Single-particle tracking data reveal anticorrelated fractional Brownian motion in crowded fluids

Anomalous diffusion with a sublinear growth of the particles' mean-square displacement (subdiffusion) has been observed frequently in crowded fluids, e.g., in the cytoplasm of living cells or in artificial solutions. Based on a recently reported set of single-particle tracking data, it is shown here that trajectories of nanoparticles immersed in artificial crowded fluids display all signatures of anticorrelated fractional Brownian motion. Moreover, the trajectories' power spectrum follows a scaling that reports on the fluid's viscoelasticity. Macromolecular crowding therefore renders fluids viscoelastic which in turn leads to subdiffusion of immersed tracer particles with all the characteristics of fractional Brownian motion.

Figure 1

Representative examples of the cumulative distribution of squared displacements [Eq. (1)] for sucrose and dextran solutions (full and open symbols, respectively). Data are shown as $-\ln [1-P\left(r_{\tau }^2\right)]$ vs $r_{\tau }^2$ in double-logarithmic style. A linear scaling (dashed lines) is anticipated for random walks with a Gaussian PDF of increments. All data sets follow this scaling over several orders of magnitude ($\tau =0.1,0.4,1.6\mathrm{s}$ shown as circles, diamonds, and squares, respectively).

Figure 2

Rescaling the squared increments of Fig. 1 with their respective MSD, $x=r_{\tau }^2/\langle r_{\tau }^2\rangle$, leads to a collapse of all data (colored lines) to a single, parameter-free master curve $P\left(x\right)=1-e^{-x}$, i.e., all data follow the prediction $-\ln [1-P(x\left)\right]=x$ over several orders of magnitude (indicated by open circles). Thus, all trajectories have a Gaussian PDF of increments irrespective of the lag time $\tau$. Inset: The MSD used for rescaling shows the previously reported subdiffusion [8]. For sucrose solutions (open symbols), $\langle r_{\tau }^2\rangle /\tau =\mathrm{const}$ (full line); for dextran solutions (closed symbols) $\langle r_{\tau }^2\rangle /\tau \sim 1/{\tau }^{0.2}$ (dash-dotted line). Beyond $\tau \sim 1$ s, data for dextran solutions converge towards normal diffusion (dashed line). Data for dextran solutions have been shifted up by a factor 5 for better visibility.

Figure 3

(a) The correlation function of particle displacements, $C_{\tau }\left(t\right)$ [Eq. (2)], shows a rapid decay from unity towards zero in sucrose solutions, in agreement with simulation results on normal Brownian motion (inset). Colored lines represent time lags $\tau =0.1,0.2,0.4,0.8,1.6\mathrm{s}$ (shown left to right, respectively, as black, green, red, blue, and grey). (b) In contrast, data for dextran solutions show a pronounced anticorrelation [$C_{\tau }\left(t\right)<0$] after an initial correlation decay towards zero. This result is in favorable agreement with simulation data on FBM (inset), therefore confirming that trajectories in dextran solutions have all the features of an anticorrelated FBM. Here, $\Delta t=4$ ms is the trajectories' temporal resolution.

Figure 4

The number of distinct sites visited within a period $\tau$ divided by the MSD (averaged over three trajectories) shows a scaling $s_{\tau }/\langle r_{\tau }^2\rangle \sim {\tau }^{\delta }$ with $\delta =0$ for both sets of experimental data, hence confirming FBM in dextran solutions. In contrast, simulations of OD show $\delta <0$ as expected [15].

Figure 5

(a) The power spectrum $S\left(\omega \right)$ (averaged over three individual trajectories) in sucrose (blue) agrees well with the prediction for purely viscous fluids, $S\left(\omega \right)\sim 1/{\omega }^2$ (dashed line). In contrast, data for dextran solutions show a clear indication for fractional Gaussian noise and viscoelasticity in the high-frequency regime [dash-dotted line; $S\left(\omega \right)\sim 1/{\omega }^{1.7}$]. For better visibility, $\omega$ has been normalized by the smallest possible frequency ${\omega }_0$, and $S\left({\omega }_0\right)$ has been set to unity. (b) The power spectrum of simulated CTRW trajectories (blue) follow the scaling of a normal random walk, whereas obstructed diffusion at the percolation threshold (black) also yields a nontrivial scaling $S\left(\omega \right)\sim 1/{\omega }^{1.7}$.