Anomalous diffusion of self-propelled particles in directed random environments

We theoretically study the transport properties of self-propelled particles on complex structures, such as motor proteins on filament networks. A general master equation formalism is developed to investigate the persistent motion of individual random walkers, which enables us to identify the contributions of key parameters: the motor processivity, and the anisotropy and heterogeneity of the underlying network. We prove the existence of different dynamical regimes of anomalous motion, and that the crossover times between these regimes as well as the asymptotic diffusion coefficient can be increased by several orders of magnitude within biologically relevant control parameter ranges. In terms of motion in continuous space, the interplay between stepping strategy and persistency of the walker is established as a source of anomalous diffusion at short and intermediate time scales.

Figure 1

(a) A typical sample trajectory (red lines) on a random filamentous structure. Inset: Possible choices at the nodes of the directed network. (b) Path of the walker during two consecutive steps, described by Eq. (1).

Figure 2

(a) Evolution of $\lambda$ with $\mathcal{F}\left(\ell \right)$. (b) Examples of $R\left(\varphi \right)$ and their corresponding anisotropy parameter $\mathcal{R}$. Insets: Black arrows show the possible directions of motion at the next step, with length being proportional to the probability. (c) Asymptotic diffusion coefficient vs $p$ at different $\mathcal{R}$, from simulations (symbols) and analytical predictions via Eq. (3) (curves). The red (blue) color corresponds to $\lambda =1\left(6\right)$. Insets: Typical trajectories at $p=\mathcal{R}=0$. (d) 2D profiles of $D$ in the $p-\mathcal{R}$ plane for $\lambda =1$ (left) and $\lambda =6$ (right).

Figure 3

(a) MSD vs $n$ at different $\lambda ,p$, and $\mathcal{R}$, from simulations (symbols) and theory via Eq. (2) (curves). (b) Schematic representation of the possible regimes of motion and the crossover times. (c) The crossover time $n_{\mathrm{c},1}$ vs $p$ for different values of $\mathcal{R}$, for $\lambda =1$ (red) and $\lambda =6$ (blue).

Figure 4

2D profiles of the phase diagram of short-time dynamics in the $p-\mathcal{R}$ plane for $\lambda =1$ (left) and $\lambda =6$ (right). The color intensity reflects the magnitude of anomalous exponent $\alpha$, with red (blue) meaning subdiffusion (superdiffusion). The white plus symbols and hatched lines denote the ballistic and oscillatory subdomains. The circles mark the regions of the $p-\mathcal{R}$ parameter space where the slope of $\langle {\mathbf{r}}^2\rangle$ initially grows with time (regime III). The blue-red (red-oscillatory) interface is specified by $p=\frac{\mathcal{R}}{\mathcal{R}-1} \left(p+\mathcal{R}-p\mathcal{R}=-\lambda /2\right)$.

Figure 5

MSD vs time for a paramagnetic tracer particle in the two-state flashing potential. The analytical (solid lines) and simulation (dashed lines) fits are compared with experimental data (symbols) taken from Fig. 2 of Ref. [28]. Top inset: Schematic picture of two consecutive disordered lattices in simulations with $\delta =10\%$. Bottom inset: The normalized step-step correlation function $C_s$ vs time.