Shock formation in an exclusion process with creation and annihilation

Totally asymmetric simple exclusion process with local resetting in a resource-constrained environment

Inspired by the process of mRNA translation, in which the stochastic degradation of mRNA-ribosome machinery is modeled by the resetting dynamics, we study an open totally asymmetric simple exclusion process with local resetting at the entry site in a resource-constrained environment. The effect of constrained resources on the stationary properties of the system has been comprehended in the form of the filling factor. The mean-field approximations are utilized to obtain stationary state features, such as density profiles and phase diagrams. The phase diagram possesses pure phases as well as coexisting phases, including a low-density-high-density phase separation, which did not manifest under periodic boundary conditions despite the system being closed there as well. The role of the resetting rate has been investigated on the stationary properties of the system, depending on how the filling factor scales with the system size. In contrast to the resetting model for infinite resources, two distinct phase transitions are observed for the smaller values of the filling factor leading to a change in the topology of the phase diagram. The impact of the resetting rate along with the finite-size effect has also been examined on the shock dynamics. All the mean-field results are found in remarkable agreement with the Monte Carlo simulations.

Multiple reentrance transitions in exclusion process with finite reservoir

The proposed study is motivated by the scenario of two-way vehicular traffic. We consider a totally asymmetric simple exclusion process in the presence of a finite reservoir along with the particle attachment, detachment, and lane-switching phenomena. The various system properties in terms of phase diagrams, density profiles, phase transitions, finite size effect, and shock position are analyzed, considering the available number of particles in the system and different values of coupling rate, by employing the generalized mean-field theory and the obtained results are detected to be a good match with the Monte Carlo simulation outcomes. It is discovered that the finite resources significantly affect the phase diagram for different coupling rate values, which leads to nonmonotonic changes in the number of phases in the phase plane for comparatively minor lane-changing rates and produces various exciting features. We calculate the critical value of the total number of particles in the system at which the multiple phases in the phase diagram appear or disappear. The competition between the limited particles, bidirectional motion, Langmuir kinetics, and particle lane-shifting behavior yields unanticipated and unique mixed phases, including the double shock phase, multiple reentrance and bulk-induced phase transitions, and phase segregation of the single shock phase.

Asymmetric coupling induces two-directional reentrance transition in three-lane exclusion process

Inspired by vehicular traffic phenomena, we study a three-lane open totally asymmetric simple exclusion process with both-sided lane switching in the companionship of Langmuir kinetics. We calculate the phase diagrams, density profiles, and phase transitions through mean-field theory and successfully validate these findings with Monte Carlo simulation results. It has been found that both the qualitative and quantitative topology of phase diagrams crucially rely on the ratio of lane-switching rates called coupling strength. The proposed model has various unique mixed phases, including a double shock resulting in bulk-induced phase transitions. The interplay between both-sided coupling, third lane, and Langmuir kinetics produces unusual features, including a back-and-forth phase transition, also called a reentrance transition, in two directions for relatively nominal values of coupling strength. The presence of reentrance transition and peculiar phase boundaries leads to a rare type of phase division in which one phase lies entirely within another region. Moreover, we scrutinize the shock dynamics by analyzing four different types of shock and finite-size effects.

Role of extended coupling in bidirectional transport system

Motivated by vehicular traffic phenomena, we study a bidirectional two-lane open totally asymmetric simple exclusion process with extended symmetric coupling conditions in the presence of Langmuir kinetics. The phase diagrams and density profiles are calculated utilizing mean-field theory for different lane-changing rates and are found to be in a good match with Monte Carlo simulation results. It has been observed that the qualitative topology of phase diagrams depends on the lane-switching rate significantly, resulting in nonmonotonic variations in the number of steady-state phases. The proposed model provides various mixed phases leading to bulk induced phase transitions. The interplay between bidirectional movement, extended coupling conditions, and Langmuir kinetics produces unusual phenomena, including a back-and-forth phase transition and partial phase division of the shock region for comparatively smaller values of the lane-changing rate. Moreover, we analyze the shock dynamics and calculate critical values for the lane-changing rate at which the phases appear or disappear.

Exclusion process with scaled resources: Delocalized shocks and interplay of reservoirs

In this paper we study a conserved system comprised of two directed lanes having identical dynamics and two reservoirs with scaled resources that are strategically connected to the boundaries of the lanes, forming a ringlike structure. The steady-state properties of the system have been analyzed in the framework of mean-field theory. Our findings display a rich behavior, emphasizing the nontrivial effects of incorporating two reservoirs. As a consequence, two distinct phases that admit delocalized shocks emerge and occupy a significant region in the phase diagram. Moreover in one of theses phases, each lane admits a delocalized shock whose movements are perfectly synchronized. In another phase, the single shock in the system may traverse both lanes or remain restricted to a single lane, depending upon the size of the system. All the findings are validated by Monte Carlo simulations.

