Asymmetric Simple Exclusion Process Research Articles

Power series method for solving TASEP-based models of mRNA translation

We develop a method for solving mathematical models of messenger RNA (mRNA) translation based on the totally asymmetric simple exclusion process (TASEP). Our main goal is to demonstrate that the method is versatile and applicable to realistic models of translation. To this end we consider the TASEP with codon-dependent elongation rates, premature termination due to ribosome drop-off and translation reinitiation due to circularisation of the mRNA. We apply the method to the model organism Saccharomyces cerevisiae under physiological conditions and find an excellent agreement with the results of stochastic simulations. Our findings suggest that the common view on translation as being rate-limited by initiation is oversimplistic. Instead we find theoretical evidence for ribosome interference and also theoretical support for the ramp hypothesis which argues that codons at the beginning of genes have slower elongation rates in order to reduce ribosome density and jamming.

Exact solution of the facilitated totally asymmetric simple exclusion process

We obtain the exact solution of the facilitated totally asymmetric simple exclusion process (F-TASEP) in 1D. The model is closely related to the conserved lattice gas (CLG) model and to some cellular automaton traffic models. In the F-TASEP a particle at site j in jumps, at integer times, to site j + 1, provided site j − 1 is occupied and site j + 1 is empty. When started with a Bernoulli product measure at density , the system approaches a stationary state. This non-equilibrium steady state (NESS) has phase transitions at and . The different density regimes , , and exhibit many surprising properties; for example, the pair correlation satisfies, for all , , with k = 2 when , k = 6 when , and k = 3 when . The quantity , where is the variance in the number of particles in an interval of length L, jumps discontinuously from to 0 when and when .

Statistics of TASEP with Three Merging Characteristics

In this paper we consider the totally asymmetric simple exclusion process, with non-random initial condition having three regions of constant densities of particles. From left to right, the densities of the three regions are increasing. Consequently, there are three characteristics which meet, i.e. two shocks merge. We study the particle fluctuations at this merging point and show that they are given by a product of three (properly scaled) GOE Tracy-Widom distribution functions. We work directly in TASEP without relying on the connection to last passage percolation.

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.

Cluster mean field analysis of spontaneous symmetry breaking in totally asymmetric simple exclusion processes on two intersected lattices

This paper studies totally asymmetric simple exclusion processes (TASEPs) on two intersected lattices. There are four types of particles in the model, and the particles have pre-defined destinations. Spontaneous symmetry breaking (SSB) phenomenon exists in the model. When parameter γ<1, simple mean field analysis fails to describe the four kinds of particles due to neglect of correlation of sites. In this paper, we carry out a cluster mean field analysis for the asymmetric phase boundaries and density profiles, the correlation of three sites is considered in the analysis. It is found that analytical results are in good agreement with Monte Carlo simulation ones. When γ=1, the simple mean field analysis can be adopted in the analysis, and the cluster mean field analytical results are closer to the simulation ones than that of simple mean field analysis.

Kardar-Parisi-Zhang universality of the Nagel-Schreckenberg model.

Dynamical universality classes are distinguished by their dynamical exponent z and unique scaling functions encoding space-time asymmetry for, e.g., slow-relaxation modes or the distribution of time-integrated currents. So far the universality class of the Nagel-Schreckenberg (NaSch) model, which is a paradigmatic model for traffic flow on highways, was not known. Only the special case v_{max}=1, where the model corresponds to the totally asymmetric simple exclusion process, is known to belong to the superdiffusive Kardar-Parisi-Zhang (KPZ) class with z=3/2. In this paper, we show that the NaSch model also belongs to the KPZ class for general maximum velocities v_{max}>1. Using nonlinear fluctuating hydrodynamics theory we calculate the nonuniversal coefficients, fixing the exact asymptotic solutions for the dynamical structure function and the distribution of time-integrated currents. The results of large-scale Monte Carlo simulations match the exact asymptotic KPZ solutions without any fitting parameter left. Additionally, we find that nonuniversal early-time effects or the choice of initial conditions might have a strong impact on the numerical determination of the dynamical exponent and therefore lead to inconclusive results. We also show that the universality class is not changed by extending the model to a two-lane NaSch model with lane-changing rules.

