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Collective dynamics of an inhomogeneous two-channel exclusion process: Theory and Monte Carlo simulations

This work is devoted to the development of a novel theoretical approach, named hybrid approach, to handle a localized bottleneck in a symmetrically coupled two-channel totally asymmetric simple exclusion process with Langmuir kinetics. The hybrid approach is combined with singular perturbation technique to get steady-state phase diagrams and density profiles. We have thoroughly examined the role played by the strength of bottleneck, binding constant and lane-changing rate in the system dynamics. The appearances of bottleneck-induced shock, a bottleneck phase and Meissner phase are explained. Further, the critical values of bottleneck rate are identified, which signify the changes in the topology of phase diagram. It is also found that increase in lane-changing rate as well as unequal attachment, detachment rates weaken the bottleneck effect. Our theoretical arguments are in good agreement with extensively performed Monte Carlo simulations.


Introduction
In protein synthesis, the genetic information is deciphered into proteins by molecular machines called ribosomes that attach themselves at the start end of mRNA, move along the chain in a unidirectional manner and finally detach at the stop end [2]. Each translation step requires the binding of a freely diffusing transfer-RNA (tRNA) molecule, carrying the amino acid specific to each codon [2,30]. The important factor affecting the ribosome translation rate is relative concentrations of tRNA, which may vary from codon to codon. The codons with lower concentrations of tRNA reduce the protein synthesis rate and thus plays the role of an inhomogeneity in a homogeneous system [32,33,34]. Apart from this, inhomogeneities also occur naturally in many other transport systems such as vehicular traffic [11], blood flow [24] and flow of data in a Von Neumann architecture [20]. In traffic flow, the ongoing construction on roads, a slow moving vehicle or an accident can lead to slow down the flow rate on highways and can lead to congestion.
Further, the separation of the CPU and the memory in computers creates Von Neumann bottleneck, which limits the performance of the computer via limited bandwidth between the CPU and the memory.
Totally asymmetric simple exclusion process (TASEP) [3,21] is well known to be a paradigmatic model for studying stochastic transport in many-particle systems. Both single-channel and multi-channel TASEPs have been well explored theoretically as well as using Monte Carlo simulations [5,9,19,27,28,31]. Further, one has to take into account the fact that the proteins as molecular motors can also attach from the bulk reservoir or detach from it (Langmuir Kinetics (LK)) [13]. In contrast to TASEP, where total number of particles remain conserved, the additional attachmentdetachment dynamics (LK dynamics) violate the particle conservation and leads to many interesting phenomena [25]. In literature, the consequences of coupling of the two different dynamics : TASEP and LK have been well analyzed in single-channel [22,23,25] as well as two-channel homogeneous systems [6,10,17,36]. The idea of slowing down of particles at certain defect positions can be incorporated in the form of a set of inhomogeneous lattice sites (bottleneck), either as a single unit or randomly distributed over the whole lattice in TASEP. This type of disorder is known as site-wise disorder. Another type of disorder studied in literature is particle-wise, where a slow moving particle itself acts as an inhomogeneity in the system. The present work focuses on site-wise disorder which is suitable to model the inhomogeneities present in transport of molecular motors.
Although a lot of work has been done on homogeneous TASEPs, the effect of disorder on the steady-state dynamics of such systems is not well understood. Several studies have been performed on single-channel inhomogeneous TASEPs with [26,29] as well as without LK [2,7,8,14,15,18,30,35].
While investigating the role of a bottleneck in a closed TASEP [14,15], it was found that even the presence of a single bottleneck site can produce shock profile and a plateau in the fundamental current-density relation. Later, Kolomeisky [18] examined even richer case of single-channel open system and explored the consequences of the inhomogeneity with both faster and slower transition rates. He divided the system into two homogeneous TASEPs coupled at the single bottleneck site (defect mean-field theory (DMFT)) and proved analytically that a fast site has no effect on the phase diagram; whereas a slow site leads to shifting of the phase boundaries only.
He also tested the theoretical results with Monte Carlo simulations and found a good agreement in low density (LD) and high density (HD) phases; while a little deviation in maximal current (MC) phase. Another analytical approach namely finite-segment mean-field theory (FSMFT) was introduced by Chou and Laktaos [2] to study clusters of slow codons in protein synthesis. They found that ribosome density profiles near neighboring clusters of slow codons suppress the proteins synthesis. Dong et al. [7] generalized the DMFT to study the effect of two bottlenecks on the protein production rate and also performed extensive Monte Carlo simulations to conclude that the location as well as spacing between the bottlenecks affect the production rate, pointing out an important phenomenon namely edge effect. The investigation into edge effects was further carried out by Greulich and Scadshneider [8] to generate the phase diagrams of inhomogeneous TASEP using interacting subsystem approximation (ISA). They could successfully explain the interactions of defects with the boundaries of the single-channel system. The more complex case of inhomogeneous TASEP in the presence of Langmuir kinetics was studied by Qiu et al. [29]. They calculated phase diagrams and density profiles by adopting the concept of DMFT and also studied the effect of slow hopping rate and detachment rate on the phase diagram. Pierobon et al. [26] provided a detailed study on the role of a bottleneck in a TASEP with LK, using an effective mean-field theory and Monte Carlo simulations. They introduced the concept of carrying capacity to identify various novel phases called bottleneck phases. Importantly, all of the above studies focused on single-channel inhomogeneous systems.
Parallel to the inhomogeneous single-channel systems, particles in multichannel transport systems [6,9,10,16,28,31] may also confront a bottleneck, present in either one or in more than one channels. The importance of studying the multi-channel system lies in the fact that it can act as a framework for extending the analysis to networks. Due to the complexity in dynamics generating from the interactions between different channels, it is difficult to examine the effects of inhomogeneity in a multi-channel open system. Up to our knowledge, the only contribution in this direction has been made by Wang et al. [37], which explored the effect of a local inhomogeneity in one of the lanes of a two-lane TASEP with LK under a symmetric lane changing rule. They extended the DMFT to a two-channel system by incorporating the concept of effective injection and removal rates at the inhomogeneous lattice site. Despite a good agreement between the solution of the mean-field equations and Monte Carlo simulations in ref. [37], this approach fails to produce the steady-state phase diagrams of the two-channel system. Moreover, it is not feasible to analyze the role played by various parameters such as lane-changing rate, attachment-detachment rate and strength of bottleneck on the steady-state dynamics. One can infer from here that though DMFT is capable to provide analytic solutions for single-channel inhomogeneous TASEPs with [26,29] as well as without LK [18], it lacks some important ingredients to generate a complete picture of the dynamics of the corresponding two-channel system as discussed in section 3. This motivates us to develop a new approach to handle the bottleneck in a two-channel TASEP with LK, which not only overcomes the existing limitations, but also demonstrates the unexplored dynamics of two-channel inhomogeneous systems.
The objective of the proposed study is two-fold, to develop a general theoretical approach, which is capable to produce the phase diagrams and to study the effect of various system parameters on the steady-state phases.
In this paper, we attempt to provide a complete picture of the dynamics of two-channel symmetrically coupled TASEP with LK in the presence of a single localized bottleneck in one of the two channels by adopting a new and simplified approach, called the hybrid approach. We have also validated the theoretical results with Monte Carlo simulations. The paper is organized as follows. In section 2, we define the model under examination and the governing dynamical rules. We briefly discuss the limitations of the earlier approaches in sec. 3. The theoretical hybrid approach and Monte Carlo simulations are covered in section 4 and section 5. A thorough analysis of the stationary properties of the model is discussed in section 6. In the concluding section 7, we summarize the results and future prospectives of our work.