Large-scale mRNA translation and the intricate effects of competition for the finite pool of ribosomes

We present a new theoretical framework for large-scale mRNA translation using a network of models called the ribosome flow model with Langmuir kinetics (RFMLK), interconnected via a pool of free ribosomes. The input to each RFMLK depends on the pool density, and it affects the initiation rate and potentially also the internal ribosome entry rates along each RFMLK. Ribosomes that detach from an RFMLK owing to termination or premature drop-off are fed back into the pool. We prove that the network always converges to a steady state, and study its sensitivity to variations in the parameters. For example, we show that if the drop-off rate at some site in some RFMLK is increased then the pool density increases and consequently the steady-state production rate in all the other RFMLKs increases. Surprisingly, we also show that modifying a parameter of a certain RFMLK can lead to arbitrary effects on the densities along the modified RFMLK, depending on the parameters in the entire network. We conclude that the competition for shared resources generates an indirect and intricate web of mutual effects between the mRNA molecules that must be accounted for in any analysis of translation.

Two-lane totally asymmetric simple exclusion process with extended Langmuir kinetics

Multilane totally asymmetric simple exclusion processes with interactions between the lanes have recently been investigated actively. This paper proposes a two-lane model with extended Langmuir kinetics on a periodic lattice. Both bidirectional and unidirectional flows are investigated. In our model, the hopping, attachment, and detachment rates vary depending on the state of the corresponding site in the other lane. We obtain a theoretical expression for the global density of the system in the steady state from three kinds of mean-field analyses [(1×1)-, (2×1)-, and (2×2)-cluster cases]. We verify that the (2×2)-cluster mean-field analysis reproduces the differences between the two directional flows and approximates well the results of computer simulations for some cases. We observe that (2×1)-cluster mean-field analyses are already good approximations of the simulation results for unidirectional flows; on the other hand, the accuracy of the approximations much improves by (2×2)-cluster one for bidirectional flows. We explain the phenomena in a qualitative manner by a simple analysis of correlations. We expect these findings to give informative suggestions for actual traffic systems.

Self-Organized Lane Formation in Bidirectional Transport by Molecular Motors

Within cells, vesicles and proteins are actively transported several micrometers along the cytoskeletal filaments. The transport along microtubules is propelled by dynein and kinesin motors, which carry the cargo in opposite directions. Bidirectional intracellular transport is performed with great efficiency, even under strong confinement, as for example in the axon. For this kind of transport system, one would expect generically cluster formation. In this Letter, we discuss the effect of the recently observed self-enhanced binding affinity along the kinesin trajectories on the microtubule. We introduce a stochastic lattice-gas model, where the enhanced binding affinity is realized via a floor field. From Monte Carlo simulations and a mean-field analysis we show that this mechanism can lead to self-organized symmetry breaking and lane formation that indeed leads to efficient bidirectional transport in narrow environments.

Crowd Control: Effects of Physical Crowding on Cargo Movement in Healthy and Diseased Neurons

High concentration of cytoskeletal filaments, organelles, and proteins along with the space constraints due to the axon's narrow geometry lead inevitably to intracellular physical crowding along the axon of a neuron. Local cargo movement is essential for maintaining steady cargo transport in the axon, and this may be impeded by physical crowding. Molecular motors that mediate active transport share movement mechanisms that allow them to bypass physical crowding present on microtubule tracks. Many neurodegenerative diseases, irrespective of how they are initiated, show increased physical crowding owing to the greater number of stalled organelles and structural changes associated with the cytoskeleton. Increased physical crowding may be a significant factor in slowing cargo transport to synapses, contributing to disease progression and culminating in the dying back of the neuronal process. This review explores the idea that physical crowding can impede cargo movement along the neuronal process. We examine the sources of physical crowding and strategies used by molecular motors that might enable cargo to circumvent physically crowded locations. Finally, we describe sub-cellular changes in neurodegenerative diseases that may alter physical crowding and discuss the implications of such changes on cargo movement.

Cooperative motor action to regulate microtubule length dynamics

Motivated by the recent experimental observations on motor induced cooperative mechanism controlling the length dynamics of microtubules (MTs), we examine how plus-end-targeted proteins of the kinesin family regulate MT polymerization and depolymerization routines. Here, we study a stochastic mathematical model capturing the unusual form of collective motor interaction on MT dynamics originating due to the molecular traffic near the MT tip. We provide an extensive analysis of the joint effect of motor impelled MT polymerization and complete depolymerization. The effect of the cooperative action is included by modifying the intrinsic depolymerization rate. We analyze the model within the framework of continuum mean-field theory and the resultant steady-state analytic solution is expressed in terms of Lambert W functions. Four distinct steady-state phases including a shock phase have been reported. The significant features of the shock including its position and height have been analyzed. Theoretical outcomes are supported by extensive Monte Carlo simulations. To explore the system alterations between the regime of growth and shrinkage phase, we consider kymographs of the microtubule along with the length distributions. Finally, we investigated the dependence of MT length kinetics both on modifying factor of depolymerization rate and motor concentration. The overall extensive study reveals that the flux of molecular traffic at the microtubule plus end initiates a cooperative mechanism, resulting in significant change in MT growth and shrinkage regime as also observed experimentally.

