Totally asymmetric exclusion process with extended objects: a model for protein synthesis

Determinants of translation elongation speed and ribosomal profiling biases in mouse embryonic stem cells

Ribosomal profiling is a promising approach with increasing popularity for studying translation. This approach enables monitoring the ribosomal density along genes at a resolution of single nucleotides.

In this study, we focused on ribosomal density profiles of mouse embryonic stem cells. Our analysis suggests, for the first time, that even in mammals such as M. musculus the elongation speed is significantly and directly affected by determinants of the coding sequence such as: 1) the adaptation of codons to the tRNA pool; 2) the local mRNA folding of the coding sequence; 3) the local charge of amino acids encoded in the codon sequence. In addition, our analyses suggest that in general, the translation velocity of ribosomes is slower at the beginning of the coding sequence and tends to increase downstream.

Finally, a comparison of these data to the expected biophysical behavior of translation suggests that it suffers from some unknown biases. Specifically, the ribosomal flux measured on the experimental data increases along the coding sequence; however, according to any biophysical model of ribosomal movement lacking internal initiation sites, the flux is expected to remain constant or decrease. Thus, developing experimental and/or statistical methods for understanding, detecting and dealing with such biases is of high importance.

Gene translation is the process by which ribosomes translate mRNA molecules to proteins, a central process in all living organisms. Thus, understanding the biophysics of gene translation and the way its efficiency is encoded in the different features of the coding sequence has ramifications to every biomedical discipline. Recently, a new large-scale experimental approach named ‘ribosomal profiling’, has been developed for monitoring the ribosomal density at a resolution of single nucleotides. In this study, we analyzed ribosomal profiling data of mouse embryonic stem cells. These data enabled us to directly show that translation velocity is affected by the adaptation of codons to the tRNA pool, local mRNA folding of coding sequence, and local charge of the amino acids encoded in the coding sequence. In addition, our analyses suggest that ribosomal speed tends to be slower at the beginning of the coding sequence. Finally, we report possible biases in the ‘ribosomal profiling’ procedure that should be considered in future studies utilizing this method.

On the steady-state distribution in the homogeneous ribosome flow model

A central biological process in all living organisms is gene translation. Developing a deeper understanding of this complex process may have ramifications to almost every biomedical discipline. Reuveni et al. recently proposed a new computational model of gene translation called the Ribosome Flow Model (RFM). In this paper, we consider a particular case of this model, called the Homogeneous Ribosome Flow Model (HRFM). From a biological viewpoint, this corresponds to the case where the transition rates of all the coding sequence codons are identical. This regime has been suggested recently based on experiments in mouse embryonic cells. We consider the steady-state distribution of the HRFM. We provide formulas that relate the different parameters of the model in steady state. We prove the following properties: 1) the ribosomal density profile is monotonically decreasing along the coding sequence; 2) the ribosomal density at each codon monotonically increases with the initiation rate; and 3) for a constant initiation rate, the translation rate monotonically decreases with the length of the coding sequence. In addition, we analyze the translation rate of the HRFM at the limit of very high and very low initiation rate, and provide explicit formulas for the translation rate in these two cases. We discuss the relationship between these theoretical results and biological findings on the translation process.

Entrainment and unit velocity: surprises in an accelerated exclusion process

We introduce a class of distance-dependent interactions in an accelerated exclusion process inspired by the observation of transcribing RNA polymerase speeding up when "pushed" by a trailing one. On a ring, the accelerated exclusion process steady state displays a discontinuous transition, from being homogeneous (with augmented currents) to phase segregated. In the latter state, the holes appear loosely bound and move together, much like a train. Surprisingly, the current-density relation is simply J=1-ρ, signifying that the "hole train" travels with unit velocity.

Stability analysis of the ribosome flow model

Gene translation is a central process in all living organisms. Developing a better understanding of this complex process may have ramifications to almost every biomedical discipline. Recently, Reuveni et al. proposed a new computational model of this process called the ribosome flow model (RFM). In this study, we show that the dynamical behavior of the RFM is relatively simple. There exists a unique equilibrium point e and every trajectory converges to e. Furthermore, convergence is monotone in the sense that the distance to e can never increase. This qualitative behavior is maintained for any feasible set of parameter values, suggesting that the RFM is highly robust. Our analysis is based on a contraction principle and the theory of monotone dynamical systems. These analysis tools may prove useful in studying other properties of the RFM as well as additional intracellular biological processes.

