Recent developments in totally asymmetric simple exclusion processes with local inhomogeneity

A totally asymmetric simple exclusion process (TASEP) has become an essential tool in modeling and analyzing non-equilibrium systems. A wide variety of TASEP models have been developed that are motivated by real-world tra ﬃ c, biological transport and by the dynamics of the process itself. This paper provides an overview of recent developments in TASEP with inhomogeneity. Some important generalizations and extensions of inhomogeneous TASEP models are reviewed, and several popular mean- ﬁ eld techniques used to analyze the inhomogeneous TASEP models are summarized. A comparison between similar TASEP models under di ﬀ erent updating procedures is given. Phase separations in such disordered systems have been identi ﬁ ed. The present status of the inhomogeneous TASEP models and areas for future investigations are also described.


Introduction
A totally asymmetric simple exclusion process (TASEP) has been acknowledged as a paradigmatic model for nonequilibrium systems. A TASEP is a one-dimensional lattice model in which particles hop to the nearest-neighbor sites in a preferred direction and interact through hard-core exclusion (i.e. each site can be occupied by no more than one particle at any given time). TASEP was introduced originally in 1968 as a theoretical model for describing ribosome motion along mRNA [1].
Several review articles on TASEP models have been published from the perspective of applied biophysics [12,13] as well as from a purely theoretical viewpoint [14,15]. We aim to provide an overview from a different perspective based on local inhomogeneities. In this paper, we give a brief review of some generalizations and extensions of the inhomogeneous TASEPs, and we summarize several popular mean-field techniques used to analyze inhomogeneous TASEP models. Phase separations previously observed in such disordered systems have been identified as well. Finally, we describe the present status of inhomogeneous TASEP models and areas for future investigations. TASEP variants have been successfully applied to modeling real-world complex systems in biology, physics and chemistry. However, because of the limited space in this paper, we do not review such applications in detail. There exists a large body of literature on TASEP-related applications. We refer the reader to review articles [12,13] and references therein for biological transport, and review articles [10,11] and references therein for vehicular traffic.

TASEP with local inhomogeneities
Local inhomogeneities are also referred to as site-wise inhomogeneities. They can be divided into single inhomogeneity and multiple inhomogeneities. An inhomogeneous site is normally characterized by a different hopping probability from that of homogeneous sites. In other words, if the hopping probability is denoted by p, p = 1 represents normal sites, while p 1 corresponds to inhomogeneous sites. In reality, local inhomogeneities may be involved in many biological transport processes [16] as well as in vehicular and pedestrian traffic [17]. For instance, the local inhomogeneity of immunoreactivity may lead to a high susceptibility to respiratory infection [16], while high-density (e.g. congested) vehicular traffic may be attributed to some local inhomogeneities, e.g. on-ramps, lane reductions or temporary road works [17]. Moreover, it has been proposed that the crowding of molecular motors may be a source of several human disorders including neurodegenerative and kidney diseases [18].