Two-channel inhomogeneous TASEP with LK
We define our model in a two-channel (L, 2) lattice, where L is the length of a channel. The two channels are denoted by A and B, in which particles are distributed under hard-core exclusion principle (See Fig. 1). We adopt random-sequential update rules for the dynamical evolution of the system. For each time step, a lattice site (i, j); i = 1, 2, 3, .....L; j = A, B is randomly chosen. The state of the system is characterized by a set of occupation numbers τ i,j (i = 1, 2, 3, .....L; j = A, B), each of which is either zero (vacant site) or one (occupied site). At entrance (i = 1), a particle can enter the lattice with a rate α provided τ 1,j = 0; and at exit (i = L), a particle can leave the lattice with a rate β when τ L,j = 1. In the bulk, if τ i,j = 1, then the particle at the site (i, j) firstly tries to detach itself from the system with a rate ω d (detachment rate) and if it fails then it moves forward to site (i+1, j) with a rate p i,j provided τ i+1,j = 0; otherwise it attempts to shift to other lane with a rate ω, only if the target site is vacant. On the other hand, if τ i,j = 0 ; i = 2, 3, ...., L − 1, a particle can attach to the site (i, j) with a rate ω a (attachment rate). Here, horizontal transition rate p i,j is inhomogeneous and is given by the following binary distribution It is clear from Eq. (1) that the bottleneck connects the sites at i = m and i = m + 1 in lane A (Fig. 1). Here, q denotes the transition rate of a particle on passing through the bottleneck and will be called as bottleneck rate throughout this paper. We wish to analyze the effects of a localized bottleneck in the bulk with no boundary interactions of the open system.
This adds up to the assumption viz., 1 << m << L.