Two-channel totally asymmetric simple exclusion process with site-wise dynamic disorder

Being inspired by the biological process of gene transcription, we investigate the phenomenology of transport on two-channel TASEP in the presence of dynamical defects under symmetric coupling conditions. The continuum mean-field equations are derived and then solved numerically to obtain steady-state phase diagrams. The role of various parameters namely defect (un-)binding constants, lane-changing rates has been thoroughly examined. The defect (un-)binding rates are found to affect the phase boundaries in the phase plane. The effect of system size on the steady-state dynamics of the system has also been analyzed. The theoretical results are validated by performing extensive Monte Carlo simulations.

Limit fluctuations for density of asymmetric simple exclusion processes with open boundaries

Nous étudions les fluctuations de la densité de particules dans un processus d’exclusion simple asymétrique sous la distribution stationnaire (ou état stable), vues comme un processus stochastique indexé par $[0,1]$. Pour trois des phases du modèle et à leurs frontières nous obtenons une description complète des limites d’échelles de ces fluctuations lorsque le nombre de sites tend vers l’infini. Dans la phase de courant maximal, la limite est la somme de deux processus indépendants : un mouvement brownien et une excursion brownienne. Ce résultat étend celui obtenu précédemment par Derrida et al. (J. Statist. Phys. 115 (2004) 365–382) pour le processus d’exclusion simple totalement asymétrique et dans la même phase. Dans les phases de fortes et faibles densités, les limites sont des mouvements browniens. De façon plus intéressante, à la frontière de la phase de courant maximal, la limite est la somme de deux processus indépendants : un mouvement brownien et un méandre brownien (ou, selon la partie de la frontière, un méandre brownien renversé en temps). Nos démonstrations reposent sur une représentation des fonctions génératrices des lois fini-dimensionnelles du processus d’exclusion simple asymétrique en termes de moments joints d’un processus de Markov construit à partir de mesures rendant orthogonaux les polynômes d’Askey–Wilson.

Totally asymmetric simple exclusion processes on two intersected lanes

This paper carries out a cluster mean field analysis for spontaneous symmetry breaking in a two-lane totally asymmetric exclusion processes with an intersection. We find that the boundaries of the asymmetric phase are determined by differences of upstream segment densities and downstream segment flow rates of two lanes. The spontaneous symmetry breaking phenomenon exists when the interaction of particles is strong enough. The critical values, beyond which the phenomenon disappears, are identified through simulation and analysis, and they are in excellent agreement. The analytical results of the asymmetric phase boundaries are closer to simulation ones than those of simple mean field analysis. The analytical results of density profiles and the simulation ones are also in excellent agreement.

Dynamical Transitions in a One-Dimensional Katz–Lebowitz–Spohn Model

Dynamical transitions, already found in the high- and low-density phases of the Totally Asymmetric Simple Exclusion Process and a couple of its generalizations, are singularities in the rate of relaxation towards the Non-Equilibrium Stationary State (NESS), which do not correspond to any transition in the NESS itself. We investigate dynamical transitions in the one-dimensional Katz–Lebowitz–Spohn model, a further generalization of the Totally Asymmetric Simple Exclusion Process where the hopping rate depends on the occupation state of the 2 nodes adjacent to the nodes affected by the hop. Following previous work, we choose Glauber rates and bulk-adapted boundary conditions. In particular, we consider a value of the repulsion which parameterizes the Glauber rates such that the fundamental diagram of the model exhibits 2 maxima and a minimum, and the NESS phase diagram is especially rich. We provide evidence, based on pair approximation, domain wall theory and exact finite size results, that dynamical transitions also occur in the one-dimensional Katz–Lebowitz–Spohn model, and discuss 2 new phenomena which are peculiar to this model.

Integral Formulas and Antisymmetrization Relations for the Six-Vertex Model

We study the relationship between various integral formulas for nonlocal correlation functions of the six-vertex model with domain wall boundary conditions. Specifically, we show how the known representation for the emptiness formation probability can be derived from that for the so-called row configuration probability. A crucial ingredient in the proof is a relation expressing the result of antisymmetrization of some given function with respect to permutations in two sets of its variables in terms of the Izergin-Korepin partition function. This relation generalizes another one obtained by Tracy and Widom in the context of the asymmetric simple exclusion process.

Occupancy correlations in the asymmetric simple inclusion process.

