Totally asymmetric exclusion process on chains with a double-chain section in the middle: computer simulations and a simple theory

Exclusion processes on a roundabout traffic model with constrained resources

Motivated by the vehicular traffic phenomenon at roundabouts, we examine how the limited availability of resources affects the movement of two distinct types of particles on bidirectional lanes connected by two bridges, with each bridge specifically designated for the transportation of one species. To provide a theoretical ground for our findings, we employ a mean-field framework and successfully validate them through dynamic Monte Carlo simulations. Based on the theoretical analysis, we analytically derive various stationary properties, such as the particle densities, phase boundaries, and particle currents, for all the possible symmetric as well as asymmetric phases. The qualitative as well as quantitative behavior of the system is significantly affected by the constraint on the number of resources. The complexity of the phase diagram shows a nonmonotonic behavior with an increasing number of particles in the system. Analytical arguments enable the identification of several critical values for the total number of particles, leading to a qualitative change in the phase diagrams. The interplay of the finite resources and the bidirectional transport yields unanticipated and unusual features such as back-and-forth transition, the presence of two congested phases where particle movement is halted, as well as shock phases induced by boundaries and the bulk of the system. Also, it is found that spontaneous symmetry-breaking phenomena are induced even for very few particles in the system. Moreover, we thoroughly examine the location of shocks by varying the parameters controlling the system's boundaries, providing insights into possible phase transitions.

Metastability due to a branching-merging structure in a simple network of an exclusion process

We investigate a simple network, which has a branching-merging structure, using the totally asymmetric simple exclusion process, considering conflicts at the merging point. For both periodic and open boundary conditions, the system exhibits metastability. Specifically, for open boundary conditions, we observe two types of metastability: hysteresis and a nonergodic phase. We analytically determine the tipping points, that is, the critical conditions under which a small disturbance can lead to the collapse of metastability. Our findings provide insights into metastability induced by branching-merging structures, which exist in all network systems in various fields.

Single-file transport in periodic potentials: The Brownian asymmetric simple exclusion process

Single-file Brownian motion in periodic structures is an important process in nature and technology, which becomes increasingly amenable for experimental investigation under controlled conditions. To explore and understand generic features of this motion, the Brownian asymmetric simple exclusion process (BASEP) was recently introduced. The BASEP refers to diffusion models where hard spheres are driven by a constant drag force through a periodic potential. Here we derive general properties of the rich collective dynamics in the BASEP. Average currents in the steady state change dramatically with the particle size and density. For an open system coupled to particle reservoirs, extremal current principles predict various nonequilibrium phases, which we verify by Brownian dynamics simulations. For general pair interactions we discuss connections to single-file transport by traveling-wave potentials and prove the impossibility of current reversals in systems driven by a constant drag and by traveling waves.

Braess paradox in a network of totally asymmetric exclusion processes

We study the Braess paradox in the transport network as originally proposed by Braess with totally asymmetric exclusion processes (TASEPs) on the edges. The Braess paradox describes the counterintuitive situation in which adding an edge to a road network leads to a user optimum with higher travel times for all network users. Travel times on the TASEPs are nonlinear in the density, and jammed states can occur due to the microscopic exclusion principle, leading to a more realistic description of trafficlike transport on the network than in previously studied linear macroscopic mathematical models. Furthermore, the stochastic dynamics allows us to explore the effects of fluctuations on network performance. We observe that for low densities, the added edge leads to lower travel times. For slightly higher densities, the Braess paradox occurs in its classical sense. At intermediate densities, strong fluctuations in the travel times dominate the system's behavior due to links that are in a domain-wall state. At high densities, the added link leads to lower travel times. We present a phase diagram that predicts the system's state depending on the global density and crucial path-length ratios.