Inadequacies of DMFT
The only step to handle a single-site bottleneck in a two-channel TASEP with LK was put forth by Wang et al. [37]. They generalized the concept of DMFT from a single-channel inhomogeneous TASEP with LK to the cor-  ing each lane (inhomogeneous as well as homogeneous) into subsystems, its generalization either to multi-channel system or to include multiple defect sites would be a cumbersome job to handle.

Theoretical approach
As discussed in the previous section that DMFT is incapable of explaining certain important aspects, we introduce a novel hybrid approach to deal with Firstly, we compute the temporal evolution of occupation probabilities τ i,j in the bulk (1 < i < L, j = A, B) from the following set of master equations: where · · · denotes the statistical average and last term on right-hand side takes a negative and a positive sign for lane A and B, respectively. At the boundaries, the particle densities evolve according to Factorizing the correlations using mean-field approximation, we get τ i,j τ i+1,j = τ i,j τ i+1,j .

Continuum mean-field equations
To find the continuum limit of the master equations, we define lattice constant ǫ = 1/L, rescale the time and other kinetic rates as t ′ = t/L, Ω a = ω a L, Ω d = ω d L and Ω = ωL [10,25].
Writing x = (i−1)ǫ , we replace binary discrete variables τ i,j with continuous variables ρ i,j ∈ [0, 1] and retain the terms up to second-order in Taylor's series expansion to obtain

Continuum mean-field equation for lane B
Since lane B is free from any inhomogeneity, we can drop the subscript i and compute the following time evolution equation from master equation (2) for average density in lane B, denoted by ρ B .
where K = Ω a /Ω d is the binding constant and an important parameter in analysing the system dynamics [25].

Hybrid mean-field system of equations for lane A
We divide lane A into two sublattices viz., 1 < i < m and m + 1 < i < L, and (m + 1, A) sites intact after rescaling of aforementioned variables, we obtain the following hybrid system of equations for time evolution of average density in lane A, denoted by ρ A .

Hybrid system
Discrete part: In the continuum limit, the boundary equations (3) and (4) Along with these boundary conditions, the system of equations (6), (7) and (8) is consistent for which, a unique feasible solution always exists. With this favour, our new approach overcomes the limitation of DMFT, which was unable to guarantee a unique feasible solution to the nonlinear system of equations [37]. Moreover, no approximation, other than mean-field, has been used to generate the hybrid system, which is again an advantage over the methodology of DMFT [37] for a two-channel inhomogeneous TASEP with LK. It can be easily seen that the hybrid approach only segments the inhomogeneous lane into two parts, without disturbing the homogeneous lane. This idea makes it applicable to more general systems such as multi-channel or networks and those with multiple bottleneck sites.

