Elucidating the Origin of Anomalous Diffusion in Crowded Fluids

Anomalous diffusion in crowded fluids, e.g., in the cytoplasm of living cells, is a frequent phenomenon. So far, however, the associated stochastic process, i.e., the propagator of the random walk, has not been uncovered. Here we show by means of fluorescence correlation spectroscopy and simulations that the properties of crowding-induced subdiffusion are consistent with the predictions for fractional Brownian motion or obstructed (percolationlike) diffusion, both of which have stationary increments. In contrast, our experimental results cannot be explained by a continuous time random walk with its distinct non-Gaussian propagator.

Figure 1

Representative (normalized) FCS curve for apoferritin (a) in PBS ($\alpha =1$, ${\tau }_D=675\mu \mathrm{s}$) and (b) in a crowded dextran solution ($\alpha =0.81$, ${\tau }_D=97.6\mathrm{ms}$). (c) The distribution of the anomaly $p(\alpha )$ in PBS is very narrow with a mean $⟨\alpha ⟩\approx 0.99$. (d) In a crowded dextran solution, a broadening of $p(\alpha )$ and a shift of the mean ($⟨\alpha ⟩\approx 0.82$) is clearly visible.

Figure 2

Mean square displacement $⟨r(t{)}^2⟩$ for obstructed diffusion (36% obstacle concentration), FBM, and CTRW (from top). Open symbols denote the ensemble-averaged MSD, and crosslike symbols denote the time-averaged MSD for a representative trajectory. While both approaches coincide for FBM and obstructed diffusion, the curves differ for the CTRW due to weak ergodicity breaking. For better visibility, MSD curves for FBM and obstructed diffusion have been shifted upwards (factor 50 and 250, respectively). Full lines scale as $⟨r(t{)}^2⟩\sim t^{0.82}$; the dashed line is linear in time.

Figure 3

(a) Distribution of the anomaly $p(\alpha )$ obtained for obstructed diffusion with 35%–37% obstacle density (black line histogram). The width of the distribution agrees well with the experimental data (gray-shaded histogram; cf. Fig. 1). Data for FBM are shown as filled squares connected by full lines. While here the width of the distribution is consistent with the experimental data, a slight shift in the mean anomaly ($⟨\alpha ⟩\approx 0.84$) with respect to the value imposed in the simulation ($\alpha =0.82$) is observed. (b) The distribution obtained for a CTRW (black line histogram) yielded an average anomaly ($⟨\alpha ⟩\approx 0.59$) that was much lower than the value imposed in the waiting time statistics ($\alpha =0.82$). Moreover, the CTRW-derived distribution is much broader than the experimental distribution (gray-shaded histogram).

Figure 4

(a) The distribution $p({\tau }_D)$ of mean residence times in the confocal volume obtained for obstructed diffusion with 35%–37% obstacle density (black line histogram) is in very good agreement with the experimentally determined distribution (gray-shaded histogram). The distribution for FBM (filled squares connected by full lines) also agrees well with the experimental data. (b) The distribution of residence times obtained for a CTRW with $\alpha =0.82$ (black line histogram) is shifted towards larger values and appears quite broad as compared to the experimental data (gray-shaded histogram).