Global Density Profile For Particle Non-Conserving One Dimensional Transport From Renormalization Group Flows

The totally asymmetric simple exclusion process along with particle adsorption and evaporation kinetics is a model of boundary-induced nonequilibrium phase transition. In the continuum limit, the average particle density across the system is described by a singular differential equation involving multiple scales which lead to the formation of boundary layers (BL) or shocks. A renormalization group analysis is developed here by using the location and the width of the BL as the renormalization parameters. It not only allows us to cure the large distance divergences in the perturbative solution for the BL but also generates, from the BL solution, an analytical form for the global density profile. The predicted scaling form is checked against numerical solutions for finite systems.

Asymmetric simple exclusion process with position-dependent hopping rates: Phase diagram from boundary-layer analysis

In this paper, we study a one-dimensional totally asymmetric simple exclusion process with position-dependent hopping rates. Under open boundary conditions, this system exhibits boundary-induced phase transitions in the steady state. Similarly to totally asymmetric simple exclusion processes with uniform hopping, the phase diagram consists of low-density, high-density, and maximal-current phases. In various phases, the shape of the average particle density profile across the lattice including its boundary-layer parts changes significantly. Using the tools of boundary-layer analysis, we obtain explicit solutions for the density profile in different phases. A detailed analysis of these solutions under different boundary conditions helps us obtain the equations for various phase boundaries. Next, we show how the shape of the entire density profile including the location of the boundary layers can be predicted from the fixed points of the differential equation describing the boundary layers. We discuss this in detail through several examples of density profiles in various phases. The maximal-current phase appears to be an especially interesting phase where the boundary layer flows to a bifurcation point on the fixed-point diagram.

Relaxation dynamics of closed diffusive systems with infinitesimal Langmuir kinetics

We consider the asymmetric simple exclusion process with Langmuir kinetics in the closed boundary condition. We analytically obtain the exact stationary state and a series of excited states of the system in the limit where Langmuir kinetics is infinitesimally small. Based on this result, we propose an analytical formula for the time evolution of physical quantities of the system.

Far-from-equilibrium bidirectional transport system with constrained entrances competing for pool of limited resources

Motivated by the wide occurrence of limited resources in many real-life systems, we investigate two-lane totally asymmetric simple exclusion process with constrained entrances under finite supply of particles. We analyze the system within the framework of mean-field theory and examine various complex phenomena, including phase separation, phase transition, and symmetry breaking. Based on the theoretical analysis, we analytically derive the phase boundaries for various symmetric as well as asymmetric phases. It has been observed that the symmetry-breaking phenomenon initiates even for very small number of particles in the system. The phases with broken symmetry originates as shock-low density phase under limited resources, which is in contrast to the scenario with infinite number of particles. As expected, the symmetry breaking continues to persist even for higher values of system particles. Seven stationary phases are observed, with three of them exhibiting symmetry-breaking phenomena. The critical values of a total number of system particles, beyond which various symmetrical and asymmetrical phases appear and disappear are identified. Theoretical outcomes are supported by extensive Monte Carlo simulations. Finally, the size-scaling effect and symmetry-breaking phenomenon on the simulation results have also been examined based on particle density histograms.

A deterministic mathematical model for bidirectional excluded flow with Langmuir kinetics

In many important cellular processes, including mRNA translation, gene transcription, phosphotransfer, and intracellular transport, biological "particles" move along some kind of "tracks". The motion of these particles can be modeled as a one-dimensional movement along an ordered sequence of sites. The biological particles (e.g., ribosomes or RNAPs) have volume and cannot surpass one another. In some cases, there is a preferred direction of movement along the track, but in general the movement may be bidirectional, and furthermore the particles may attach or detach from various regions along the tracks. We derive a new deterministic mathematical model for such transport phenomena that may be interpreted as a dynamic mean-field approximation of an important model from mechanical statistics called the asymmetric simple exclusion process (ASEP) with Langmuir kinetics. Using tools from the theory of monotone dynamical systems and contraction theory we show that the model admits a unique steady-state, and that every solution converges to this steady-state. Furthermore, we show that the model entrains (or phase locks) to periodic excitations in any of its forward, backward, attachment, or detachment rates. We demonstrate an application of this phenomenological transport model for analyzing ribosome drop off in mRNA translation.

Renormalization group analysis for an asymmetric simple exclusion process

A perturbative renormalization group method is used to obtain steady-state density profiles of a totally asymmetric simple exclusion process with particle adsorption and evaporation. This method allows us to obtain a globally valid solution for the density profile without the asymptotic matching of bulk and boundary layer solutions. In addition, we show a nontrivial scaling of the boundary layer width with the system size close to specific phase boundaries.

Exclusion and Hierarchy of Time Scales Lead to Spatial Segregation of Molecular Motors in Cellular Protrusions

Molecular motors that carry cargo along biopolymer filaments within cells play a crucial role in the functioning of the cell. In particular, these motors are essential for the formation and maintenance of the cellular protrusions that play key roles in motility and specific functionalities, such as the stereocilia in hair cells. Typically, there are several species of motors, carrying different cargos, that share the same track. Furthermore, it was observed that in the mature stereocilia, the different motors occupy well-segregated bands as a function of distance from the tip. We use a totally asymmetric exclusion process model with two- and three-motor species, to study the conditions that give rise to such spatial patterns. We find that the well-segregated bands appear for motors with a strong hierarchy of attachment or detachment rates. This is a striking example of pattern formation in nonequilibrium, low-dimensional systems.

