniani [12] applied to fractal surfaces. In the field of nano-technology nano rods as catalyst were examined by Kwon et al [15], Such et al [29] experimentally examined diffusion and adsorption of single molecules on the KBr(001) surfaces and for bio-chemical applications like anomalous diffusion in crowded environments see [19], [9]. For recent applications on the nano scale see the compilation of Quirke [22].…”