Bottleneck-induced transitions in a minimal model for intracellular transport

We consider the influence of disorder on the nonequilibrium steady state of a minimal model for intracellular transport. In this model particles move unidirectionally according to the totally asymmetric exclusion process (TASEP) and are coupled to a bulk reservoir by Langmuir kinetics. Our discussion focuses on localized point defects acting as a bottleneck for the particle transport. Combining analytic methods and numerical simulations, we identify a rich phase behavior as a function of the defect strength. Our analytical approach relies on an effective mean-field theory obtained by splitting the lattice into two subsystems, which are effectively connected exploiting the local current conservation. Introducing the key concept of a carrying capacity, the maximal current that can flow through the bulk of the system (including the defect), we discriminate between the cases where the defect is irrelevant and those where it acts as a bottleneck and induces various novel phases (called bottleneck phases). Contrary to the simple TASEP in the presence of inhomogeneities, many scenarios emerge and translate into rich underlying phase diagrams, the topological properties of which are discussed.

Figure 1

Schematic representation of the TASEP with on-off kinetics in the presence of a bottleneck at the site $i=k$ (thick tick). The allowed moves are: forward jump (with rate $q\ne 1$ in $i=k$ and $r=1$ elsewhere), entrance at the left boundary (with rate $\alpha$), exit at the right boundary (with rate $\beta$), attachment (with rate ${\omega }_A$), and detachment (with rate ${\omega }_D$) in the bulk.

Figure 2

(a) Phase diagram of TASEP/LK for $K=1$. One recognizes seven phases: in addition to the TASEP LD, HD, and MC phases, there are four more coexistence phases, namely, the LD/HD, LD/MC, MC/HD, and LD/MC/HD phases. The shaded region highlights the LD/HD coexistence where a localized domain wall appears. (b) Typical density profile in the LD/HD phase and (c) the corresponding current profile. At the matching point $x_w$ between the left $\left(j_{\alpha }\right)$ and right $\left(j_{\beta }\right)$ currents a domain wall develops and connects the left $\left({\rho }_{\alpha }\right)$ and right $\left({\rho }_{\beta }\right)$ density profile.

Figure 3

Transition between a spike (a) and a step (b) in the density profile for the TASEP with defect. Stochastic simulations are performed on systems of size $N=128$ with parameters $q=0.1$, $x_d=0.5$, (a) $(\alpha ,\beta )=(0.05,0.8)$, (b) $(\alpha ,\beta )=(0.8,0.8)$.

Figure 4

Transition between spike (a) and step (b) for the TASEP/LK with defect. Stochastic simulations are performed on systems of size $N=128$ with parameters $q=0.1$, $x_d=0.5$, $\Omega =0.1$, (a) $(\alpha ,\beta )=(0.05,0.8)$, (b) $(\alpha ,\beta )=(0.8,0.8)$.

Figure 5

Schematic representation of the division into two subsystems allowing to apply a mean-field theory. The last step (bottom of the figure), illustrates the continuum limit that we are considering (see the main text).

Figure 6

Sketch of the signature of the strong defect (zigzag shape) in the density (a) and current (b) profile.

Figure 7

The four typical carrying capacity profiles $\mathcal{C}\left(x\right)$ displayed by the system (parameters $\Omega =\{0.3,1.5,0.5,0.5\}$, $x_d=\{0.5,0.5,0.7,0.3\}$, $q=0.3$). Depending on the defect strength $q$, on the position $x_d$ of the defect, and on the rate $\Omega$, the defect-imposed current $j_d$ combines in four different ways with the maximal current $j^{*}$. Each of these profiles induces a topologically distinct phase diagram (see the text).

Figure 8

Construction of the current profile. Top: For given carrying capacity ${\mathcal{C}}_1\left(x\right)$ (solid line) and fixed exit rate $\beta =0.2$, various scenarios arise when the entrance rate $\alpha =\{0.05,0.2,0.6\}$ is varied. Bottom: Three different profiles are shown in the small graphs (solid line: global current profile, dashed line: defect and boundary currents) emerging from three different left boundary conditions. Parameters are $x_d=1∕2$, $q=0.3$, $\Omega =0.3$, $\beta =0.2$, $\alpha =\{0.05,0.2,0.6\}$.