Assembly of bipolar microtubule structures by passive cross-linkers and molecular motors

During cell division, sister chromatids are segregated by the mitotic spindle, a bipolar assembly of interdigitating antiparallel polar filaments called microtubules. The spindle contains the midzone, a stable region of overlapping antiparallel microtubules, that is essential for maintaining bipolarity. Although a lot is known about the molecular players involved, the mechanism underlying midzone formation and maintenance is still poorly understood. We study the interaction of polar filaments that are cross-linked by molecular motors moving directionally and by passive cross-linkers diffusing along microtubules. Using a particle-based stochastic model, we find that the interplay of motors and passive cross-linkers can generate a stable finite overlap between a pair of antiparallel polar filaments. We develop a mean-field theory to study this mechanism in detail and investigate the influence of steric interactions between motors and passive cross-linkers on the overlap dynamics. In the presence of interspecies steric interactions, passive cross-linkers mimic the behavior of molecular motors and stable finite overlaps are generated even for non-cross-linking motors. Finally, we develop a mean-field theory for a bundle of aligned polar filaments and show that they can self-organize into a spindlelike pattern. Our work suggests possible ways as to how cells can generate spindle midzones and control their extensions.

Relaxation dynamics of the asymmetric simple exclusion process with Langmuir kinetics on a ring

We consider the asymmetric simple exclusion process with Langmuir kinetics on a periodic lattice. We analytically obtain the exact time evolution of correlation functions with arbitrary length starting from the initial state with no particle in the system. The exact stationary state of this model has been known for the totally asymmetric case. We propose a basis transformation which simplifies the proof of the stationarity of this state and enables the generalization to the partially asymmetric case. Moreover, we construct low-energy excitations and obtain the exact relaxation time.

On-site residence time in a driven diffusive system: Violation and recovery of a mean-field description

We investigate simple one-dimensional driven diffusive systems with open boundaries. We are interested in the average on-site residence time defined as the time a particle spends on a given site before moving on to the next site. Using mean-field theory, we obtain an analytical expression for the on-site residence times. By comparing the analytic predictions with numerics, we demonstrate that the mean-field significantly underestimates the residence time due to the neglect of time correlations in the local density of particles. The temporal correlations are particularly long-lived near the average shock position, where the density changes abruptly from low to high. By using domain wall theory, we obtain highly accurate estimates of the residence time for different boundary conditions. We apply our analytical approach to residence times in a totally asymmetric exclusion process (TASEP), TASEP coupled to Langmuir kinetics (TASEP+LK), and TASEP coupled to mutually interactive LK (TASEP+MILK). The high accuracy of our predictions is verified by comparing these with detailed Monte Carlo simulations.

Motor protein traffic regulation by supply-demand balance of resources

In cells and in in vitro assays the number of motor proteins involved in biological transport processes is far from being unlimited. The cytoskeletal binding sites are in contact with the same finite reservoir of motors (either the cytosol or the flow chamber) and hence compete for recruiting the available motors, potentially depleting the reservoir and affecting cytoskeletal transport. In this work we provide a theoretical framework in which to study, analytically and numerically, how motor density profiles and crowding along cytoskeletal filaments depend on the competition of motors for their binding sites. We propose two models in which finite processive motor proteins actively advance along cytoskeletal filaments and are continuously exchanged with the motor pool. We first look at homogeneous reservoirs and then examine the effects of free motor diffusion in the surrounding medium. We consider as a reference situation recent in vitro experimental setups of kinesin-8 motors binding and moving along microtubule filaments in a flow chamber. We investigate how the crowding of linear motor proteins moving on a filament can be regulated by the balance between supply (concentration of motor proteins in the flow chamber) and demand (total number of polymerized tubulin heterodimers). We present analytical results for the density profiles of bound motors and the reservoir depletion, and propose novel phase diagrams that present the formation of jams of motor proteins on the filament as a function of two tuneable experimental parameters: the motor protein concentration and the concentration of tubulins polymerized into cytoskeletal filaments. Extensive numerical simulations corroborate the analytical results for parameters in the experimental range and also address the effects of diffusion of motor proteins in the reservoir. We then propose experiments for validating our models and discuss how the 'supply-demand' effects can regulate motor traffic also in in vivo conditions.

Effect of coupling strength on a two-lane partially asymmetric coupled totally asymmetric simple exclusion process with Langmuir kinetics

We analyze an open system comprised of two parallel totally asymmetric simple exclusion processes with particle attachment and detachment in the bulk under partially asymmetric coupling conditions. The phase diagrams are obtained using boundary layer analysis of continuum mean-field equations and characterized for different values of lane-changing rates. The structure of the phase diagram remains qualitatively the same as the one in fully asymmetric coupling conditions up to a certain critical order of lane-changing rates, after which significant changes are found in the phase diagram. The effect of system size on the steady-state dynamics has also been examined. To validate theoretical findings, extensive Monte Carlo simulations are carried out.

