Revealing nonergodic dynamics in living cells from a single particle trajectory

We propose the improved ergodicity and mixing estimators to identify nonergodic dynamics from a single particle trajectory. The estimators are based on the time-averaged characteristic function of the increments and can thus capture additional information on the process as compared to the conventional time-averaged mean-square displacement. The estimators are first investigated and validated for several models of anomalous diffusion, such as ergodic fractional Brownian motion and diffusion on percolating clusters, and nonergodic continuous-time random walks and scaled Brownian motion. The estimators are then applied to two sets of earlier published trajectories of mRNA molecules inside live Escherichia coli cells and of Kv2.1 potassium channels in the plasma membrane. These statistical tests did not reveal nonergodic features in the former set, while some trajectories of the latter set could be classified as nonergodic. Time averages along such trajectories are thus not representative and may be strongly misleading. Since the estimators do not rely on ensemble averages, the nonergodic features can be revealed separately for each trajectory, providing a more flexible and reliable analysis of single-particle tracking experiments in microbiology.

Figure 1

The mean (a, c) and standard deviation (b, d) of the mixing (a, b) and ergodicity (c, d) estimators at $\omega =10$ as a function of $n/N$ for CTRW with $\alpha =0.7,\sigma =1$, and three trajectory lengths $N=100,1000,10000$ (symbols). For comparison, solid line shows the theoretical limits (13) and (14) of the mean estimators for CTRW and the standard deviation for Brownian motion (given by square roots of Eqs. (11) and (12) with $N=100)$.

Figure 2

The mean mixing estimator $\langle {\widehat{E}}_{\omega }(1,1000)\rangle$ as a function of $\omega$ for CTRW with $\alpha =0.7$ and $\sigma =1$, and four levels of white Gaussian noise: ${\sigma }_{\varepsilon }=0$ (circles) and ${\sigma }_{\varepsilon }=0.01,0.1,1$ (lines). Note that the plateau at 0.6 is smaller than the value $\alpha =0.7$ expected from Eq. (13) because of the finite length effect $\left(N=1000\right)$.

Figure 3

(Top) Examples of trajectories of four anomalous diffusion models: fBm, CTRW, diffusion on percolating clusters, and sBm, with the same scaling exponent $\alpha =0.7$ and one-step size $\sigma =1$. Each trajectory was corrupted by the white Gaussian noise of level ${\sigma }_{\varepsilon }=0.2$ and then renormalized by the empirical standard deviation of its increments. (Bottom) The real part of the mixing and ergodicity estimators at $\omega =2$ applied to these four trajectories.

Figure 4

The real part of the mixing estimator at $\omega =2$ for ten simulated trajectories from two anomalous diffusion models: ergodic fBm with $2H=0.7$ (a) and nonergodic CTRW with the same exponent $\alpha =0.7$ (b) (in both cases, we set $\sigma =1$ and $N=500)$. All trajectories were corrupted by white Gaussian noise with standard deviation ${\sigma }_{\varepsilon }=0.1$. Each trajectory is renormalized by the empirical standard deviation of its increments. Light-gray shadowed region delimits the typical range of fluctuations for Brownian motion, i.e., the mean plus and minus the standard deviation.

Figure 5

The real part of the ergodicity estimator at $\omega =2$ for ten simulated trajectories from four anomalous diffusion models with the same exponent $\alpha =2H=0.7$: fBm (a), CTRW (b), diffusion on percolating cluster (c), and sBm (d) (in all cases, we set $\sigma =1$ and $N=500)$. All trajectories were corrupted by white Gaussian noise with standard deviation ${\sigma }_{\varepsilon }=0.1$. Each trajectory was renormalized by the empirical standard deviation of its increments.

Figure 6

The real part of the mixing (a) and ergodicity (b) estimators at $\omega =2$ as a function of $n$ for two experimental trajectories of around $N=1500$ points: the motion of mRNA molecule inside live E. coli cell [33] and Kv2.1 potassium channel anomalous dynamics in the plasma membrane [1]. Both trajectories (shown in insets) were renormalized by the empirical standard deviation of their increments. Light-gray shadowed region delimits the typical range of fluctuations of estimators for Brownian motion, i.e., the mean plus and minus the standard deviation.