TASEP with single inhomogeneity
TASEP with single inhomogeneity in a one-dimensional lattice under open boundary conditions is illustrated in Figure 1. Such a system can be decomposed into two steps: (i) a single system with inhomogeneity is divided into two homogeneous subsystems, each of which is treated as a normal TASEP; (ii) the two subsystems are connected by this single inhomogeneous site. This approach is known as "Segmented Meanfield Approximation" (SMFA). The decomposition approach assumes that there are no correlations in the probability of finding two particles at any two sites, i.e.
where ρ k and ρ k+1 represent statistical densities at sites k and k + 1, respectively. ... is statistical average. This approach was first used to investigate TASEP with single inhomogeneity and random update [19]. In Figure 1, β e f f , α e f f and the current J k,k+1 between sites k and k+1 can be written as β e f f and α e f f are the exit rate and injection rate at the left and right subsystems, respectively. ρ k and ρ k+1 are densities at sites k and k + 1. p is the inhomogeneous hopping probability between sites k and k+1. Theoretical analysis of the model in [19] indicates that the phase diagram is similar to that of the normal TASEP (i.e. the LD, HD and MC phases are similar between the models) [2] with shifts of phase boundaries. Table 1 lists the relationship between current and bulk density of different phases in the TASEP with single inhomogeneity and random update. Similarly, TASEP with single inhomogeneity and parallel update has been studied in [20]. Parallel update exhibits strong interaction between particles, therefore, the model in [20] may be more suitable for car traffic, but less suitable for molecular motors or other biological transport. More importantly, eq. (2) cannot be used for theoretical analysis of parallel update. However, J k,k+1 can be written as Applying the SMFA approach, TASEP with single inhomogeneity and parallel update can be solved analytically. Table  2 shows the corresponding density profiles and system currents in the LD, HD and MC phases. Compared to Table  1, it is found that for the same p, current with parallel update is larger than that with random update. It is also shown that different updating procedures produce different dynamical properties even for the same system. TASEP with single inhomogeneity and large particles has been investigated by Shaw et al. [21]. As expected, the phase diagrams are still qualitatively identical to the standard TASEP [2] with different phase boundaries. Large particles, also referred to as extended objects in the literature, can occupy several sites (i.e. particle size > 1). The counterparts of large particles in nature can be long-size vehicles, ribosomes, dimers, multiple-motor cooperative motion, large molecules or vesicles.
Foulaadvand et al. [22] studied a TASEP with variable hopping probability. The hopping probability is assumed to follow a binary or uniform distribution. These researchers found that the impact of disorder depends greatly upon the boundary conditions. TASEP with local inhomogeneity coupled with Langmuir kinetics (often abbreviated to LK) in a one-dimensional lattice has been investigated in [23]. LK is widely used to describe absorption-desorption processes in surface chemistry. In biophysics, it is used to describe particles randomly attaching to and detaching from a lattice. In [23], a novel phase, termed the bottleneck phase (BP), is introduced to describe Table 1 Relation between current and bulk density of different phases in the TASEP with local inhomogeneity and random update [19] Phase Conditions Current (J) Bulk density (ρ) Table 2 Relation between current and bulk density of different phases in the TASEP with local inhomogeneity and parallel update [20] Phase Conditions Current (J) Bulk density (ρ) the current independently of boundary conditions. Several rich bottleneck-induced mixed phases (e.g. LD-BP, LD-MC-BP, MC-BP-MC) are reported to result from the BP. Apart from the study of the TASEP in a one-lane system, the effects of local inhomogeneity on two-lane TASEP coupled with LK are also considered [24]. A single inhomogeneity is assumed to be located in one of two lanes. It is found that the local inhomogeneity effect can be observed in both lanes because of particles changing to the other channel when the hopping probability of the inhomogeneous site is small. This effect can be reduced by increasing the hopping probability or channel-changing rate.
Theoretically, an inhomogeneous site can be found at any position of the lattice. In the above-mentioned investigations, the inhomogeneous site was assumed to be the middle of the lattice. It is expected that the phase diagram and density profiles are qualitatively the same for central and off-center inhomogeneous sites, provided the off-center site is distant from the boundaries.

