Obstructed diffusion propagator analysis for single-particle tracking

We describe a method for the analysis of the distribution of displacements, i.e., the propagators, of single-particle tracking measurements for the case of obstructed subdiffusion in two-dimensional membranes. The propagator for the percolation cluster is compared with a two-component mobility model against Monte Carlo simulations. To account for diffusion in the presence of obstacle concentrations below the percolation threshold, a propagator that includes the transient motion in finite percolation clusters and hopping between obstacle-induced compartments is derived. Finally, these models are shown to be effective in the analysis of Kv2.1 channel diffusive measurements in the membrane of living mammalian cells.

Figure 1

Propagator analysis of diffusion of a representative trajectory in a percolation cluster close to criticality. The data were generated from the random walk simulations on a square lattice with an obstacle concentration of 41$\%$, i.e., $c/c_P=1.007$. (a) CDF at $t$ $=$ 10 000 with fits to percolation cluster as a dashed red line [Eq. (6)], normal diffusion as a solid blue line (Gaussian), and two-component mobility as a green line [light gray, Eq. (2)]. (b) Histogram of displacements. The continuous lines show the renormalized propagators $P\left(r,t\right)\Delta V$, obtained from the fit to the CDF. Each Gaussian of the two-component mobility model is shown as a green line (light gray) independently for clarity and the sum of the two components is shown as a black dashed line. (c) Probability density function obtained by normalizing the distribution of displacements for $t$ $=$ 10 000. The distribution is normalized by the volume $\Delta V_{2\mathrm{D}}=2\pi r\Delta r$, where $\Delta r$ is the bin size. The propagators of percolationlike and normal diffusion, as obtained from the fits to the CDF, are shown. Because the propagator of diffusion in a percolation cluster operates in $d_f$ dimensions, the curve shown is $P(r,t)\Delta V_{d_f}/\Delta V_{2\mathrm{D}}$, where $\Delta V_{d_f}$ is the volume corresponding to a fractal dimension $d_f$, as in the text.

Figure 2

CDF of simulated obstructed diffusion with obstacle concentration $c=0.41$ at lag time $t$ $=$ 10 000. (a) The CDF of a representative trajectory is modeled by a deterministic fractal as given by Eq. (6). The best nonlinear fit is found for $d_w$ $=$ 2 and $d_f$ $=$ 1.6 (dashed red line). Note that the fit is constrained by $d_w$ ≥ 2. For comparison the biexponential fit [Eq. (2)] and the fit using $d_w$ $=$ 2.8 are also shown. (b) The diffusion coefficient $D_F$ is shown as found from fitting to the CDF at different lag times to a deterministic fractal model [Eq. (8)]. All the CDFs are described by $d_w$ $=$ 2.

Figure 3

$\langle r^2\rangle /t$ as a function of $t$ in a log-log scale for a tracer in the presence of immobile obstacles at different concentrations. $\langle r^2\rangle$ is the mean square displacement and $t$ is the lag time. All nine simulated trajectories are shown together for each obstacle concentration.

Figure 4

Percolation fit parameter $K$($t$) in Eq. (6) as a function of lag time at two moderate obstacle concentrations and close to criticality. The parameter exhibits a power law $K\left(t\right)=a/t^{1.65/d_w}$.

Figure 5

Mean squared errors of the fits to the obstructed diffusion simulations with different obstacle concentrations using the two-component mobility and the percolation cluster models.

Figure 6

Fractal dimension of the walk $d_w$ as a function of relative obstacle concentration. $d_w$ is obtained from the fit to the percolation model as shown in Fig. 4. The red line is a least square fit to $d_w$ as performed in Ref. [15]: $d_w=\left(2-3.630x+1.758x^2\right)/\left(1-1.806x+0.850x^2\right)$, where $x$ $=$ $c$/$c_P$, $c$ is the obstacle concentration, and $c_P$ is the concentration at criticality.

Figure 7

Two-component mobility model. (a) MSD of each of the two mobilities. Full symbols: fast component. Hollow symbols: slow component. (b) Subdiffusion exponent as a function of relative obstacle concentration. The red line is the function found from the least square fit in Fig. 6.

Figure 8

Analysis of experimental Kv2.1 single-particle tracking in the plasma membrane of living HEK cells. All error bars indicate standard error of the mean. (a) $\langle r^2\rangle /t$ as a function of $t$ in a log-log scale. A negative slope between 0.1 and 20 s is evident indicating anomalous subdiffusion across this time scale. (b) Mean squared errors from fitting the CDF at different lag times to three different models as described in the text: two-component mobility, percolation cluster, and obstructed diffusion. (c) $K$($t$) values as a function of $t$, obtained from fitting the experimental data to the percolation cluster and obstructed diffusion. A power law dependence is obtained from which $d_w$ is found. (d) Mean square displacements obtained by modeling the experimental data with two-component mobility [${\sigma }_1^2$ and ${\sigma }_2^2$ in Eq. (2)], and with obstructed diffusion [${\sigma }_1^2$ in Eq. (10)]. The slow-mobility component MSDs from the two different models coincide.