A max-plus model of ribosome dynamics during mRNA translation

We examine the dynamics of the translation stage of cellular protein production, in which ribosomes move uni-directionally along an mRNA strand, building amino acid chains as they go. We describe the system using a timed event graph-a class of Petri net useful for studying discrete events, which have to satisfy constraints. We use max-plus algebra to describe a deterministic version of the model, where the constraints represent steric effects which prevent more than one ribosome reading a given codon at a given time and delays associated with the availability of the different tRNAs. We calculate the protein production rate and density of ribosomes on the mRNA and find exact agreement between these analytical results and numerical simulations of the deterministic model, even in the case of heterogeneous mRNAs.

Efficient manipulations of synonymous mutations for controlling translation rate: an analytical approach

Gene translation is a central process in all living organism with important ramifications to almost every biomedical field. Previous systems evolutionary studies in the field have demonstrated that in many organisms coding sequence features undergo selection to optimize this process. In the current study, we report for the first time analytical proofs related to the various aspects of this process and its optimality. Among our results we show that coding sequences with mono- tonic increasing profiles of translation efficiency (i.e., with slower codons near the 5'UTR), mathematically optimize ribosomal allocation by minimizing the number of ribosomes needed for translating a codon per time unit. Thus, the genomic translation efficiency profile reported in previous studies for many organisms is optimal in this sense. In addition, we show that improving translation efficiency of a codon in a gene may result in a decrease in the translation rate of other genes, demonstrating that the relation between codon bias and protein translation rate is less trivial than was assumed before. Based on these observations we describe an efficient heuristic for designing coding sequences with specific translation efficiency and minimal ribosomal allocation for heterologous gene expression. We demonstrate how this heuristic can be used in biotechnology for engineering a heterologous gene before expressing it in a new host.

Weakening interaction suppresses spontaneous symmetry breaking in two-channel asymmetric exclusion processes

This paper has studied spontaneous symmetry breaking (SSB) phenomenon in two types of two-channel asymmetric simple exclusion processes (ASEPs). One common feature of the two systems is that interactions for each species of particle happen at only one site, and the system reduces to two independent ASEPs when interaction vanishes. It is shown that with the weakening of interaction, the SSB is suppressed. More interestingly, the SSB disappears before the interaction is eliminated. Our work thus indicates that local interaction has to be strong enough to produce SSB. The mean-field analysis has been carried out, and the results are consistent with the simulation ones.

Statistics of current-activity fluctuations in asymmetric flow with exclusion

We consider steady-state current activity statistics for the one-dimensional totally asymmetric simple exclusion process. With the help of the known operator algebra (for general open boundary conditions), as well as general probabilistic concepts (for the periodic case), we derive and evaluate closed-form expressions for the lowest three moments of the probability distribution function. These are confirmed, to excellent degree of accuracy, by numerical simulations. Further exact expressions and asymptotic approximations are provided for probability distributions and generating functions.

Mathematical and Computational Modelling of Ribosomal Movement and Protein Synthesis: an overview

Translation or protein synthesis consists of a complex system of chemical reactions, which ultimately result in decoding of the mRNA and the production of a protein. The complexity of this reaction system makes it difficult to quantitatively connect its input parameters (such as translation factor or ribosome concentrations, codon composition of the mRNA, or energy availability) to output parameters (such as protein synthesis rates or ribosome densities on mRNAs). Mathematical and computational models of translation have now been used for nearly five decades to investigate translation, and to shed light on the relationship between the different reactions in the system. This review gives an overview over the principal approaches used in the modelling efforts, and summarises some of the major findings that were made.