Segregation of diffusible and directionally moving particles on a polar filament

Directed transport in living cells relies on the action of motor proteins. These enzymes can transform chemical energy into mechanical work and move directionally along filamentous tracks. At the same time, these filaments serve as a substrate for the binding of proteins performing other functions, but that also obstruct the motors' motion. Motivated by the mobile cross-linker Ase1, we theoretically study a system of molecular motors in the presence of diffusible particles. Both the motors and the obstacles shuttle between the filament and its surrounding. Numerical simulations of this system show a segregation between motors and obstacles if the filament ends act as diffusion barriers for the obstacles. A phenomenological mean-field theory captures the essential effects observed in the simulations.

Asymmetric coupling in two-lane simple exclusion processes with Langmuir kinetics: phase diagrams and boundary layers

We use boundary layer analysis for an open system consisting of two parallel totally asymmetric simple exclusion processes with Langmuir kinetics under a biased lane-changing rule. The two kinds of phase transitions--bulk transitions and surface transitions--have been examined. The dynamics of shock and their dependence on the system parameters have been investigated. We find a reduction in the number of steady-state phases with increase in lane-changing rate.

Asymmetric exclusion process with global hopping

We study a one-dimensional totally asymmetric simple exclusion process with one special site from which particles fly to any empty site (not just to the neighboring site). The system attains a nontrivial stationary state with a density profile varying over the spatial extent of the system. The density profile undergoes a nonequilibrium phase transition when the average density passes through the critical value 1-[4(1-ln2)](-1)=0.185277..., viz., in addition to the discontinuity in the vicinity of the special site, a shock wave is formed in the bulk of the system when the density exceeds the critical density.

TEMPLATE-DIRECTED BIOPOLYMERIZATION: TAPE-COPYING TURING MACHINES

DNA, RNA and proteins are among the most important macromolecules in a living cell. These molecules are polymerized by molecular machines. These natural nano-machines polymerize such macromolecules, adding one monomer at a time, using another linear polymer as the corresponding template. The machine utilizes input chemical energy to move along the template which also serves as a track for the movements of the machine. In the Alan Turing year 2012, it is worth pointing out that these machines are "tape-copying Turing machines". We review the operational mechanisms of the polymerizer machines and their collective behavior from the perspective of statistical physics, emphasizing their common features in spite of the crucial differences in their biological functions. We also draw the attention of the physics community to another class of modular machines that carry out a different type of template-directed polymerization. We hope this review will inspire new kinetic models for these modular machines.

Impact of motor molecules on the dynamics of treadmilling filaments

We study a driven lattice gas model for the length dynamics of treadmilling filaments in the presence of molecular motors. A treadmilling filament grows by subunit addition at one end and shrinks by subunit removal at the other. Molecular motors can attach to the filament, move towards the shrinking end, and detach from the filament. We consider motors that are also capable of inducing subunit removal at the shrinking filament end. Stochastic simulations reveal a phase of unimodal length distribution and a phase of unbounded growth. Exploiting a condition on the motor flux, we explore the system's phase diagram. In certain limits we can define random walks that allow us to estimate the full length distribution. The width of steady state distributions decreases with increasing motor activity. Our analysis indicates possible ways that cells can use to regulate the size of cytoskeletal structures such as mitotic spindles by controlling various motor properties.

Microtubule length dependence of motor traffic in cells

Motor proteins in living cells, such as kinesin and dynein, can move processively along the microtubule (MT), and can also detach from or attach to MT stochastically. Experiments found that the traffic of motors along MT may be jammed; thus various theoretical models were designed to understand this process. However, previous studies mainly focused on motor attachment/detachment rate dependent properties. Leduc et al. recently found that the traffic jam of motor protein Kip3 depends on MT length (Proc. Natl. Acad. Sci. U.S.A. 109, 6100 (2012)). Therefore, this study discusses the MT length-dependent properties of motor traffic. The results showed that MT length has one critical value N(c); a traffic jam occurs only when MT length N > N(c). The jammed MT length increases with total MT length N , whereas the non-jammed MT length may not change monotonically with N . The critical value N(c) increases with motor detachment rate from MT, but decreases with motor attachment rate to MT. Therefore, the traffic of motors will be more likely to be jammed when the MT is long, motor detachment rate is high, and motor detachment rate is low.

Length regulation of active biopolymers by molecular motors

For biopolymers like cytoskeletal actin filaments and microtubules, assembly and disassembly are inherently dissipative processes. Molecular motors can affect the rates of subunit removal at filament ends. We introduce a driven lattice-gas model to study the effects of motor-induced depolymerization on the length of active biopolymers and find that increasing motor activity sharpens unimodal steady-state length distributions. Furthermore, for sufficiently fast moving motors, the relative width of the length distribution is determined only by the attachment rate of motors. Our results show how established molecular processes can be used to robustly regulate the size of cytoskeletal structures like mitotic spindles.

Phase diagram and density large deviations of a nonconserving ABC model

The effect of particle-nonconserving processes on the steady state of driven diffusive systems is studied within the context of a generalized ABC model. It is shown that in the limit of slow nonconserving processes, the large deviation function of the overall particle density can be computed by making use of the steady-state density profile of the conserving model. In this limit one can define a chemical potential and identify first order transitions via Maxwell's construction, similarly to what is done in equilibrium systems. This method may be applied to other driven models subjected to slow nonconserving dynamics.

