Fingerprints of viscoelastic subdiffusion in random environments: Revisiting some experimental data and their interpretations

Many experimental studies revealed subdiffusion of various nanoparticles in diverse polymer and colloidal solutions, cytosol and plasma membrane of biological cells, which are viscoelastic and, at the same time, highly inhomogeneous randomly fluctuating environments. The observed subdiffusion often combines features of ergodic fractional Brownian motion (reflecting viscoelasticity) and nonergodic jumplike non-Markovian diffusional processes (reflecting disorder). Accordingly, several theories were proposed to explain puzzling experimental findings. Below we show that some of the significant and profound published experimental results are better rationalized within the viscoelastic subdiffusion approach in random environments, which is based on generalized Langevin dynamics in random potentials, than some earlier proposed theories.

Figure 1

Experimental data extracted using Engauge Digitizer [58] from Ref. [28] and their fits. In (a) and (b), the ensemble-averaged mean-square displacement vs time is taken from Fig. 1 in [28]. In all cases, $R=0.25\mu \mathrm{m}$; however, mesh size varies from $\xi =0.75\mu \mathrm{m}$ (empty magenta triangles), through $\xi =0.55\mu \mathrm{m}$ (filled blue circles) and $\xi =0.30\mu \mathrm{m}$ (filled green squares) to $\xi =0.25\mu \mathrm{m}$ (filled red diamonds). In (a), various power-law fits are presented with the parameters given in this plot (see the main text for more detail). In (b), the same data are fitted with Eq. (1) and the parameters shown. In (c), the data on the RTD from Fig. 3(b) of [28] for $\xi =0.31\mu \mathrm{m}$ and the same $R$ are fitted by (i) power-law dependence $\psi \left(\tau \right)\propto {\tau }^{-1-\alpha }$ with $\alpha =0.33$, which was used in [28] (dashed black line), and (ii) a generalized log-normal distribution (2) (solid green line), with the parameters given in the plot.

Figure 2

Experimental data (empty black squares) on the dependence of subdiffusion exponent $\alpha$ on the ratio of the particle radius $R$ and the mesh size $\xi$ in an entangled actin meshwork and their fit (solid black line). Data are extracted using Engauge Digitizer [58] from Fig. 4 of Ref. [28]. Fit is done by $\alpha =1/[1+{(cR/\xi )}^d]$ dependence using one fitting parameter $c$ at fixed $d=4$. The optimal value is $c=1.45$. The quality of fit is not visibly improved, if to make also $d$ variable. The red open circles with their connecting dashed red line depict the value of $\alpha \approx 4/{\sigma }_{\mathrm{eff}}$ predicted by our theory using fitting values ${\sigma }_{\mathrm{eff}}$ from Fig. 1, which in turn were obtained by applying Eq. (1) to data from Ref. [28]. The predicted value of $\alpha$, which corresponds to $R/\xi =1/3$, lies outside of the figure's frame, being unphysical, as $\alpha$ cannot exceed 1. This fit is expected to work only for $\sigma \gtrsim 2$ or ${\sigma }_{\mathrm{eff}}\gtrsim 5.86$. Nevertheless, the overall agreement between the theory and experimental data is convincing. The inset shows fitting values $t_0$ from Fig. 1 and their exponential fit.

Figure 3

Experimental RTD and their fits. Data for two values $R_H\approx 22.36\mathrm{nm}$ (filled black squares) and $R_H\approx 44.72\mathrm{nm}$ (empty green triangles) are extracted using Engauge Digitizer [58] from Fig. 4 in [48] for clustered Kv2.1 ion channels initially confined to a circle of radius $R_H$. Different fits are presented: (i) power law with $\alpha =0.9$ (dashed red line) following [48], for $R_H\approx 22.36\mathrm{nm}$; (ii) generalized log-normal distributions (2) (dash-dotted light-green and solid black lines), with the parameters shown. The second fit is evidently better than the power-law fit, which covers one time decade at best and is hampered by a strong data noise for $\tau >4$ s. Moreover, for $R_H\approx 22.36\mathrm{nm}$ the RTD is most obviously not a power law asymptotically—the largest experimental residence time is about 4 s only.

Figure 4

Displacement distribution of RNA-protein particles in the cytosol of (a) $E$. coli cells and (b) $S$. cerevisiae cells normalized by the standard deviation of displacement for two instants of time in each case. Data are taken from Fig. 4 of [39] using Engauge Digitizer [58]. Fits [solid blue (dark) and orange (light) lines] are done using Eq. (4) with fitting parameters shown in the plots. Variable values $1<\chi <2$ allow for visibly better fits of the central part than the scaled and normalized Laplace distribution $P\left(x\right)=exp(-\sqrt{2}\left|x\right|)/\sqrt{2}$ (dashed black lines) [39], also depicted for comparison.