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/ T(m) between the measurement time T and the meaningful time T(m), 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.

Fig. (1)

The minima of the variations $\left|\stackrel{\sim }{\mathit{\gamma }}-{〈\stackrel{\sim }{\mathit{\gamma }}〉}_{\mathit{sub}}\right|$ are shown as a function of the number of single molecule tracks $N_{\ell }\in \mathbb{N}$. In these graphs, the lower bound ${\stackrel{\sim }{\mathit{\gamma }}}_{\min }$ is fixed to the value ${\stackrel{\sim }{\mathit{\gamma }}}_{\min }$ =0.243 representing the optimal choice for the experimental $\stackrel{\sim }{\mathit{\gamma }}$= 0.689. The curves are assigned to the upper bounds of $\stackrel{\sim }{\mathit{\gamma }}$ from top to bottom. On the right side of the figure we have ${\stackrel{\sim }{\mathit{\gamma }}}_{\max } = \left\{0.785,0.795,0.797,\mathrm{0.799.0.8}\right\}$. We identify that for a given $\stackrel{\sim }{\mathit{\gamma }}$= 0.689 the interval for selecting the random number of single-molecule tracks from the total ensemble of singlemolecule macromolecules in the subpopulation (the total ensemble of single-molecule tracks) should range from $\left[{\stackrel{\sim }{\mathit{\gamma }}}_{\min },{\stackrel{\sim }{\mathit{\gamma }}}_{\max }\right]$ = [0.243, 0.799]; for this range ${〈\stackrel{\sim }{\mathit{\gamma }}〉}_{\mathit{sub}}\in \left[{\stackrel{\sim }{\mathit{\gamma }}}_{min},{\stackrel{\sim }{\mathit{\gamma }}}_{max}\right]$, we find the minimal variation if the number of randomly selected single-molecule tracks is $N_{{\ell }_{max}} = 32$. All other values of $N_{\ell }$ deliver only a local minimum instead of a global minimum. The graphs also show that the variation approaches a stable value if $N_{\ell }$ approaches large values; i.e. only a small subpopulation of single molecules delivers the minimal variation.

Fig. (2)

The variation of the $\stackrel{\sim }{\mathit{\gamma }}$-interval $\left[{\stackrel{\sim }{\mathit{\gamma }}}_{min},{\stackrel{\sim }{\mathit{\gamma }}}_{max}\right]$ with respect to the MSD scaling behavior $\stackrel{\sim }{\mathit{\gamma }}$ in the optimization process for deriving sub ${〈\stackrel{\sim }{\mathit{\gamma }}〉}_{\mathit{sub}}$ by using Eqs. (12) and (13). In the graph we can distinguish different domains for$\stackrel{\sim }{\mathit{\gamma }}$. The first domain ranges from 0 < $\stackrel{\sim }{\mathit{\gamma }}$ < 0.392… allowing a fixed interval for the limits $\left[{\stackrel{\sim }{\mathit{\gamma }}}_{min},{\stackrel{\sim }{\mathit{\gamma }}}_{max}\right]$ = [0, 1/2]. For values 0.392…< $\stackrel{\sim }{\mathit{\gamma }}$ < 0.878…, we observe a highly structured set of intervals where the upper and lower limit reaches some minimal or maximal value and allows the whole range [0, 1] for specific values. The optimal number of tracks is for both $\stackrel{\sim }{\mathit{\gamma }}$-intervals equal to 32. For the last interval 0.878…< $\stackrel{\sim }{\mathit{\gamma }}$ < 1, we observe the maximal value ${\stackrel{\sim }{\mathit{\gamma }}}_{max}\approx 1$ while the lower limit ${\stackrel{\sim }{\mathit{\gamma }}}_{\min }$ varies between 0 < ${\stackrel{\sim }{\mathit{\gamma }}}_{\min }$ <1/2 . For this last domain of $\stackrel{\sim }{\mathit{\gamma }}$, the number of tracks decreases to a smaller value $N_{{\ell }_{max}}$ < 32. The resolution in $\stackrel{\sim }{\mathit{\gamma }}$ to derive the shown plot was Δ$\stackrel{\sim }{\mathit{\gamma }}$ =0.00078125.

Fig. (3)

The graph shows a random selection of $N_{{\ell }_{max}}$ = 32 single-molecule tracks (dots) within the bounds $\left[{\stackrel{\sim }{\mathit{\gamma }}}_{\min },{\stackrel{\sim }{\mathit{\gamma }}}_{\max }\right]$ determined in the optimization for a given value of $\stackrel{\sim }{\mathit{\gamma }}$= 0.689. The graph is using the data listed in Fig. 1. The solid green line corresponds to the average sub ${〈\stackrel{\sim }{\mathit{\gamma }}〉}_{\mathit{sub}}$ over the 32 randomly selected single-molecule tracks.

Fig. (4)

A single HeLa cell was optically sectioned by two-photon imaging after transfection with an Alexa488-labeled short RNA duplex (SQ-dsCon2) in order to visualize the geometrical dimension of the cell nucleus, i.e. its measured geometrical size. Two-photon imaging is described elsewhere [40].