One-dimensional deterministic transport in neurons measured by dispersion-relation phase spectroscopy

We studied the active transport of intracellular components along neuron processes using a new method developed in our laboratory: dispersion-relation phase spectroscopy. This method is able to quantitatively map spatially the heterogeneous dynamics of the concentration field of the cargos at submicron resolution without the need for tracking individual components. The results in terms of density correlation function reveal that the decay rate is linear in wavenumber, which is consistent with a narrow Lorentzian distribution of cargo velocity.

Multishocks in driven diffusive processes: insights from fixed-point analysis of the boundary layers

Boundary-induced phase transitions in a driven diffusive process can be studied through a phase-plane analysis of the boundary-layer equations. In this paper, we generalize this approach further to show how various shapes including multishocks and downward shocks in the bulk particle density profile can be understood by studying the dependence of the fixed points of the boundary-layer equation on an appropriate parameter. This is done for a particular driven interacting particle system as a prototypical example. The present analysis shows the special role of a specific bifurcation of the fixed points in giving rise to different kinds of shocks.

Active transport and cluster formation on 2D networks

We introduce a model for active transport on inhomogeneous networks embedded in a diffusive environment which is motivated by vesicular transport on actin filaments. In the presence of a hard-core interaction, particle clusters are observed that exhibit an algebraically decaying distribution in a large parameter regime, indicating the existence of clusters on all scales. The scale-free behavior can be understood by a mechanism promoting preferential attachment of particles to large clusters. The results are compared with a diffusion-limited aggregation model and active transport on a regular network. For both models we observe aggregation of particles to clusters which are characterized by a finite size scale if the relevant time scales and particle densities are considered.

Dynamical phase transition of a one-dimensional transport process including death

Motivated by biological aspects related to fungus growth, we consider the competition of growth and corrosion. We study a modification of the totally asymmetric exclusion process, including the probabilities of injection alpha and death of the last particle delta . The system presents a phase transition at deltac(alpha), where the average position of the last particle L grows as sqrt[t]. For delta>deltac, a nonequilibrium stationary state exists while for delta<deltac the asymptotic state presents a low density and max current phases. We discuss the scaling of the density and current profiles for parallel and sequential updates.

The origins of time-delay in template biopolymerization processes

Time-delays are common in many physical and biological systems and they give rise to complex dynamic phenomena. The elementary processes involved in template biopolymerization, such as mRNA and protein synthesis, introduce significant time delays. However, there is not currently a systematic mapping between the individual mechanistic parameters and the time delays in these networks. We present here the development of mathematical, time-delay models for protein translation, based on PDE models, which in turn are derived through systematic approximations of first-principles mechanistic models. Theoretical analysis suggests that the key features that determine the time-delays and the agreement between the time-delay and the mechanistic models are ribosome density and distribution, i.e., the number of ribosomes on the mRNA chain relative to their maximum and their distribution along the mRNA chain. Based on analytical considerations and on computational studies, we show that the steady-state and dynamic responses of the time-delay models are in excellent agreement with the detailed mechanistic models, under physiological conditions that correspond to uniform ribosome distribution and for ribosome density up to 70%. The methodology presented here can be used for the development of reduced time-delay models of mRNA synthesis and large genetic networks. The good agreement between the time-delay and the mechanistic models will allow us to use the reduced model and advanced computational methods from nonlinear dynamics in order to perform studies that are not practical using the large-scale mechanistic models.

Traffic by multiple species of molecular motors

We study the traffic of two types of molecular motors using the two-species asymmetric simple exclusion process (ASEP) with periodic boundary conditions and with attachment and detachment of particles. We determine characteristic properties such as motor densities and currents by simulations and analytical calculations. For motors with different unbinding probabilities, mean-field theory gives the correct bound density and total current of the motors, as shown by numerical simulations. For motors differing in their stepping probabilities, the particle-hole symmetry of the current-density relationship is broken and mean-field theory fails drastically. The total motor current exhibits exponential finite-size scaling, which we use to extrapolate the total current to the thermodynamic limit. Finally, we also study the motion of a single motor in the background of many nonmoving motors.

Competition between multiple totally asymmetric simple exclusion processes for a finite pool of resources

Using Monte Carlo simulations and a domain-wall theory, we discuss the effect of coupling several totally asymmetric simple exclusion processes (TASEPs) to a finite reservoir of particles. This simple model mimics directed biological transport processes in the presence of finite resources such as protein synthesis limited by a finite pool of ribosomes. If all TASEPs have equal length, we find behavior which is analogous to a single TASEP coupled to a finite pool. For the more generic case of chains with different lengths, several unanticipated regimes emerge. A generalized domain-wall theory captures our findings in good agreement with simulation results.

