Anomalous diffusion due to hindering by mobile obstacles undergoing Brownian motion or Orstein-Ulhenbeck processes

In vivo measurements of the passive movements of biomolecules or vesicles in cells consistently report “anomalous diffusion,” where mean-squared displacements scale as a power law of time with exponent $\alpha <1$ (subdiffusion). While the detailed mechanisms causing such behaviors are not always elucidated, movement hindrance by obstacles is often invoked. However, our understanding of how hindered diffusion leads to subdiffusion is based on diffusion amidst randomly located immobile obstacles. Here, we have used Monte Carlo simulations to investigate transient subdiffusion due to mobile obstacles with various modes of mobility. Our simulations confirm that the anomalous regimes rapidly disappear when the obstacles move by Brownian motion. By contrast, mobile obstacles with more confined displacements, e.g., Orstein-Ulhenbeck motion, are shown to preserve subdiffusive regimes. The mean-squared displacement of tracked protein displays convincing power laws with anomalous exponent $\alpha$ that varies with the density of Orstein-Ulhenbeck (OU) obstacles or the relaxation time scale of the OU process. In particular, some of the values we observed are significantly below the universal value predicted for immobile obstacles in two dimensions. Therefore, our results show that subdiffusion due to mobile obstacles with OU type of motion may account for the large variation range exhibited by experimental measurements in living cells and may explain that some experimental estimates are below the universal value predicted for immobile obstacles.

Figure 1

Computer simulations of subdiffusion due to immobile obstacles in two dimensions. The time course of the ratio between the mean-squared displacement and time, $\langle R^2\left(t\right)\rangle /t$, is shown for increasing obstacle densities. Each color codes for a different obstacle density, expressed here as the excluded volume fraction $\theta$, i.e., the fraction of the accessible surface occupied by obstacles. Here, $\theta =0$, 0.197, 0.355, 0.395, 0.441, and 0.484 (from top to bottom). The black dashed line locates the power law $y\propto t^{-0.34}$, yielding percolation threshold ${\theta }_c=0.441$ and anomalous diffusion exponent $\alpha =0.66$. The insets show representative trajectories (red) of protein amidst obstacles (green disks) at the indicated densities. The black disks locate the initial and final position of the proteins. Parameters: protein radius $r_{\mathrm{w}}=2.0\mathrm{nm}$, obstacle radius $r_{\mathrm{obs}}=5.0\mathrm{nm}$, protein diffusion constant $D_{\mathrm{RW}}=1.0\mu {\mathrm{m}}^2/\mathrm{s}$, time step $\Delta t=0.25\mu \mathrm{s}$, space step $\Delta r=1\mathrm{nm}$, and total domain size $L_x=L_y=5.0\mu \mathrm{m}$. All data are averages of the motion of 10 proteins per obstacle configurations and 500 obstacle configurations.

Figure 2

Protein diffusion in two dimensions in the presence of mobile obstacle with Brownian motion. The time courses of $\langle R^2\left(t\right)\rangle /t$ are shown for increasing values of the diffusion constant of the obstacles $D_{\mathrm{obs}}=0$, $2\times {10}^{-4}$, $5\times {10}^{-3}$, 0.125, and 1.0 $\mu {\mathrm{m}}^2/\mathrm{s}$ (from bottom to top). The black dashed line locates the critical regime $y\propto t^{-0.34}$ for immobile obstacles. The insets show representative trajectories of the obstacles (Brownian motion) at the indicated diffusion constants. The black bars indicate the spatial scale of the trajectories and the green disks locate the starting and ending positions (the radii of the green disks are not to scale). The obstacle density was set at the percolation threshold for immobile obstacles, i.e., $\theta =0.441$. Data are averaged over 10 proteins per obstacle configurations and 200 obstacle configurations. All other parameters are as in Fig. 1, including the protein diffusion constant $D_{\mathrm{RW}}=1.0\mu {\mathrm{m}}^2/\mathrm{s}$.