Steady-state solution
We introduce a new space variable x to merge the equations (6),(7) and (8) into a single system, which further reduces to the following system in steady-state.
Since it is difficult to solve the system (9) analytically due to its hybrid nature, we propose to find the steady-state solution of the hybrid system using the singular perturbation technique [4]. Recently, this technique has been successfully applied for studying two-channel homogeneous TASEP with LK [6,10]. Singular perturbation technique typically involves obtaining solutions of the differential equations describing the boundary layer region and the bulk solution, separately and then matching both the solutions to get the global solution. The bulk part of the solution, known as outer solution, is found in the limit ǫ → 0. We propose a generalized methodology to find the outer solution of the hybrid system (9). Instead of solving the system (9) explicitly, we find the steady-state outer solution by capturing the long time solution of (6), (7) and (8) using the scheme discussed below [6,10]. We discretize the continuum part of model equations using finite-difference and replace the time derivative in discrete equations at sites (m, A) and (m+1, A) with forward-difference formula. For 1 < i < m, m + 1 < i < L with j = A and 1 < i < L with j = B, the following scheme is applied to model equations (6), (7) and (8) with the boundary conditions ρ The solution is captured in the limit n → ∞ to ensure the occurrence of a steady-state. Clearly, the above system is consistent and gives a unique steady-state outer solution. It is easy to check the consistency of the above system of equations by analyzing the system in the limit q → 1.
To satisfy the boundary conditions at both the ends, the density profiles incurs a crossover narrow regime in the form of either a boundary layer or a shock. This solution is known as inner solution and is found by rescaling the space variable in the neighbourhood of x d as x = x−x d ǫ , where x d is the position of boundary layer. This rescaling eliminates the non-conservative source terms, which are formed by lane-changing transitions and attachmentdetachment dynamics, in the system (9). In terms of x, the inner solution ρ j,in is given by Here, the integration constant a j is computed from the matching condition of outer and inner solution. For example, let the boundary layer appears at Here, ρ j,o is value of the outer solution in lane-j at x = 1. Clearly, ρ j,o is a function of system parameters Ω d , Ω a , and Ω A .
where ξ j = tanh −1 The solution given by eq. (12) represents a right boundary layer (rbl) in lane j with positive slope (tanh −r). When β < ρ j,o , the inner solution fails to satisfy the right boundary condition Here, the change in the slope of boundary layer describes a surface transition, which does not affect bulk density profile.
Analyzing, in a similar fashion, we can find mathematically all the bulk as well as surface transition lines in the α − β plane and generate the phase diagram for different parameters. For a more detailed solution methodology of singular perturbation technique, we refer [6].

Monte Carlo simulations
To testify the outcomes of our theoretical mean-field approach, we simulate the two-channel model using Monte Carlo simulations (MCS) [1]. We adopt random sequential update dynamical rules, which is the realization of the usual master equation in continuous time. Moreover, random-sequential type update rule is appropriate to model the dynamics of intracellular protein transport [25,6]. We run the Monte Carlo simulations for 10 10 − 10 11 time steps and first 5% steps are ignored to ensure the occurrence of steady-state.
The densities in both the lanes have been computed by taking time averages over an interval of 10L. Since the size of a real system is normally not very large, it is reasonable to simulate the system for a lattice size up to L = 1000.
Note that the dynamical rules involve many kinds of particle hoppings such as horizontal hoppings, vertical transitions and attachment-detachment dynamics, it is computationally expensive to calculate steady-state densities.
As we proceed in the next section, it is found that the results of theoretical mean-field approach agree to a good extent with Monte Carlo simulations.

Results and discussion
In this section, we provide a detailed discussion of the steady-state prop- When there is no bottleneck, i.e. q = 1, the present model reduces to the homogeneous two-channel TASEP with LK. Under symmetric coupling conditions, the two-channel homogeneous TASEP with LK is analogous to its single-channel counterpart, which has been studied well in literature [36]. For q > 1, we call the defect to be fast defect as the bottleneck rate is more than the usual horizontal particle hopping rate in the system. Clearly, a single fast defect does not affect the steady-state dynamics as the rate-limiting hoppings occurring with rate 1 dominate the whole lattice [18]. Thus, it is appropriate to focus on the situation where the bottleneck acts as a slow defect i.e. q < 1.
It is well known that the steady-state dynamics of exclusion process are significantly affected by the binding constant. So, we proceed by segmenting the analysis into two subcases viz. (i) K = 1, and (ii) K = 1. The other parameters are kept fixed to Ω d = 0.2 and L = 1000.

