Bidirectional transport on a dynamic lattice

Bidirectional variants of stochastic many particle models for transport by molecular motors show a strong tendency to form macroscopic clusters on static lattices. Inspired by the fact that the microscopic tracks for molecular motors are dynamical, we study the influence of different types of lattice dynamics on stochastic bidirectional transport. We observe a transition toward efficient transport (corresponding to the dissolution of large clusters) controlled by the lattice dynamics.

Figure 1

Schematic representation of the particle dynamics (lattice dynamics is not included in this figure). Arrows indicate possible moves with corresponding rates which are symmetric for both particle species. We impose hard-core interaction on the lower lane (filament), while the particles on the upper lane are noninteracting. Periodic boundary conditions are considered.

Figure 2

(Color online) Distribution of cluster sizes in a system of size $L=1000$ with simple lattice dynamics (scenario 1). Recreation of lattice sites occurs at $k_p=1$. The black line corresponds to a static lattice.

Figure 3

(Color online) Flux of positive particles along the filament with simple lattice dynamics at $k_p=1$ for different system sizes $L$ and for the same density. If depolymerization is weak, the flux depends on the system size, but changes to a density-dependent state as large clusters disappear (see also Fig. 2). The inset shows the same data divided by the flux in the smallest system $\left(L=100\right)$.

Figure 4

(Color online) Space-time plots of the filament occupation for a static (left) and a dynamic (right) filament in a system of size $L=1000$ and density ${\rho }_{tot}^{\pm }=0.5$. Red, respectively, green dots are particles with moving direction to the right, respectively, left, white spaces are unoccupied filament sites. The filament dynamics clearly induces a transition from a condensated to a homogenous phase. Note that considering a finite capacity of the diffusive lane would reinforce the jamming.

Figure 5

(Color online) Maximum flux $j_b\left[k_p,k_{d,\text{max}}\left(k_p\right)\right]$ (red squares) and optimal depolymerization rate $k_{d,\text{max}}\left(k_p\right)$ (black circles) at given polymerization rates $k_p$ in a system of size $L=1000$. For comparison, the flux in a system with a static lattice is drawn as a solid red line. Note that the horizontal axis is in logarithmic scale and, therefore, both quantities seem to saturate at high polymerization rates.