Enhanced diffusion of a tracer particle in a lattice model of a crowded active system

Living systems at the subcellular, cellular, and multicellular levels are often crowded systems that contain active particles. The active motion of these particles can also propel passive particles, which typically results in enhanced effective diffusion of the passive particles. Here we study the diffusion of a passive tracer particle in such a dense system of active crowders using a minimal lattice model incorporating particles pushing each other. We show that the model exhibits several regimes of motility and quantify the enhanced diffusion as a function of density and activity of the active crowders. Moreover, we demonstrate an interplay of tracer diffusion and clustering of active particles, which suppresses the enhanced diffusion. Simulations of mixtures of passive and active crowders show that a rather small fraction of active particles is sufficient for the observation of enhanced diffusion.

Figure 1

Lattice model of a dense system of active particles (blue) containing a passive tracer particle (red). (a) Overview of the model on a hexagonal lattice. (b) Active particles perform run-and-tumble motion and move persistently in the same direction with probability $1-\alpha$ and randomize that direction (tumble) with probability $\alpha$ (i.e., $\alpha /6$ for each direction). (c) The passive tracer particle (red) performs a simple random walk; its diffusion coefficient is modulated by the parameter $\beta$. (d) Collision rules among particles.

Figure 2

Diffusion of tracer particles with different diffusion coefficients in a system of active crowders. (a)–(c) The mean-square displacement (thick blue lines) shows diffusive motion for $\beta =0$ and anomalous diffusion for $\beta =0.9,1$. The red lines indicate the diffusive motion of the tracer in an otherwise empty lattice in (a) and (b); in (c), the tracer would not move in that case. The dashed lines indicate the scaling behavior of the different regimes by showing the power laws fitted to the mean-square displacement. (d)–(f) show trajectories ($t_i$ and $t_f$ indicate initial and final time, respectively, in the trajectory plots), the figures in the last row show the zoomed view of the trajectories at the early stages. Each column corresponds to the value of $\beta$ indicated above it. The other parameters are $\rho =0.025, \alpha =0.0001$.

Figure 3

Snapshots of the simulation for $\rho =0.1$ and $\alpha =0.0001$: the active particles are shown in blue and the tracer particle in purple. (a) The tracer (purple) is next to an active particle (blue), which pushes it. (b) The tracer is trapped in empty space. (c) The tracer is trapped by active particles.

Figure 4

Trapping analysis. (a) The sketch shows the tracer (red) surrounded by different number of active particles (blue). (b)–(d) show the probability that $n$ neighbor sites are occupied (for $n=0,\cdots 6$). The red arrows indicate the numbers of occupied neighbors most relevant for the different regimes in the MSD plots in Fig. 2. The probabilities were determined by averaging over 1000 simulation steps (starting at time 0 for the early state, at time 1000 for the intermediate state, and 1000 steps before time ${10}^6$ for the late state) as well as over 1000 realizations of the simulation (with random initial conditions).

Figure 5

The relative diffusion coefficient of a tracer with $\beta =0.9$, normalized to the diffusion constant $D_0$ of an isolated tracer.

Figure 6

Cluster analysis: (a), (b) Average cluster size as a function of time (averaged over 1000 realizations) for different densities. The numbers above the curves indicate the corresponding density. (c), (d) Fraction of particles assembled in clusters in the last snapshot of simulation for the same densities. The dashed lines mark the density in which the tracer exhibits the maximum diffusion coefficient. Each column corresponds to the value of $\alpha$ indicated above it.

Figure 7

Cluster analysis for different fractions ${\chi }_A$ of active particles at fixed $\rho =0.05$. Cluster size and fraction of particles in clusters (inset) as in Fig. 6 for different ${\chi }_A$ in a binary mixture of active and passive crowders. The numbers above each curve indicate the fraction of active particles (${\chi }_A$).

Figure 8

The diffusion coefficient of a tracer with $\beta =0.9$ as a function of the density $\rho$ and of the fraction of active row ${\chi }_A$ in a binary mixture.

Figure 9

MSD as a function of time plotted for different simulation box size ($L=50$, 100, 256 and 512) for densities (a) $\rho =0.02$ and (b) $\rho =0.4$, both with $\alpha =0.0001$ (c) $\rho =0.02$ and (d) $\rho =0.4$, both with $\beta =0.9$ and $\alpha =0.01$. $\beta =0.9$ in all cases. All the plots are averaged over 1000 realizations for each box size.

Figure 10

The mean-square displacement of a tracer with $\beta =1.0$ as a function of time for different densities and tumbling probabilities (a) $\alpha =0.0001$, (b) $\alpha =0.001$, (c) $\alpha =0.01$, (d) $\alpha =0.1$.