K = 1
We examine the steady-state dynamics for different strengths of bottleneck by gradually decreasing the value of q. No significant qualitative changes are found in the phase diagram till q c,1 ≈ 0.75, below which one notices the effect of bottleneck more clearly. Fig. 2 shows the phase diagram of the system with q = 0.75 and Ω = 1. Specifically, fig. 2(a) shows the complete characterization of various phases in terms of the boundary layers. The solid and dashed lines denote the bulk and surface transitions, respectively. As an illustration, the notation (tanh-r,tanh-r) signifies that there is a right boundary layer in lane A as well as lane B of the kind tanh-r, given by eq. (12) and S denotes the shock phase. For a detailed understanding of the bulk and surface transitions, we refer the readers to references [6,23]. As discussed in previous section that a surface transition is responsible for the change in slope of the boundary layer only and does not affect the bulk density, henceforth, our focus will be only on bulk transitions to avoid any complexities in the phase diagrams. Fig. 2(b) represents the bulk phase transitions only corresponding to fig. 2(a).
To understand the dynamics in a better way, we introduce an appropriate notation (P h A(lef t) − P h A(right) , P h B ) denoting the kind of stationary phase  rant (α ≥ 0.5, β ≥ 0.5, shaded blue), the bulk densities in each lane becomes independent of entrance as well as exit rates. This effect is similar to the behavior of the Meissner phase found in superconducting materials [12]. Therefore, we also adopt the name Meissner phase (denoted by M) to identify boundary independent behavior of two-channel system. It is important to mention that Meissner phase is actually a part of certain existing phase, denoted by phase 7 in fig. 2(b). Meissner phase is also reported in the literature for a homogeneous single-channel TASEP with LK [25]. Note that the density profiles get significantly influenced by bottleneck for a specific range of α, β. The corresponding region in phase-plane is marked with red dots and referred as bottleneck phase throughout this paper. Similar results also exist for singlechannel inhomogeneous TASEP with LK, in which bottleneck phase is identified and analyzed in terms of carrying capacity [26]. In contrast, for low entrance and exit rates one recovers the TASEP/LK density profiles perturbed by a local spike or a dip, which will be discussed as we proceed in this paper.

Phase transitions and mixed phases:
To gain an intuitive understanding of the steady-state dynamics of the system, it is important to analyze the phase transitions. Across the bulk transition line between (LD-LD,LD)( fig. 3(a)) and (LD-S,S) ( fig. 3(b)) phases, one finds the formation of a shock through deconfinement of the right boundary layer [10]. It is evident from fig. 2(b) that β = α + Ω d = α + 0.2 and β = α − Ω d = α − 0.2 for α, β ≤ 1/2 are the lines of deconfinement of shock from right boundary layer (LD phase) and left boundary layer (HD phase), respectively. Interestingly, these are also the corresponding lines of deconfinement in analogous single-channel homogeneous system [23]. From here, one can infer that the phase diagram under a weak bottleneck has almost similar topological structure as for the one with no bottleneck (q = 1) [25,36] though some of the phases differ in their qualitative nature. Actually, a weak bottleneck is incapable of affecting the steady-state dynamics of a symmetrically coupled twochannel TASEP with LK system globally. The only effects can be seen in the vicinity of the bottleneck. This point is evident from the density profiles in fig. 3, which will be discussed in detail subsequently.
As one can see from fig. 3(b), a shock is present in both the lanes in  fig. 3(b)) in HD phase. This can be attributed to particle-hole symmetry. When a particle comes out of the bottleneck in a highly packed situation, it experiences a sudden increase in its hopping rate. This leads to formation of a lower density region just downstream to the bottleneck. Similar findings have been reported in literature [26,37] fig. 3(f)), the left one being bottleneckinduced; whereas the right one is formed due to deconfinement of right boundary layer.

Effect of bottleneck strength
We propose a terminology to identify the strength of the bottleneck as shown in table 1. We also provide the number of steady-state phases observed under each case, the details of which follow in the text.
Weak bottleneck: So far, we have discussed the steady-state dynamics for a weak bottleneck and pointed out important features such as appearance of Meissner phase, bottleneck phase and bottleneck-induced shock etc. Clearly, the number of steady-state phases under a weak bottleneck is more than in homogenous case, which is due to the appearance of mixed phases. Now, we will gradually change the strength of bottleneck from weak to moderate and then to strong and see whether the same trend continues or not?
Moderate strength: In the range of moderate strength of bottleneck, we  fig. 4(a)) is qualitatively as well as quantitatively different from the one with weaker bottleneck (See fig. 2(b)). An important distinguishing feature of the phase diagram is the  Strong bottleneck: Fig. 4(c) shows the phase diagram for the present system with q = 0.1, a strong bottleneck. It can be seen that topology of the phase diagram is significantly changed in comparison to the one with moderate bottleneck. Here, the bottleneck effect is sufficiently strong to increase the density difference between two lanes up to a level that we have a complex phase diagram structure as shown in fig. 4(c). One can see that  A further decrease in value of q pulls the shock leftwards.