Mixed population of competing totally asymmetric simple exclusion processes with a shared reservoir of particles

We introduce a mean-field theoretical framework to describe multiple totally asymmetric simple exclusion processes (TASEPs) with different lattice lengths and entry and exit rates, competing for a finite reservoir of particles. We present relations for the partitioning of particles between the reservoir and the lattices: These relations allow us to show that competition for particles can have nontrivial effects on the phase behavior of individual lattices. For a system with nonidentical lattices, we find that when a subset of lattices undergoes a phase transition from low to high density, the entire set of lattice currents becomes independent of total particle number. We generalize our approach to systems with a continuous distribution of lattice parameters, for which we demonstrate that measurements of the current carried by a single lattice type can be used to extract the entire distribution of lattice parameters. Our approach applies to populations of TASEPs with any distribution of lattice parameters and could easily be extended beyond the mean-field case.

Positive congestion effect on a totally asymmetric simple exclusion process with an adsorption lane

We propose a type of asymmetric simple exclusion process by introducing an adsorption lane. In the system, each particle can be adsorbed by the adsorption lane only once per travel. The adsorption and desorption probabilities control the density of adsorption sites and the number of adsorbed particles. The ratio of the adsorbed particles shows reversal dynamics for congestion of the system, which is called the "positive congestion effect." We analyze this phenomenon by simulations and an approximation and successfully derive its critical condition. The spatial distribution of particles controlled by the adsorption and desorption probabilities is also investigated through balance equations of particles.

Simulations of driven and reconstituting lattice gases

We discuss stationary aspects of a set of driven lattice gases in which hard-core particles with spatial extent, covering more than one lattice site, diffuse and reconstruct in one dimension under nearest-neighbor interactions. As in the uncoupled case [M. Barma et al., J. Phys.: Condens. Matter 19, 065112 (2007)], the dynamics of the phase space breaks up into an exponentially large number of mutually disconnected sectors labeled by a nonlocal construct, the irreducible string. Depending on whether the particle couplings are taken attractive or repulsive, simulations in most of the studied sectors show that both steady state currents and pair correlations behave quite differently at low temperature regimes. For repulsive interactions an order-by-disorder transition is suggested.

The dynamics of supply and demand in mRNA translation

We study the elongation stage of mRNA translation in eukaryotes and find that, in contrast to the assumptions of previous models, both the supply and the demand for tRNA resources are important for determining elongation rates. We find that increasing the initiation rate of translation can lead to the depletion of some species of aa-tRNA, which in turn can lead to slow codons and queueing. Particularly striking “competition” effects are observed in simulations of multiple species of mRNA which are reliant on the same pool of tRNA resources. These simulations are based on a recent model of elongation which we use to study the translation of mRNA sequences from the Saccharomyces cerevisiae genome. This model includes the dynamics of the use and recharging of amino acid tRNA complexes, and we show via Monte Carlo simulation that this has a dramatic effect on the protein production behaviour of the system.

In this paper we show that the rate at which proteins are produced can be controlled at the elongation stage of mRNA translation. Regulation of translation initiation has been a focus of much study, but the subsequent effect of changes in the initiation rate on the overall translation rate, and the role of slow and fast codon usage in mRNA sequences is still not fully understood. We consider a model of elongation in which the dynamics of tRNA use and recharging are considered for real mRNA sequences. We find that the balance between the demand for, and supply of tRNAs is crucial in determining translation rates. Particularly interesting “competition” effects are observed when the simultaneous translation of multiple mRNA is considered. We show indeed that, via the choice of slow or fast codons, it is in principle possible to control how variation of the supply and demand for tRNA resources changes the rate of protein production from different mRNAs.

Driven k-mers: correlations in space and time

Steady-state properties of hard objects with exclusion interaction and a driven motion along a one-dimensional periodic lattice are investigated. The process is a generalization of the asymmetric simple exclusion process (ASEP) to particles of length k, and is called the k-ASEP. Here, we analyze both static and dynamic properties of the k-ASEP. Density correlations are found to display interesting features, such as pronounced oscillations in both space and time, as a consequence of the extended length of the particles. At long times, the density autocorrelation decays exponentially in time, except at a special k-dependent density when it decays as a power law. In the limit of large k at a finite density of occupied sites, the appropriately scaled system reduces to a nonequilibrium generalization of the Tonks gas describing the motion of hard rods along a continuous line. This allows us to obtain in a simple way the known two-particle distribution for the Tonks gas. For large but finite k, we also obtain the leading-order correction to the Tonks result.