The asymmetric simple inclusion process (ASIP)-a lattice-gas model for unidirectional transport with irreversible aggregation-has been proposed as an inclusion counterpart of the asymmetric simple exclusion process and as a batch service counterpart of the tandem Jackson network. To date, the analytical tractability of the model has been limited: while the average particle density in the model is easy to compute, very little is known about the joint occupancy distribution. To partially bridge this gap, we study occupancy correlations in the ASIP. We take an analytical approach to this problem and derive an exact formula for the covariance matrix of the steady-state occupancy vector. We verify the validity of this formula numerically in small ASIP systems, where Monte Carlo simulations can provide reliable estimates for correlations in reasonable time, and further use it to draw a comprehensive picture of spatial occupancy correlations in ASIP systems of arbitrary size.

Dependence of the transportation time on the sequence in which particles with different hopping probabilities enter a lattice.

Smooth transportation has drawn the attention of many researchers and practitioners in several fields. In the present study, we propose a modified model of a totally asymmetric simple exclusion process (TASEP), which includes multispecies of particles and takes into account the sequence in which the particles enter a lattice. We investigate the dependence of the transportation time on this "entering sequence" and show that, for a given collection of particles, group sequence in some cases minimizes the transportation time better than a random sequence. We also introduce the "sorting cost" necessary to transform a random sequence into a group sequence and show that when this is included a random sequence can become advantageous in some conditions. We obtain these results not only from numerical simulations but also by theoretical analyses that generalize the simulation results for some special cases.

Stochastic transport on flexible lattice under limited resources

Many theoretical models have tried to study the biological transport systems with the unbending paths in the framework of totally asymmetric simple exclusion processes. However, in most cases, these models ignore the three-dimensional (3D) conformation of the intracellular highways, in particular, microtubules and mRNAs and it is assumed that particles perform a motion along one-dimensional rigid paths connected with the reservoir of the infinite number of particles. In this work, we generalize the standard single lane exclusion process to analyze the transport of particles moving along a polymer-like flexible lattice plunged in a 3D reservoir with the finite number of particles which is a more realistic case both in physical as well as biological systems. We investigate the system dynamics by obtaining phase diagrams and density profiles with respect to the total number of particles. Moreover, we also study the effect of flexibility and finite resources on the steady-state system properties. It is also analyzed how the lattice occupancy affixes the 3D shape of the considered lattice under the influence of limited resources. Four stationary phases, including a shock phase, are reported. The interplay between the finite supply of particles and 3D environment with flexible lattice produces a novel feature in the form of back-and-forth phase transition. The study reveals that as a non-trivial effect on the system dynamics, the 3D environment counters the existence of bottleneck like situation arising through jamming of many particles on the lattice.

Both on-ramp and off-ramp on the boundaries of totally asymmetric simple exclusion

In this paper, both on-ramp and off-ramp on the boundaries of totally asymmetric simple exclusion process (TASEP) were considered. Here, particles can enter and leave the one dimensional lattice on the same site of boundaries by attachment and detachment. The changes of the phase diagrams and density profiles are given by theoretical calculations and the extensive Monte Carlo computer simulations. The result shows that the off-ramp and on-ramp can vary the diagram very obviously.

Totally asymmetric simple exclusion process with two consecutive ramps

In this paper, the effect of totally asymmetric simple exclusion process (TASEP) with two consecutive ramps (in the bulk far away boundaries) has been investigated. The phase diagram changes with the off-ramp rate p and on-ramp rate q have been given. In addition, the main boundary expressions are also obtained by mean field approximation (MFA). The LD (Low Density)/LD and HD (High Density)/HD phases still vanish with p > 1/2 and q > 1/2, respectively. In particular, the MC/MC phase exists in the system with the relationship p/(1+2p) < q < p/(1-2p). Specially, when p and q satisfy p >1/2 and q > 1/2, the phase diagram is divided into four intervals with same area by the LD/MC (Maximum Current), LD/HD, MC/MC and MC/HD phases.