Obstacles on the microtubule reduce the processivity of Kinesin-1 in a minimal in vitro system and in cell extract

Inside cells, a multitude of molecular motors and other microtubule-associated proteins are expected to compete for binding to a limited number of binding sites available on microtubules. Little is known about how competition for binding sites affects the processivity of molecular motors and, therefore, cargo transport, organelle positioning, and microtubule organization, processes that all depend on the activity of more or less processive motors. Very few studies have been performed in the past to address this question directly. Most studies reported only minor effects of crowding on the velocity of motors. However, a controversy appears to exist regarding the effect of crowding on motor processivity. Here, we use single-molecule imaging of mGFP-labeled minimal dimeric kinesin-1 constructs in vitro to study the effects of competition on kinesin's processivity. For competitors, we use kinesin rigor mutants as static roadblocks, minimal wild-type kinesins as motile obstacles, and a cell extract as a complex mixture of microtubule-associated proteins. We find that mGFP-labeled kinesin-1 detaches prematurely from microtubules when it encounters obstacles, leading to a strong reduction of its processivity, a behavior that is largely independent of the type of obstacle used here. Kinesin has a low probability to wait briefly when encountering roadblocks. Our data suggest, furthermore, that kinesin can occasionally pass obstacles on the protofilament track.

Disordered driven lattice gases with boundary reservoirs and Langmuir kinetics

The asymmetric simple exclusion process with additional Langmuir kinetics, i.e., attachment and detachment in the bulk, is a paradigmatic model for intracellular transport. Here we study this model in the presence of randomly distributed inhomogeneities ("defects"). Using Monte Carlo simulations, we find a multitude of coexisting high- and low-density domains. The results are generic for one-dimensional driven diffusive systems with short-range interactions and can be understood in terms of a local extremal principle for the current profile. This principle is used to determine current profiles and phase diagrams as well as statistical properties of ensembles of defect samples.

Pathlines in exclusion processes

Trajectory data from observations of a random-walk process are often used to characterize macroscopic transport coefficients and to make inferences about motility mechanisms. Continuum equations describing the average moments of the position of an agent in an exclusion process are derived and validated with simulation data. Unlike standard noninteracting random walks, the moment equations for the exclusion process explicitly represent the interaction of agents since they depend on the averaged macroscopic agent density. Key issues associated with the validity of the continuum equations and interpretation of experimental data are discussed.

Effects of the chemomechanical stepping cycle on the traffic of molecular motors

We discuss effects of the stepping kinetics of molecular motors on their traffic behavior on crowded filaments using a simple two-state chemomechanical cycle. While the general traffic behavior is quite robust with respect to the detailed kinetics of the step, a few observable parameters exhibit a strong dependence on these parameters. Most strikingly, the effective unbinding rate of the motors may both increase and decrease with increasing traffic density, depending on the details of the motor step. Likewise the run length either exhibits a strong decrease or almost no dependence on the traffic density. We compare our theoretical results with recent experimental observations on motor traffic.

Domain wall delocalization, dynamics and fluctuations in an exclusion process with two internal states

We investigate the delocalization transition appearing in an exclusion process with two internal states, respectively on two parallel lanes. At the transition, delocalized domain walls form in the density profiles of both internal states, in agreement with a mean-field approach. Remarkably, the topology of the system's phase diagram allows for the delocalization of a (localized) domain wall when approaching the transition. We quantify the domain wall's delocalization close to the transition by analytic results obtained within the framework of the domain wall picture. Power law dependences of the domain wall width on the distance to the delocalization transition as well as on the system size are uncovered, they agree with numerical results.

Asymmetric simple exclusion process with periodic boundary driving

We consider the asymmetric simple exclusion process (ASEP) on a semi-infinite chain which is coupled at the end to a reservoir with a particle density that changes periodically in time. It is shown that the density profile assumes a time-periodic sawtoothlike shape. This shape does not depend on initial conditions and is found analytically in the hydrodynamic limit. In a finite system, the stationary state is shown to be governed by effective boundary densities and the extremal flux principle. Effective boundary densities are determined numerically via Monte Carlo simulations and compared with those given by mean-field approach and numerical integration of the hydrodynamic limit equation which is the Burgers equation. Our results extend straightforwardly beyond the ASEP to a wide class of driven diffusive systems with one conserved particle species.

Theoretical investigation of synchronous totally asymmetric exclusion processes on lattices with multiple-input-single-output junctions

In this paper, we investigate the dynamics of synchronous totally asymmetric exclusion processes on lattices with a multiple-input-single-output (MISO) junction, which consists of m subchains for the input and one main chain for the output. A MISO junction is a type of complex geometry that is relevant to many biological processes as well as vehicular and pedestrian traffic flow. A mean-field approach is developed to deal with the junction that connects the subchains and the main chain. Theoretical results for stationary particle currents, density profiles, and a phase diagram have been obtained. It is found that the phase boundary moves toward the left in the phase diagram with an increase of the number of subchains. The nonequilibrium stationary states, stationary-state phases, and phase boundaries are determined by the boundary conditions of the system as well as by the number of subchains. The density profiles obtained from computer simulations show very good agreement with our theoretical analysis.

Traffic of single-headed motor proteins KIF1A: effects of lane changing

KIF1A kinesins are single-headed motor proteins which move on cylindrical nanotubes called microtubules (MTs). A normal MT consists of 13 protofilaments on which the equispaced motor binding sites form a periodic array. The collective movement of the kinesins on a MT is, therefore, analogous to vehicular traffic on multilane highways where each protofilament is the analog of a single lane. Does lane changing increase or decrease the motor flux per lane? We address this fundamental question here by appropriately extending a recent model [P. Greulich, Phys. Rev. E 75, 041905 (2007)]. By carrying out analytical calculations and computer simulations of this extended model, we predict that the flux per lane can increase or decrease with the increasing rate of lane changing, depending on the concentrations of motors and the rate of hydrolysis of ATP, the "fuel" molecules. Our predictions can be tested, in principle, by carrying out in vitro experiments with fluorescently labeled KIF1A molecules.