Figure 3

Protein diffusion in two dimensions in the presence of mobile obstacles with Ornstein-Uhlenbeck motion. The time courses of $\langle R^2\left(t\right)\rangle /t$ are shown for the same values (same color code) of the obstacle diffusion constant as in Fig. 2: $D_{\mathrm{obs}}=0$, $2\times {10}^{-4}$, $5\times {10}^{-3}$, 0.125, and 1.0 $\mu {\mathrm{m}}^2/\mathrm{s}$ (from bottom to top), except that the obstacle motion here is simulated by an Ornstein-Uhlenbeck process with relaxation constant $\tau =1\mu \mathrm{s}$. All other parameters are as in Fig. 2, including the obstacle density $\theta =0.441$.

Figure 4

Protein diffusion amidst Ornstein-Uhlenbeck mobile obstacles at suprathreshold densities. The density of Ornstein-Uhlenbeck obstacles increases from top to bottom as $\theta =0.441$, 0.501, 0.548, and 0.606 (density is expressed as the excluded volume fraction of an equivalent number of immobile obstacles). The slopes of the $\langle R^2\left(t\right)\rangle /t$ curves at long times range from 0.23 (intermediate regime for $\theta =0.501$) to 0.43 (long times for $\theta =0.606$), yielding estimates for the apparent anomalous exponents $\alpha \in [0.57–0.77]$. Insets are illustrative snapshots of obstacle locations at a time point of the simulation, intended to illustrate the difference in excluded fractions. Parameters of the obstacle OU motion: diffusion constant $D_{\mathrm{obs}}=1.0\mu {\mathrm{m}}^2/\mathrm{s}$ and relaxation constant $\tau =1\mu \mathrm{s}$. All other parameters are as in Fig. 2, including the protein diffusion constant $D_{\mathrm{RW}}/D_{\mathrm{obs}}=1.0$.

Figure 5

Continuum of obstacle movements from Ornstein-Uhlenbeck to Brownian motion. The time courses $\langle R^2\left(t\right)\rangle /t$ are shown for different values of the relaxation time of OU mobile obstacles, $\tau =1,10,100$, and 1000 $\mu \mathrm{s}$ (from bottom to top). The topmost (pale blue) curve is for Brownian motion in the same conditions (formally corresponding to $\tau =\infty$). The straight lines illustrate the corresponding values of the anomalous exponent $\alpha$ when $\tau$ varies, from $\alpha =0.71$ (dashed line) to $0.88$ (dashed-dotted line). The insets show representative trajectories at the indicated values of $\tau$. The black bars indicate the spatial scale of the trajectories and the green disks locate the starting and ending positions (the radii of the green disks are not to scale). Obstacle diffusion constant $D_{\mathrm{obs}}=0.125\mu {\mathrm{m}}^2/\mathrm{s}$. All other parameters are as in Fig. 3.

Figure 6

Excluded fraction depends on the variance of the obstacle size. Each curve shows the evolution of the excluded fraction (for immobile obstacles) with the standard deviation SD of the obstacle radius. Obstacle size was drawn from a normal distribution with mean 5.0 nm and variance ${\left(\text{SD}\right)}^2$. The number of randomly located obstacles was (from bottom to top)$172\times {10}^3,177\times {10}^3,182\times {10}^3,183.5\times {10}^3,\mathrm{and}185\times {10}^3$. The dashed line locates the excluded fraction at the percolation threshold (0.441).

Figure 7

Protein diffusion amidst polydisperse Ornstein-Uhlenbeck mobile obstacles. The size of the mobile obstacles was a random variable drawn according to normal distribution with mean $5.0\mathrm{nm}$ and standard deviation $\left(\text{SD}\right)=0$, 0.45, 0.80, 1.04, and 1.41 nm (from bottom to top). Parameters of the OU motion for the obstacles $D_{\mathrm{obs}}=5\times {10}^{-3}\mu {\mathrm{m}}^2/\mathrm{s}$ and $\tau =1\mu \mathrm{s}$. For each value of the standard deviation, the total number of obstacles was adjusted so that the excluded fraction was kept to the critical threshold of immobile obstacles $\theta =0.441$ (see text and Fig. 6). All other parameters are as in Fig. 3.