Asymmetric inclusion process

We introduce and explore the asymmetric inclusion process (ASIP), an exactly solvable bosonic counterpart of the fermionic asymmetric exclusion process (ASEP). In both processes, random events cause particles to propagate unidirectionally along a one-dimensional lattice of n sites. In the ASEP, particles are subject to exclusion interactions, whereas in the ASIP, particles are subject to inclusion interactions that coalesce them into inseparable clusters. We study the dynamics of the ASIP, derive evolution equations for the mean and probability generating function (PGF) of the sites' occupancy vector, obtain explicit results for the above mean at steady state, and describe an iterative scheme for the computation of the PGF at steady state. We further obtain explicit results for the load distribution in steady state, with the load being the total number of particles present in all lattice sites. Finally, we address the problem of load optimization, and solve it under various criteria. The ASIP model establishes bridges between statistical physics and queueing theory as it represents a tandem array of queueing systems with (unlimited) batch service, and a tandem array of growth-collapse processes.

Genome-scale analysis of translation elongation with a ribosome flow model

We describe the first large scale analysis of gene translation that is based on a model that takes into account the physical and dynamical nature of this process. The Ribosomal Flow Model (RFM) predicts fundamental features of the translation process, including translation rates, protein abundance levels, ribosomal densities and the relation between all these variables, better than alternative ('non-physical') approaches. In addition, we show that the RFM can be used for accurate inference of various other quantities including genes' initiation rates and translation costs. These quantities could not be inferred by previous predictors. We find that increasing the number of available ribosomes (or equivalently the initiation rate) increases the genomic translation rate and the mean ribosome density only up to a certain point, beyond which both saturate. Strikingly, assuming that the translation system is tuned to work at the pre-saturation point maximizes the predictive power of the model with respect to experimental data. This result suggests that in all organisms that were analyzed (from bacteria to Human), the global initiation rate is optimized to attain the pre-saturation point. The fact that similar results were not observed for heterologous genes indicates that this feature is under selection. Remarkably, the gap between the performance of the RFM and alternative predictors is strikingly large in the case of heterologous genes, testifying to the model's promising biotechnological value in predicting the abundance of heterologous proteins before expressing them in the desired host.

Distribution of dwell times of a ribosome: effects of infidelity, kinetic proofreading and ribosome crowding

Ribosome is a molecular machine that polymerizes a protein where the sequence of the amino acid residues, the monomers of the protein, is dictated by the sequence of codons (triplets of nucleotides) on a messenger RNA (mRNA) that serves as the template. The ribosome is a molecular motor that utilizes the template mRNA strand also as the track. Thus, in each step the ribosome moves forward by one codon and, simultaneously, elongates the protein by one amino acid. We present a theoretical model that captures most of the main steps in the mechanochemical cycle of a ribosome. The stochastic movement of the ribosome consists of an alternating sequence of pause and translocation; the sum of the durations of a pause and the following translocation is the time of dwell of the ribosome at the corresponding codon. We derive the analytical expression for the distribution of the dwell times of a ribosome in our model. Wherever experimental data are available, our theoretical predictions are consistent with those results. We suggest appropriate experiments to test the new predictions of our model, particularly the effects of the quality control mechanism of the ribosome and that of their crowding on the mRNA track.

Facilitated asymmetric exclusion

We introduce a class of facilitated asymmetric exclusion processes in which particles are pushed by neighbors from behind. For the simplest version in which a particle can hop to its vacant right neighbor only if its left neighbor is occupied, we determine the steady-state current and the distribution of cluster sizes on a ring. We show that an initial density downstep develops into a rarefaction wave that can have a jump discontinuity at the leading edge, while an upstep results in a shock wave. This unexpected rarefaction wave discontinuity occurs generally for facilitated exclusion processes.