Phase transitions and optimal transport in stochastic roundabout traffic

We study traffic in a roundabout model, where the dynamics along the interior lane of the roundabout are described by the totally asymmetric simple exclusion process (TASEP). Vehicles can enter the interior lane or exit from it via S intersecting streets with given rates, and locally modified dynamics at the junctions take into account that collisions of entering vehicles with vehicles approaching the entrance point from the interior lane should be avoided. A route matrix specifies the probabilities for vehicles to arrive from and to exit to certain intersecting streets. By subdividing the interior lane into segments between consecutive intersecting streets with effective entrance and exit rates, a classification of the stationary roundabout traffic in terms of TASEP multiphases is given, where each segment can be in either the low-density, high-density, or maximum current TASEP phase. A general methodology is developed, which allows one to calculate the multiphases and optimal throughput conditions based on a mean-field treatment. Explicit analytical results from this treatment are derived for equivalent interesting streets. The results are shown to be in good agreement with kinetic Monte Carlo simulations.

Effects of junctional correlations in the totally asymmetric simple exclusion process on random regular networks

We investigate the totally asymmetric simple exclusion process on closed and directed random regular networks, which is a simple model of active transport in the one-dimensional segments coupled by junctions. By a pair mean-field theory and detailed numerical analyses, it is found that the correlations at junctions induce two notable deviations from the simple mean-field theory, which neglects these correlations: (1) the narrower range of particle density for phase coexistence and (2) the algebraic decay of density profile with exponent 1/2 even outside the maximal-current phase. We show that these anomalies are attributable to the effective slow bonds formed by the network junctions.

Asymmetric simple exclusion process on chains with a shortcut

We consider the asymmetric simple exclusion process (TASEP) on an open network consisting of three consecutively coupled macroscopic chain segments with a shortcut between the tail of the first segment and the head of the third one. The model was introduced by Y.-M. Yuan et al. [J. Phys. A 40, 12351 (2007)] to describe directed motion of molecular motors along twisted filaments. We report here unexpected results which revise the previous findings in the case of maximum current through the network. Our theoretical analysis, based on the effective rates' approximation, shows that the second (shunted) segment can exist in both low- and high-density phases, as well as in the coexistence (shock) phase. Numerical simulations demonstrate that the last option takes place in finite-size networks with head and tail chains of equal length, provided the injection and ejection rates at their external ends are equal and greater than one-half. Then the local density distribution and the nearest-neighbor correlations in the middle chain correspond to a shock phase with completely delocalized domain wall. Upon moving the shortcut to the head or tail of the network, the density profile takes a shape typical of a high- or low-density phase, respectively. Surprisingly, the main quantitative parameters of that shock phase are governed by a positive root of a cubic equation, the coefficients of which linearly depend on the probability of choosing the shortcut. Alternatively, they can be expressed in a universal way through the shortcut current. The unexpected conclusion is that a shortcut in the bulk of a single lane may create traffic jams.

Role of network junctions for the totally asymmetric simple exclusion process

We study the effect of local regulation mechanisms on stochastic network traffic, based on simple examples. Using the totally asymmetric simple exclusion process on a multiloop structure in which several segments share a single junction, we illustrate several mechanisms: (i) additional segments improve transport but the effect saturates due to blockage, (ii) bias reduces the overall transport and leads to several regimes, (iii) "pumping" particles out of the junctions, via a locally increased hopping rate, allows us to compensate the bottlenecks but becomes futile beyond a characteristic rate which we determine. We provide a generic discussion of combinations of these effects, including phase diagrams in terms of the control parameters.

Phase diagrams of the Katz-Lebowitz-Spohn process on lattices with a junction

This paper studies the Katz-Lebowitz-Spohn (KLS) process on lattices with a junction, where particles move on parallel lattice branches that combine into a single lattice at the junction. It is shown that 11 kinds of phase diagrams could be observed, depending on the two parameters ε and δ in the KLS process. We have investigated the phase diagrams as well as bulk density analytically based on flow rate conservation and the extremal current principle. Extensive Monte Carlo computer simulations are performed, and it is found that they are in excellent agreement with theoretical prediction.