Figure 7

The real part of the ergodicity estimator at $\omega =2$ for ten experimental trajectories of two samples: the motion of mRNA molecules inside live E. coli cells [33] and Kv2.1 potassium channel anomalous dynamics in the plasma membrane [1]. Each trajectory is renormalized by the empirical standard deviation of its increments. The trajectory length ranges between 400 and 500 points for the first set and around 1500 points for the second set. Light-gray shadowed region delimits the typical range of fluctuations for Brownian motion, i.e., the mean plus and minus the standard deviation.

Figure 8

The real parts of the mixing (a) and ergodicity (b) estimators at $\omega =1$ applied to the increments of a single CTRW trajectory (dashed line) and to the positions of the same trajectory (solid line), with $N=1000,\alpha =0.7$, and $\sigma =1$. In both cases, $X\left(n\right)$ are renormalized by the empirical standard deviation of the increments. Thin dash-dotted line presents the mean estimators from Eqs. (13) and (14).

Figure 9

The real part of the original ergodicity estimator, $\mathrm{Re}\left\{n^{-1}{\sum }_{k=0}^{n-1}\widehat{E}\left(k\right)\right\}$ from Ref. [14], applied to ten experimental trajectories of two samples: (a) the motion of mRNA molecules inside live E. coli cells [33] and (b) Kv2.1 potassium channel anomalous dynamics in the plasma membrane [1]. The estimator is applied to the increments of each trajectory, which are renormalized by their empirical standard deviation. The trajectory length ranges between 400 and 500 points for the first set and around 1500 points for the second set.

Figure 10

The real part of the ergodicity estimator as a function of $n$ for a single fGn trajectory with $N=500,H=0.9$, and $\sigma =1$. At $\omega =1$ (solid line), the estimator deviates from zero that may wrongly suggest nonergodic behavior. Increasing $\omega$, one can eliminate this false conclusion.

Figure 11

The mean mixing (a) and ergodicity (c) estimators as a function of $n$ for a Brownian motion trajectory with $N=1000$ and ${\sigma }^2=0.04$. At $\omega =1$ (solid line), both estimators are not small that may wrongly suggest nonmixing or nonergodic behavior. Increasing $\omega$, one can eliminate this false conclusion. The same conclusions can be made from the real part of the mixing (b) and ergodicity (d) estimators for a single Brownian motion trajectory with the same parameters.

Figure 12

The real part of the mixing (a) and ergodicity (b) estimators at $\omega =1$ as a function of $n$ for an original Brownian motion trajectory with $N=1000$ and $\sigma =1$ (solid line) and for the same trajectory corrupted by resetting ten randomly chosen points to 0 (dashed line). In both cases, the trajectory was renormalized by the empirical standard deviation of its increments. Light-gray shadowed region delimits the mean plus and minus the standard deviation of each estimator computed for Brownian motion according to Eqs. (8), (9), and (11).

Figure 13

The standard deviation of the mixing (a) and ergodicity (b) estimators at several values of $\omega$ for Brownian motion with $N=100$ and $\sigma =1$. The large $\omega$ asymptotic limits given by square roots of Eqs. (11) are shown by solid lines.

Figure 14

The mean mixing (a) and ergodicity (b) estimators at $\omega =1$ as a function of $n$ for one coordinate $X\left(n\right)$ of two-dimensional random walk of length $N=500$ on a critical percolating cluster on a $1000\times 1000$ square lattice, with the lattice step $\sigma =1$. Each curve presents the mean computed by averaging over 1000 trajectories generated on one cluster. The curves obtained for ten random clusters are almost identical. For comparison, light-gray shadowed region delimits the mean plus and minus the standard deviation of each estimator computed for Brownian motion according to Eqs. (8), (9), and (11).

Figure 15

The mean (a, c) and standard deviation (b, d) of the mixing (a, b) and ergodicity (c, d) estimators at $\omega =1$ (lines) and $\omega =10$ (symbols) for CTRW with $N=1000,\alpha =0.7,\sigma =1,{\delta }_s=1$, and exponential cutoff at $T_c=10,{10}^2,{10}^3$.

Figure 16

The mean mixing (a) and ergodicity (b) estimators at $\omega =1$ as a function of $n$ for geometric Brownian motion with $\mu =0$ and $\sigma =0.01,\sigma =0.1$, and $\sigma =1$. Each generated trajectory was renormalized by the empirical standard deviation of its increments in order to get comparable results for different $\sigma$. The mean was computed numerically by averaging over 1000 trajectories.