Scaling properties of the asymmetric exclusion process with long-range hopping

The exclusion process in which particles may jump any distance l > or = 1 with the probability that decays as l;{-(1+sigma)} is studied from the coarse-grained equation for density profile in the limit when the lattice spacing goes to zero. For 1<sigma<2 , the usual diffusion term of this equation is replaced by the fractional one, which affects dynamical-scaling properties of the late-time approach to the stationary state. When applied to an open system with totally asymmetric hopping, this approach gives two results: First, it accounts for the sigma -dependent exponent that characterizes the algebraic decay of a density profile in the maximum-current phase for 1<sigma<2 , and second, it shows that in this region of sigma the exponent is of the mean-field type.

Weak and strong coupling in a two-lane asymmetric exclusion process

This paper studies a two-lane totally asymmetric simple exclusion process, in which particles could jump between the two lanes with asymmetric rates. In the weak coupling situation, the rates are inversely proportional to system size L . The appearance of localized shock in one lane and the discontinuous phase transition as revealed by Juhász [Phys. Rev. E 76, 021117 (2007)] are also reproduced. The density profiles and phase diagrams are constructed in the hydrodynamic limit, by numerically solving the steady state equations. The phase diagram in our model exhibits asymmetry, which is different from the symmetric one in Juhász's model. We have studied the phase boundary and discontinuous line analytically. The analytical results are in good agreement with that obtained from numerical integration. We also study the strong coupling situation, in which the lane changing rates are independent of L . Results completely different from that arising from weak coupling are presented. Furthermore, features different from that of the model presented by Pronina and Kolomeisky [Physica A 372, 12 (2006)] are revealed.

Interacting RNA polymerase motors on a DNA track: effects of traffic congestion and intrinsic noise on RNA synthesis

RNA polymerase (RNAP) is an enzyme that synthesizes a messenger RNA (mRNA) strand which is complementary to a single-stranded DNA template. From the perspective of physicists, an RNAP is a molecular motor that utilizes chemical energy input to move along the track formed by DNA. In many circumstances, which are described in this paper, a large number of RNAPs move simultaneously along the same track; we refer to such collective movements of the RNAPs as RNAP traffic. Here we develop a theoretical model for RNAP traffic by incorporating the steric interactions between RNAPs as well as the mechanochemical cycle of individual RNAPs during the elongation of the mRNA. By a combination of analytical and numerical techniques, we calculate the rates of mRNA synthesis and the average density profile of the RNAPs on the DNA track. We also introduce, and compute, two different measures of fluctuations in the synthesis of RNA. Analyzing these fluctuations, we show how the level of intrinsic noise in mRNA synthesis depends on the concentrations of the RNAPs as well as on those of some of the reactants and the products of the enzymatic reactions catalyzed by RNAP. We suggest appropriate experimental systems and techniques for testing our theoretical predictions.

Dynamic boundaries in asymmetric exclusion processes

We investigate the dynamics of a one-dimensional asymmetric exclusion process with Langmuir kinetics and a fluctuating wall. At the left-hand boundary, particles are injected onto the lattice; from there, the particles hop to the right. Along the lattice, particles can adsorb or desorb, and the right-hand boundary is defined by a wall particle. The confining wall particle has intrinsic forward and backward hopping, a net leftward drift, and cannot desorb. Performing Monte Carlo simulations and using a moving-frame finite segment approach coupled to mean field theory, we find the parameter regimes in which the wall acquires a steady-state position. In other regimes, the wall will either drift to the left and fall off the lattice at the injection site, or drift indefinitely to the right. Our results are discussed in the context of nonequilibrium phases of the system, fluctuating boundary layers, and particle densities in the laboratory frame versus the frame of the fluctuating wall.

Weakly coupled, antiparallel, totally asymmetric simple exclusion processes

We study a system composed of two parallel totally asymmetric simple exclusion processes with open boundaries, where the particles move in the two lanes in opposite directions and are allowed to jump to the other lane with rates inversely proportional to the length of the system. Stationary density profiles are determined and the phase diagram of the model is constructed in the hydrodynamic limit, by solving the differential equations describing the steady state of the system, analytically for vanishing total current and numerically for nonzero total current. The system possesses phases with a localized shock in the density profile in one of the lanes, similarly to exclusion processes endowed with nonconserving kinetics in the bulk. Besides, the system undergoes a discontinuous phase transition, where coherently moving delocalized shocks emerge in both lanes and the fluctuation of the global density is described by an unbiased random walk. This phenomenon is analogous to the phase coexistence observed at the coexistence line of the totally asymmetric simple exclusion process, however, as a consequence of the interaction between lanes, the density profiles are deformed and in the case of asymmetric lane change, the motion of the shocks is confined to a limited domain.