Position dependence of the particle density in a double-chain section of a linear network in a totally asymmetric simple exclusion process

We report here results on the study of the totally asymmetric simple exclusion process, defined on an open network, consisting of head and tail simple-chain segments with a double-chain section inserted in between. Results of numerical simulations for relatively short chains reveal an interesting feature of the network. When the current through the system takes its maximum value, a simple translation of the double-chain section forward or backward along the network leads to a sharp change in the shape of the density profiles in the parallel chains, thus affecting the total number of particles in that part of the network. In the symmetric case of equal injection and ejection rates α=β>1/2 and equal lengths of the head and tail sections, the density profiles in the two parallel chains are almost linear, characteristic of the coexistence line (shock phase). Upon moving the section forward (backward), their shape changes to the one typical for the high- (low-) density phases of a simple chain. The total bulk density of particles in a section with a large number of parallel chains is evaluated too. The observed effect might have interesting implications for the traffic flow control as well as for biological transport processes in living cells. An explanation of this phenomenon is offered in terms of a finite-size dependence of the effective injection and ejection rates at the ends of the double-chain section.

Motor-mediated bidirectional transport along an antipolar microtubule bundle: a mathematical model

Long-distance bidirectional transport of organelles depends on the coordinated motion of various motor proteins on the cytoskeleton. Recent quantitative live cell imaging in the elongated hyphal cells of Ustilago maydis has demonstrated that long-range motility of motors and their endosomal cargo occurs on unipolar microtubules (MTs) near the extremities of the cell. These MTs are bundled into antipolar bundles within the central part of the cell. Dynein and kinesin-3 motors coordinate their activity to move early endosomes (EEs) in a bidirectional fashion where dynein drives motility towards MT minus ends and kinesin towards MT plus ends. Although this means that one can easily assign the drivers of bidirectional motion in the unipolar section, the bipolar orientations in the bundle mean that it is possible for either motor to drive motion in either direction. In this paper we use a multilane asymmetric simple exclusion process modeling approach to simulate and investigate phases of bidirectional motility in a minimal model of an antipolar MT bundle. In our model, EE cargos (particles) change direction on each MT with a turning rate Ω and there is switching between MTs in the bundle at the minus ends. At these ends, particles can hop between MTs with rate q(1) on passing from a unipolar to a bipolar section (the obstacle-induced switching rate) or q(2) on passing in the other direction (the end-induced switching rate). By a combination of numerical simulations and mean-field approximations, we investigate the distribution of particles along the MTs for different values of these parameters and of Θ, the overall density of particles within this closed system. We find that even if Θ is low, the system can exhibit a variety of phases with shocks in the density profiles near plus and minus ends caused by queuing of particles. We discuss how the parameters influence the type of particle that dominates active transport in the bundle.

Totally asymmetric simple exclusion process on networks

We study the totally asymmetric simple exclusion process (TASEP) on complex networks, as a paradigmatic model for transport subject to excluded volume interactions. Building on TASEP phenomenology on a single segment and borrowing ideas from random networks we investigate the effect of connectivity on transport. In particular, we argue that the presence of disorder in the topology of vertices crucially modifies the transport features of a network: irregular networks involve homogeneous segments and have a bimodal distribution of edge densities, whereas regular networks are dominated by shocks leading to a unimodal density distribution. The proposed numerical approach of solving for mean-field transport on networks provides a general framework for studying TASEP on large networks, and is expected to generalize to other transport processes.

Understanding totally asymmetric simple-exclusion-process transport on networks: generic analysis via effective rates and explicit vertices

In this paper we rationalize relevant features of totally asymmetric simple-exclusion processes on topologies more complex than a single segment. We present a mean-field framework, exploiting the previously introduced notion of effective rates, which we express in terms of the average particle density on explicitly introduced junction sites. It allows us to construct the phase behavior as well as the current-density characteristic from well-known results for a linear totally asymmetric simple-exclusion-process segment in a very systematic and generic way. We validate the approach by studying a fourfold vertex in all variations in the number of entering/exiting segments and compare our predictions to simulation data. Generalizing the notion of particle-hole symmetry to take into account the topology at a junction shows that the average particle density at the junction constitutes a relevant directly observable parameter which gives detailed insight into the transport process. This is illustrated by a complete study of a simple network with figure-of-eight topology. Finally we generalize the approach to handle rate bias at a junction and discuss the surprisingly rich phenomenology of a biased figure-of-eight structure. This example highlights that the proposed framework is generic and readily extends to other topologies.

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.