Driven diffusive systems with mutually interactive Langmuir kinetics

We investigate the simple one-dimensional driven model, the totally asymmetric exclusion process, coupled to mutually interactive Langmuir kinetics. This model is motivated by recent studies on clustering of motor proteins on microtubules. In the proposed model, the attachment and detachment rates of a particle are modified depending upon the occupancy of neighboring sites. We first obtain continuum mean-field equations and in certain limiting cases obtain analytic solutions. We show how mutual interactions increase (decrease) the effects of boundaries on the phase behavior of the model. We perform Monte Carlo simulations and demonstrate that our analytical approximations are in good agreement with the numerics over a wide range of model parameters. We present phase diagrams over a selective range of parameters.

Figure 1

A graphic representation of the TASEP model with Langmuir dynamics. Particles are injected at the first site with rate $\alpha$ and extracted at the last site with rate $\beta$. The particles move exclusively to the right, with unitary rate, if the next site is empty. In this model the only interaction between the particles is the hard-core repulsion, which means that hopping over particles and multiple occupation are not allowed. The injection, extraction, and hopping to the next site constitute the TASEP model. The Langmuir dynamics consists of the detachment and attachment from and to the background. The attachment and detachment rates are ${\omega }_A$ and ${\omega }_D$, respectively. For analysis of this model see, for example, Refs. [5, 6]. Mutual interactions are incorporated by modifying the attachment and detachment rates; see Fig. 2.

Figure 2

The mutual interaction are incorporated in the TASEP model with LK dynamics by modifying the attachment and detachment rates if neighboring sites are occupied. For each occupied neighboring site the attachment rate is multiplied by $\delta$ and the detachment rate is multiplied by $\gamma$.

Figure 3

Five different phase diagrams for $\Omega =0.3$ and different values of $\eta$: (a) $\eta =0$, (b) $\eta =0.5$, (c) $\eta =-0.5$, (d) $\eta =2.0$, and (e) $\eta =-0.9$. (a) Corresponds to the case without mutual interaction. The influence of $\Omega$ is analyzed in Ref. [6]. From (b) and (d) it becomes clear that due to the increase in $\eta$ the HD region disappears more quickly from the phase diagram than the LD phase. This is explained by the fact that the mutual interactions are most apparent in HD regions. In cases (c) and (e) $\eta$ is decreased. Again it becomes clear from these figures that a change in $\eta$ has more influence on the phases containing HD regions than on phases containing LD regions.

Figure 4

Density profiles for $\Omega =0.3$ and $\eta =2$; this corresponds to the phase diagram in Fig. 3. The boundary conditions are (a) $\alpha =0.01$ $\beta =0.01$, (b) $\alpha =0.2$ $\beta =0.2$, (c) $\alpha =0.8$ $\beta =0.1$, (d) $\alpha =0.8$ $\beta =0.8$, and (e) $\alpha =0.1$ $\beta =0.8$. The solid lines are the analytical solutions for the density profiles [Eqs. (14) and (15)], the constant solution ${\rho }_l$ is included as a dash-dot line. The solid lines with noise are the result of the Monte Carlo simulations with a lattice of 1000 sites, averaged over 2000 simulations. The analytical solutions for $\Omega =0.3$ and the same boundary conditions but without MIs ($\eta =0$) are included as a dotted line to emphasize the effect of the mutual interactions. (a) The MIs do change the density profile significantly, but do not change the phase. (b) Due to the MIs the LK dynamics increases, and the profile changes from a LD-HD phase for $\eta =0$ to a LD-MC-HD phase. (c) The MIs cause a phase change from the HD phase to the MC-HD phase, due to the increased LK dynamics. (d) There is no difference between the solution with or without MIs. (e) The MIs induce a phase change from the LD phase to the LD-MC phase, due to the increased LK dynamics. There are two types of discrepancies between the analytical results and the Monte Carlo simulations. Boundary layers are formed at the left boundary (c), at the right boundary (e), and at both boundaries (d). Other discrepancies between the analytical result and the simulations occur where there is a transition between the ${\rho }_{\alpha }$, the ${\rho }_{\beta }$, and/or the ${\rho }_l$ solution. Both types of discrepancies can be explained by the finite size of the lattice used in the simulations.