Passive tracer dynamics in slow-bond problem

Asymptotic Kardar–Parisi–Zhang (KPZ) properties are investigated in the totally asymmetric simple exclusion process (TASEP) with a localized geometric defect. In particular, we focus on the universal nature of nonequilibrium steady states of the modified TASEP. Since the original TASEP belongs to the KPZ universality class, it is mathematically and physically a quite interesting question whether the localized columnar defect, the slow bond (SB), is really always relevant to the KPZ universality or not. However, it is numerically controversial to address the possibility of the non-queued SB phase in the weak-strength SB limit. Based on the detailed statistical analysis of KPZ-type growing interfaces, we present a comprehensive view of the non-queue SB phase, compared to finite-size crossover effects that reported in our earlier work (Soh et al 2017 Phys. Rev. E 95 042123). Moreover, we employ two types of passive tracer dynamics as the probe of the SB dynamics. Finally, we provide intuitive arguments for additional clues to resolve the controversy of the SB problem.

One-dimensional discrete aggregation-fragmentation model.

We study here one-dimensional model of aggregation and fragmentation of clusters of particles obeying the stochastic discrete-time kinetics of the generalized totally asymmetric simple exclusion process (gTASEP) on open chains. The gTASEP is essentially the ordinary TASEP with backward-ordered sequential update (BSU), however, equipped with two hopping probabilities: p and p_{m}. The second modified probability p_{m} models a special kinematic interaction between the particles of a cluster in addition to the simple hard-core exclusion interaction, existing in the ordinary TASEP. We focus on the nonequilibrium stationary properties of the gTASEP in the generic case of attraction between the particles of a cluster. In this case the particles of a cluster have higher chance to stay together than to split, thus producing higher throughput in the system. We explain how the topology of the phase diagram in the case of irreversible aggregation, occurring when the modified probability equals unity, changes sharply to the one, corresponding to the ordinary TASEP with BSU, as soon as the modified probability becomes less than unity and aggregation-fragmentation of clusters appears. We estimate various physical quantities in the system and determine the parameter-dependent injection and ejection critical values by extensive computer simulations. With the aid of random walk theory, supported by the Monte Carlo simulations, the properties of the phase transitions between the three stationary phases are assessed.

On Matrix Product Ansatz for Asymmetric Simple Exclusion Process with Open Boundary in the Singular Case

We study a substitute for the matrix product ansatz for asymmetric simple exclusion process with open boundary in the “singular case” $$\alpha \beta =q^N\gamma \delta ,$$ when the standard form of the matrix product ansatz of Derrida et al. (J Phys A 26(7):1493–1517, 1993) does not apply. In our approach, the matrix product ansatz is replaced with a pair of linear functionals on an abstract algebra. One of the functionals, $$\varphi _1,$$ is defined on the entire algebra, and determines stationary probabilities for large systems on $$L\ge N+1$$ sites. The other functional, $$\varphi _0,$$ is defined only on a finite-dimensional linear subspace of the algebra, and determines stationary probabilities for small systems on $$L< N+1$$ sites. Functional $$\varphi _0$$ vanishes on non-constant Askey–Wilson polynomials and in non-singular case becomes an orthogonality functional for the Askey–Wilson polynomials.

The effect of local dissociation on dynamics of interacting molecular motors

Successful functioning of all living systems depends on several classes of active enzymatic molecules known as biological molecular motors. They are involved in processes that require the application of mechanical forces such as cellular transport, muscle functioning, synthesis of proteins and nucleic acids and many others. Experimental studies suggest that most biological molecular motors function collectively by interacting with each other and moving along linear tracks, from which they occasionally dissociate at specific locations. We develop a theoretical model to investigate the multi-particle dynamics of interacting molecular motors with local dissociations. It is specifically stimulated by ribosomes motion along ribonucleic acid (RNA) molecules during the protein synthesis when the ribosome complex might dissociate into the solution by encountering a specially localized region on RNA. In our theoretical approach, we model the dynamics of molecular motors as one-dimensional totally asymmetric simple exclusion processes for interacting particles. Using a cluster mean-field approach, which partially takes into account the correlations in the system, stationary properties such as particle currents, densities and phase diagrams are explicitly calculated. It is found that the presence of local dissociations increases the number of possible stationary phases. Furthermore, the strength of interactions between molecular motors, the modification of transition rates due to interactions and the frequency of dissociations strongly influence the dynamics of molecular motors. The microscopic origin of these observations are discussed. Our theoretical predictions are fully supported by Monte Carlo computer simulations.