TASEP with multiple inhomogeneities
Multiple inhomogeneities may include: (1) randomly distributed inhomogeneous sites; and (2) a group of continuous inhomogeneous sites. When several site-wise inhomogeneities exist, the system becomes more complex. Ref. [25] investigated dynamic properties of a TASEP with two different hopping rates p a and p b on a one-dimensional lattice. In this model, sites at rate p b are arranged with a period of T . The authors' theoretical analysis suggests that the dual-rate TASEP retains the three stationary phases (LD, HD and MC).
The effects of clustered slow codons (local inhomogeneities) in mRNA translation and protein synthesis are investigated using TASEP in [26]. Results from this study show that the clustered defects can affect ribosomal (particle) current. When the number of slow codons in a cluster exceeds 3 or 4, the ribosomal current is not significantly reduced. However, this approach does not allow segment lengths of more than 20 sites to be treated, on account of the increased numerical complexity for longer segments.
Dong et al. [27] investigated the effects of having inhomogeneities at different positions on a lattice. For instance, a slow site near the system boundaries induces a higher current than when the inhomogeneity is at the center of the lattice. Thus a concept of effective boundary rate is proposed. The corresponding effect, called "edge effect", implies a dependence of the current on the position of the inhomogeneity.
Differently from [27], [28] studied a case of a zoned inhomogeneity in the TASEP. The zoned inhomogeneity is a sequence of consecutive inhomogeneous sites with a reduced hopping probability p (0 < p < 1). These consecutive inhomogeneities are assumed to be close to one of the boundaries. Theoretical and simulation results support each other and show that the current and the phase diagram depend not only on the length of the bottleneck, but also on its position. If two sequences of consecutive inhomogeneities appear in the system, it is found that inhomogeneities of smaller length and distant from the first one (near the boundary) have no influence on the maximal current (i.e. on transport capacity). This study provides alternative support for the edge effect.
Similarly, the effect of an extended inhomogeneity on a one-dimensional TASEP with parallel update was studied by Liu et al. [29]. Here, however, the extended inhomogeneity is assumed to be located in the middle of the lattice. The study considers two lattice geometries, V and W. In Case V, the system consists of a left and right subsystem (segment I and segment II respectively). Segment I is a homogeneous lattice with hopping probability 1, while segment II is also homogeneous, but with hopping probability p. Case W is a natural extension of Case V, comprising three segments (I, II and III),corresponding to the left, middle and right subsystems, respectively.
The phase diagram of Case W is shown in Figure 2(a). The (HD/MC/LD) phase means that the three segments (I, II and III) are in the HD, MC and LD phases, respectively. α * and β * are a pair of critical points. Any change of α * or β * will lead  Table 3 for details of A, B, C, and D.
to a phase transition. The different MC regions are illustrated in Figure 2(b) and detailed in Table 3. From Figure 2(b), it can be seen that in case W the MC region reaches the maximal area. However, the maximal currents are the same for the TASEP with zoned inhomogeneities (see Table 3). Another conclusion that can be drawn is that the maximal current of the TASEP with single inhomogeneity is larger than that of the TASEP with extended inhomogeneities because p/(1 + p) > (1 − 1 − p)/2 for p < 1 (see Table 3). Note that such models with random update have been studied by Xiao et al. [30]. Models with parallel update [29] and random update [30] have the same number of steady-state phases, but possess different phase boundaries. In addition, the maximal current in [29] is (1 − 1 − p)/2, which is larger than that reported by [30], p/4. We remark that these recent investigations share two common features: (i) The presence of inhomogeneities leads to a decrease of the maximal current; however, the maximal-current regions expand, compared to the normal systems; and (ii) The occurrence of phase separations, reported in other disordered systems in a previous literature review [31], has also been observed in the above-mentioned disordered systems.
In practice, TASEP has been applied to modeling traffic flow in a single-lane highway with ramps [32]. For a single ramp (either on-ramp or off-ramp), the bottleneck phenomenon does not exist in this model. Traffic jams before the ramp and free flow after the ramp are not observed. For two consecutive ramps, the outcome changes. In particular, when an on-ramp is placed before an off-ramp, the model predicts that the bottleneck effect emerges and that the flow between the two ramps saturates. No real traffic experiments have been reported to either confirm or reject these predictions.

Summary and conclusions
The various aspects of TASEP with inhomogeneities reviewed in this paper cover a relatively small portion of the problems related to non-equilibrium processes. The phenomena of phase separations and the expansion of the maximalcurrent regions have been observed in these disordered systems. Theoretical methods (e.g. mean-field approximation) introduced here and in previous literature reviews could provide a basis for further developments.
From the perspective of mathematical modeling, many further investigations could be undertaken. On the other hand, the spatial correlations in mean-field treatments are normally neglected, resulting in deviations of theoretical results from computer simulations. More recently, the use of correlation functions to examine the spatially correlated variability, in particular for inhomogeneity cases, has been considered by Foulaadvand et al. [33]. Also recently, Sun et al. [34] investigated the effect of information feedback on traffic current in the presence of a single inhomogeneity. Such a model considers the next-step traffic and is resolved via computer simulations, the complexity of the model precluding analytical solutions. Model complexity in realistic scenarios poses a challenge to theoretical approaches. However, pursuing numerical approaches in future work is expected to reveal a rich variety of new phenomena.
In conclusion, we have highlighted the use of TASEP variants in the modeling and analysis of non-equilibrium systems, Table 3 Details of Figure 2 though the models are still oversimplified and remain unrealistic because of the lack of empirical data. Nevertheless, the simple TASEP models have proved encouraging enough for researchers to try and tackle more sophisticated models that reveal previously little unexplored phenomena. Future studies along these lines will hopefully provide deeper insight into fundamental issues as well as enhanced understanding of real-life traffic flows.