Figure 5

Three phase diagrams obtained with Eqs. (20) and (21). With $\Omega =0.3$ and (a) $\psi =0.001$, (b) $\psi =0.4$, (c) $\psi =0.8$. The phases which contain the ${\rho }_{\alpha }$ solution disappear from the phase diagram for increasing MIs, until only the M and HD phases are left. The area of the M phase in the phase diagram occupies an increasingly large part of the upper half of the phase diagram for increasing $\psi$. This stands in contrast with the MC phase in case 1, where the MC phase is confined to the upper right quarter of the phase diagram and the area is independent of any parameter; see Fig. 3.

Figure 6

Density profiles for $\Omega =0.3$ and $\psi =0.4$; this corresponds to the phase diagram in Fig. 5. The injection and extraction rates are (a) $\alpha =0.01$, $\beta =0.7$ (LD phase), (b) $\alpha =0.1,\beta =0.7$ [LD-HD(M) phase], (c) $\alpha =0.8,\beta =0.8$ (M phase), (d) $\alpha =0.05,\beta =0.1$ (LD-HD phase), and (e) $\alpha =0.3,\beta =0.2$ (HD phase). The dashed lines are the Lambert $W$ function solutions to Eq. (17) obeying the left and right boundary conditions. The parts of these solutions that make up the density profile are represented as solid black lines. The dash-dot line is the constant solution. The analytical solutions for the same values but without MIs are included as dotted lines to emphasize the influence of the MIs on the density profiles. Solid lines with noise are the results of the Monte Carlo simulations with a lattice of 1000 sites, averaged over 2000 simulations. Overall there is good agreement between the simulations and the analytical result. There are two causes for the discrepancies: the finite size of the lattice and the approximation ${\psi }^2\approx 0$. Due to the finite size of the lattice boundary layers are formed at the right end of (a), (b), and (d) and at the left end of (c) and (d); and the domain walls in (b) and (d) are not localized. The discrepancies caused by the approximation are visible in two ways: the density profile does not fully coincide with the analytical result, and due to this the position of the domain wall is shifted. This is visible in (b) and (d).

Figure 7

The same legend as in Fig. 6 is used. In order to illustrate the limits of the approximation ${\psi }^2\approx 0$ used in deriving the equations for the density profiles, four extreme cases are shown. For all figures $\Omega =0.3$ was used for (a) $\psi =-0.99,\alpha =0.1,\beta =0.1$, (b) $\psi =0.8,\alpha =0.1,\beta =0.9$, (c) $\psi =0.5,\alpha =0.9,\beta =0.1$, and (d) $\psi =0.9,\alpha =0.9,\beta =0.1$. From (a) it is clear that the approximation holds for low densities, regardless of the values of $\psi$. The analytical solution in (b) does not fully coincide with the simulations. There are two causes for the discrepancies. First, there is a boundary layer on the right side caused by the finite size of the lattice. Second, the ${\rho }_{\beta }$ solution is higher than the results from the simulation; this is caused by neglecting the ${\psi }^2$ terms. Overall the simulations are in good agreement with the analytical solutions; however, the domain of the solutions differs significantly. The analytical profile is in the HD phase, $x_w<0$. The simulation result, on the other hand, is in the LD-HD phase, $0<x_w<1$. (c) For this value of $\psi$ there is agreement between the analytical solution and the simulations, even though density is high. (d) The analytical result does not coincide with the simulations due to the combination of high density and a $\psi$ close to 1. In this case the density of the analytical result exceeds 1, which is physically impossible.

Figure 8

The density profiles for (a) ${\Omega }_A=0.5,{\Omega }_D=1,\delta =2,\gamma =0.1,\alpha =1$, and $\beta =1$, (b) ${\Omega }_A=0.5,{\Omega }_D=0.5,\delta =2,\gamma =0.5,\alpha =0$, and $\beta =1$, (c) ${\Omega }_A=0.5,{\Omega }_D=0.5,\delta =2,\gamma =0.5,\alpha =0$, and $\beta =0$, (d) ${\Omega }_A=0.5,{\Omega }_D=1,\delta =3,\gamma =0.1,\alpha =0$, and $\beta =0$. Solid line with noise is the result from the simulation, the dashed line is the solution without MIs. In all cases the attachment or detachment is increased or decreased, which results in a higher density. In (b) and (c) the density overlaps with a small part of the analytical solution without MIs. This can be explained by the in low-density regions the effect of the MIs is small. Though in (d) the solution without MIs the density is low, the lattice is almost completely filled due to the increase or decrease in attachment or detachment. All parameters in (a) and (d) are the same except for the boundary conditions, which prevents the lattice in (a) from filling up completely.