Effect of coupling strength
So far, we have investigated the system dynamics under different bottleneck strengths, but for fixed lane-changing rate at Ω = 1. Since coupling strength has always been an important parameter in two-channel systems, it is necessary to examine its role for the proposed model. In literature, it is well explored [6,9,28,36] that higher orders of lane-changing rate does not affect the dynamics of a symmetrically coupled homogeneous two-channel TASEP with/without LK because the system decouples into two independent single-channel homogeneous systems. But, in our model, this is not the case as our system is inhomogeneous. So, we thoroughly investigate the

K = 1
Since the methodology of the proposed hybrid mean-field approach is general and applicable for any value of binding constant K, we follow same lines to discuss the case where attachment and detachment rates differ, Ω a = Ω d , i.e. K = 1. Quite surprisingly, no density profile in α−β plane is significantly affected by bottleneck, excluding the presence of a downward spike at x = 1/2. This is in sharp contrast to the parallel case with K = 1 (See fig. 2(b)). Consequently, one does not get a bottleneck-induced shock here for any values of α and β.  fig. 8(b). A part of phase-plane (shaded with red dots) is strongly influenced by the bottleneck-induced shock called as bottleneck phase, which was absent for q > 0.4 (See fig. 8(a)). On the other hand, the value of q does not influence the Meissner phase.

Non-existence of bottleneck-induced shock
We have observed that an increase in lane-changing rate weakens the bottleneck effect and as a result the bottleneck affected region in α − β plane shrinks with respect to an increase in Ω. Comparing the two cases K = 1 and K = 1, one can infer that the effect of bottleneck is further reduced when K = 1. The above two statements generate interest to know whether there exist some values of K and/or Ω, at which we do not get a bottleneck-induced shock even for a strong bottleneck. A thorough analysis of our study has revealed that for K = 4 and Ω = 1000 with a very strong bottleneck (q = 0.01, almost a blockage), we get a steady-state situation free of any bottleneck-induced shock. The resulting steady-state phase diagram is shown in fig. 9(b). Although we do have a downward spike in the density profiles, yet the non-existence of a bottleneck-induced shock is an important outcome. Such dynamics further opens up new challenges to dig deeply into the shock dynamics of the proposed model, which will be undertaken in our future study.

Conclusion
In this paper, we attempted to provide a clear picture of the role played by an inhomogeneity in a two-channel TASEP with LK under a symmetric coupling environment. The location of bottleneck is fixed at middle site in lane A, while lane B is kept homogeneous. Theoretically, we have introduced a new hybrid approach and also employed singular perturbation technique to get the steady-state density profiles and phase diagrams. We have adopted random-sequential update rules to perform Monte Carlo simulations and shown a good agreement between theoretical and MCS.
The study is segmented into two subcases: K = 1 and K = 1. Based on the analysis, one finds very interesting topologies of the phase diagrams under different bottleneck rates. It is concluded that the steady-state phase diagrams for different bottleneck strengths under K = 1 are comparatively richer in their composition as compared to those under K = 1. For lower strengths of bottleneck, the phase diagram comes out to be quite similar to the corresponding homogeneous system and effects on the density profiles are only local in the vicinity of the bottleneck. The origin of upward and downward spike at bottleneck in LD and HD phases, respectively is explained. We also shed light on the emergence of bottleneck-induced shock in the system, which accounts for the global effects of bottleneck. The effect of lane-changing rate on the dynamics is also examined and found that an increase in coupling strength weakens the bottleneck effect for any value of K. This effect is supported by the reduction in height of upward and downward spikes with respect to increase in lane-changing rate. Another important observation is that unequal attachment-detachment rates are also responsible for weakening of bottleneck effect.
Here, we focused on a localized bottleneck, fixed at the middle of the lattice far away from the boundaries. The approach developed in this paper might be easily extended to analyze more general inhomogeneous systems such as those with more number of bottlenecks and also to study the influence of the location of the bottleneck in the multi-channel TASEPs. From the biological significance, one can say that the present study might help in optimizing the protein translation rate in multi-channel inhomogeneous systems.