Figure 9

Examples of density profiles in the bottleneck phases for a carrying capacity $\mathcal{C}\left(x\right)={\mathcal{C}}_1\left(x\right)$. Stochastic simulations (continuous line) are compared to analytical mean-field predictions (dashed line). The system size is $N=4096$ and the parameters are $q=0.3$, $x_d=0.5$, ${\Omega }_D=0.3$, LD-BP: $(\alpha ,\beta )=(0.3,0.6)$, BP: $(\alpha ,\beta )=(0.6,0.6)$, LD-BP-HD: $(\alpha ,\beta )=(0.3,0.3)$ and BP-HD: $(\alpha ,\beta )=(0.6,0.3)$.

Figure 10

Phase diagram for $\Omega =0.3$, $q=0.4$, and $x_d=1∕2$, i.e., for the carrying capacity ${\mathcal{C}}_1\left(x\right)$ (large screening length). Continuous lines are the phase boundaries introduced by the defect; dashed lines are the phase boundaries already present in the model without bottleneck. The shadowed region indicates the bottleneck phases where the defect is relevant, the darkest one highlights the pure bottleneck phase (for the meaning of the different phases see text).

Figure 11

Examples of density profile (top) and current (bottom) for the system in the LD-MC-BP-MC-HD phase: the defect is relevant and the carrying capacity is ${\mathcal{C}}_2\left(x\right)$. Results of stochastic simulations (continuous line) are compared to analytical mean-field predictions (dashed line). The system size is $N=4096$, $K=1$, $\Omega ={\Omega }_D=1.5$, $q=0.3$, $(\alpha ,\beta )=(0.2,0.2)$, and $x_d=1∕2$: one can clearly distinguish the various phases and note a discontinuity in the density profile in the proximity of the defect, while the current profile is continuous.

Figure 12

Phase diagram for $\Omega =0.3$, $x_d=1∕2$, and $q=0.8$, i.e., for the carrying capacity ${\mathcal{C}}_2\left(x\right)$ (short screening length). Continuous lines are the phase boundaries introduced by the defect; dashed lines are the phase boundaries already present in the model without bottleneck. The shadowed region indicates the bottleneck phases where the defect is relevant; the darkest one highlights the bottleneck phase independent of the entrance $\left(\alpha \right)$ and exit $\left(\beta \right)$ rates (see the text and Table ).

Figure 13

Cut of the phase diagram in $\delta$ and $\alpha$ (parameters: $\Omega =0.4$, $x_d=1∕2$, $\beta >1∕2$). Solid lines identify usual phase transitions, while the dashed line identifies the peculiar transition between relevant and irrelevant defect. The multiple coexistence points $\mathcal{P}$ and $\mathcal{Q}$ are shown in the graph (see the main text).

Figure 14

Phase diagram for $K=2$, ${\Omega }_D=0.1$, $x_d=1∕2$, and $q=0.3$, i.e., for the ${\mathcal{C}}_1\left(x\right)$-like carrying capacity. Continuous lines are the phase boundaries introduced by the defect (BP); dashed lines are the phase boundaries already present in the model without bottleneck. The shadowed region indicates the bottleneck phases where the defect is relevant; the darkest one highlights the pure bottleneck phase. The numbers stem for the different phases: LD-BP (1), BP (2), LD-BP-HD (3), and BP-HD (4). For the description of the phases see the text and Figs. 15.

Figure 15

Two examples of density profile and current in the LD-BP-HD for the ${\mathcal{C}}_1\left(x\right)$-like carrying density capacity (left) and BP-MC phase for the ${\mathcal{C}}_3\left(x\right)$ (right). Stochastic simulations (continuous line) are compared to analytical mean-field predictions (dashed line). The system size is $N=4096$ and the parameters are $q=0.2$, $x_d=1∕2$, $K=2$, ${\Omega }_D=0.3$, $(\alpha ,\beta )=(0.1,0.2)$ (left); $q=0.3$, $x_d=1∕2$, $K=2$, ${\Omega }_D=0.3$, $(\alpha ,\beta )=(0.8,0.8)$ (right).