Self-organized system-size oscillation of a stochastic lattice-gas model

The totally asymmetric simple exclusion process (TASEP) is a paradigmatic stochastic model for nonequilibrium physics, and has been successfully applied to describe active transport of molecular motors along cytoskeletal filaments. Building on this simple model, we consider a two-lane lattice-gas model that couples directed transport (TASEP) to diffusive motion in a semiclosed geometry, and simultaneously accounts for spontaneous growth and particle-induced shrinkage of the system's size. This particular extension of the TASEP is motivated by the question of how active transport and diffusion might influence length regulation in confined systems. Surprisingly, we find that the size of our intrinsically stochastic system exhibits robust temporal patterns over a broad range of growth rates. More specifically, when particle diffusion is slow relative to the shrinkage dynamics, we observe quasiperiodic changes in length. We provide an intuitive explanation for the occurrence of these self-organized temporal patterns, which is based on the imbalance between the diffusion and shrinkage speed in the confined geometry. Finally, we formulate an effective theory for the oscillatory regime, which explains the origin of the oscillations and correctly predicts the dependence of key quantities, such as the oscillation frequency, on the growth rate.

Figure 1

(a) Illustration of the dynamics in cellular protrusions. Movements of molecular motors are indicated by black arrows, and are restricted to the cell body and the protrusion by the cell membrane. On the filament the motors move unidirectionally towards the protrusion tip, while their motion in the surrounding cytoplasm is diffusive. (b) Illustration of the two-lane lattice-gas model. We consider a two-lane lattice-gas model consisting of a TASEP or transport lane [TL, upper lane, occupied by orange (light gray) particles] and a diffusive lane [DL, lower lane, occupied by blue (dark gray) particles] with hopping rates $\nu \equiv 1$ and $\epsilon$, respectively. The lanes are coupled by attachment and detachment kinetics at rate $\omega$, respecting exclusion for transfer from the DL to TL. Entry and exit occurs via the first DL site only, at rates $\alpha$ and $\epsilon$, respectively. The system spontaneously grows by simultaneously appending a site to the TL and DL tip at rate $\gamma$, while both lanes shrink by a site, at rate $\delta$, if the TL tip is occupied by a particle. In the latter case, particle conservation is ensured by shifting all particles of the previous TL and DL tip site to the new DL tip site. (c) Illustration of the particle currents and density profiles. The density profile on the TL (DL), $\rho \left(x\right)$ $\left[\eta \right(x\left)\right]$, is displayed in orange (light gray) in the upper panel [blue (dark gray) in the lower panel]. The density $\rho \left(x\right)$ is discontinuous at the last TL site, with ${\rho }_{-}$ referring to the left and ${\rho }_{+}$ to the right limit. The currents (black arrows) come from entry $\alpha$, exit $\epsilon {\eta }_0$, diffusion $D$, attachment and detachment $J^{D\to T}$, directed movement $J^T$, and detachment due to depolymerization ${\rho }_{+}\delta$.

Figure 2

Mean system length. Mean length of the system is plotted as a function of the growth rate $\gamma$. The analytical result (red, solid line) agrees well with the results from stochastic simulations (gray, filled circles) for small growth rates, $\gamma \ll 1$, where the adiabatic assumption is expected to hold. For increasing growth rates the numerical data show an inflection point at which they begin to deviate strongly from the results in the adiabatic limit. This indicates that the dynamics shows qualitatively new behavior for large growth rates.

Figure 3

Length histograms for different growth rates $\gamma$ and a simulation time of ${10}^7$. The larger the growth rate, the longer the average length and the broader the length distribution. The distributions are right-skewed in contrast to a Gaussian. Inset: The average length (squares), the standard deviation (bars) of the average length, and the maximum length reached (right-pointing triangles) increase nonlinearly with larger growth rates, while the minimum length attained (left-pointing triangles) remains essentially constant. The shaded areas [green (gray), orange (light gray), and red (dark gray)] correspond to the value of the growth rate $\gamma$ in the corresponding length histograms.

Figure 4

Autocorrelation function. The autocorrelation functions, each of an ensemble of 1000 runs, for several growth rates $\gamma$ are compared. The autocorrelation function for the smallest growth rate $\gamma =0.005$ (purple line with squares) almost immediately decays to zero, while the autocorrelation of $\gamma =0.14$ (blue line with triangles) oscillates with a frequency comparable to the observed length oscillations.

Figure 5

Time traces for filament length, total particle number, DL occupancy, and TL occupancy for a full simulation time of ${10}^7$. (a) System length (gray) and total particle number [red (dark gray)] dynamics for a long time interval and small growth rate $\gamma =0.01$. Both the length and the total particle number change stochastically. (b) System length (gray) and total particle number [red (dark gray)] dynamics for a large growth rate $\gamma =0.14$. We observe length oscillations and a sawtooth-shaped behavior of the total particle number. (c) Zoom-in for large $\gamma =0.14$. Upper panel: System length (gray) and total particle number [red (dark gray)] dynamics. Middle panel: Occupancy of the TL [orange (light gray), lower line] and DL [blue (dark gray), upper line] tip neighborhood, which is chosen to consist of 20 sites from the tip. Lower panel: Occupancy of the TL [orange (light gray), lower line] and DL [blue (dark gray), upper line] bulk, which corresponds to the whole lane except the tip neighborhood. We observe that the tip neighborhood is densely occupied compared with the bulk.

Figure 6

Intuitive picture for the occurrence of length oscillations. Starting from a short and empty system (a), the only two processes possible are growth and influx of particles from the reservoir into the system (b). Once attached to the TL particles start walking towards the tip, away from the reservoir, and the system grows while new particles enter (c). Since growth is slow compared with transport of particles on the TL, the particles on the TL “catch up” with the tip. Furthermore, due to the finite diffusion and the closure at the tip, the particles then begin crowding at the tip, turning the growth phase into a shrinkage phase (d). During the shrinkage phase more and more particles accumulate at the tip as new particles still enter from the reservoir on the left while the system shrinks from the right (e). Only when the system has become very short is diffusion of particles fast enough that particles which accumulate at the tip can leave the system by exiting into the reservoir (f), leaving behind a short and empty system (a), from which the next oscillation cycle can begin anew.

Figure 7

Schematic of the effective model. We split the system into four regions, and use an effective description of the DL, restricting our analysis to the TL. In the in-region we have attachment at an effective in-rate ${\alpha }_{\mathrm{eff}}$ and detachment at rate $\omega$. The bulk region links the in-region to the tip neighborhood and we assume that in the bulk attachment and detachment balance. For the tip neighborhood we assume that particles which have detached at the tip a time $1/\omega$ earlier reattach to the TL in the tip neighborhood and need another time $\mathrm{\Delta }-1/\omega$ to reach the TL tip again, yielding a recursion relation for the tip density ${\rho }_{\mathrm{it}}^{+}$. Finally, at the tip we have detachment at rate $\delta {\rho }^{+}$, and the corresponding shrinkage of the system, and spontaneous growth. The tip and the tip neighborhood are described in the comoving frame.

Figure 8

Solution of the recursion relation for the tip density (orange, solid line) and the length (gray, dashed line) as a function of the iteration step, for $\gamma =0.14$. Both show periodic behavior, but while the growth and shrinkage phases are rather symmetric for the length dynamics, the tip density exhibits a sawtooth shape.

Figure 9

Minimal and maximal length. The average minimal (maximal) length per oscillation period from stochastic simulations is compared with the prediction from the effective theory. From the stochastic simulations we determined the minimal and maximal length for each oscillation period, and the average minimal (maximal) length is depicted with orange squares (green circles), with error bars representing the corresponding standard deviation. Note that the average minimal length is approximately independent of the growth rate, in contrast to the average maximal length. The prediction from the effective theory is shown with red lines. As in the stochastic simulations the maximal length (solid line) increases with the growth rate $\gamma$, whereas the minimal length (dashed line) is only weakly dependent on the growth rate, decreasing slightly with increasing growth rate.

Figure 10

Oscillation frequency. The oscillation frequency from stochastic simulations is compared with the prediction from the effective theory. We determined the length oscillation frequency from the stochastic simulations, first, by determining the autocorrelation oscillation frequency (freq via autocorr, purple squares), second, by evaluating the distribution of duration between two adjacent minima (freq via hist, orange circles), and third, by performing a Lomb-Scargle analysis comprising a sine fit (freq via L. S., blue left-pointing triangles). For the distribution approach, where a bound on the maximal frequency is used as explained in Appendix pp2, the distribution average and standard deviation (error bars) are depicted. For the Lomb-Scargle analysis the most probable frequency is shown. Clearly, for all methods the oscillation frequency decreases with larger growth rates. For very small rates noise masks the oscillation, such that the methods employed cannot determine the frequency correctly. Thus, results from the stochastic simulations are only shown for growth rates $\ge 0.06$. The predicted oscillation frequency from the effective theory is shown as a solid red line. It displays the same qualitative behavior as the result from stochastic simulations.

Figure 11

Occupancies of the TASEP lane tip neighborhood, i.e., here the 20 last sites, at the time point or time interval when the length reaches its maximum during one oscillation period. They have been measured for several periods and are shown for a growth rate of 0.14. The occupancy at these time points (red squares) and their average (upper blue line) as well as the occupancy average (yellow right-pointing triangles) and maximum (gray left-pointing triangles) over a time period of ten events—five before and five after the maximum—is depicted. Moreover, the critical density for switching from growth to shrinkage (lower purple line) is shown.