Slow sites in an exclusion process with limited resources

We introduce slow bottleneck sites into a recent extension of the totally asymmetric exclusion process where hopping rates are allowed to vary dynamically with the availability of resources. In the context of messenger RNA (mRNA) translation in biology, this refers to the availability of amino acid-transfer-RNA (aa-tRNA) complexes which act as the source of amino acids for protein production. We study a simple designer mRNA with a single defect codon in the center. As well as the familiar queuing behavior we also observe a regime within the queuing phase where the queue becomes less severe as the aa-tRNAs become depleted.

Limited resources in a driven diffusion process

The advance of particles in many driven diffusion systems depends on the availability of resources in the surrounding environment. In the balance between supply and demand of such resources we are confronted with a regime in which, under limited resource availability, the flow is markedly reduced. In the context of mRNA translation this represents the finite availability of amino acid-tRNA molecules. In this limited resources regime a severe depletion of amino acid tRNAs is also observed. These dramatic effects are vital to our understanding of translation, and are likely to also be important for the many other applications of driven diffusion models.

Role of the particle's stepping cycle in an asymmetric exclusion process: a model of mRNA translation

Messenger RNA translation is often studied by means of statistical-mechanical models based on the asymmetric simple exclusion process (ASEP), which considers hopping particles (the ribosomes) on a lattice (the polynucleotide chain). In this work we extend this class of models and consider the two fundamental steps of the ribosome's biochemical cycle following a coarse-grained perspective. In order to achieve a better understanding of the underlying biological processes and compare the theoretical predictions with experimental results, we provide a description lying between the minimal ASEP-like models and the more detailed models, which are analytically hard to treat. We use a mean-field approach to study the dynamics of particles associated with an internal stepping cycle. In this framework it is possible to characterize analytically different phases of the system (high density, low density or maximal current phase). Crucially, we show that the transitions between these different phases occur at different parameter values than the equivalent transitions in a standard ASEP, indicating the importance of including the two fundamental steps of the ribosome's biochemical cycle into the model.

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.

A mathematical modelling framework for elucidating the role of feedback control in translation termination

Translation is the final stage of gene expression where messenger RNA is used as a template for protein polymerization from appropriate amino acids. Release of the completed protein requires a release factor protein acting at the termination/stop codon to liberate it. In this paper we focus on a complex feedback control mechanism involved in the translation and synthesis of release factor proteins, which has been observed in different systems. These release factor proteins are involved in the termination stage of their own translation. Further, mutations in the release factor gene can result in a premature stop codon. In this case translation can result either in early termination and the production of a truncated protein or readthrough of the premature stop codon and production of the complete release factor protein. Thus during translation of the release factor mRNA containing a premature stop codon, the full length protein negatively regulates its production by its action on a premature stop codon, while positively regulating its production by its action on the regular stop codon. This paper develops a mathematical modelling framework to investigate this complex feedback control system involved in translation. A series of models is established to carefully investigate the role of individual mechanisms and how they work together. The steady state and dynamic behaviour of the resulting models are examined both analytically and numerically.

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.

Stochastic kinetics of ribosomes: single motor properties and collective behavior

Syntheses of protein molecules in a cell are carried out by ribosomes. A ribosome can be regarded as a molecular motor which utilizes the input chemical energy to move on a messenger RNA (mRNA) track that also serves as a template for the polymerization of the corresponding protein. The forward movement, however, is characterized by an alternating sequence of translocation and pause. Using a quantitative model, which captures the mechanochemical cycle of an individual ribosome, we derive an exact analytical expression for the distribution of its dwell times at the successive positions on the mRNA track. Inverse of the average dwell time satisfies a "Michaelis-Menten-type" equation and is consistent with the general formula for the average velocity of a molecular motor with an unbranched mechanochemical cycle. Extending this formula appropriately, we also derive the exact force-velocity relation for a ribosome. Often many ribosomes simultaneously move on the same mRNA track, while each synthesizes a copy of the same protein. We extend the model of a single ribosome by incorporating steric exclusion of different individuals on the same track. We draw the phase diagram of this model of ribosome traffic in three-dimensional spaces spanned by experimentally controllable parameters. We suggest new experimental tests of our theoretical predictions.

Queueing phase transition: theory of translation

We study the current of particles on a lattice, where to each site a different hopping probability has been associated and the particles can move only in one direction. We show that the queueing of the particles behind a slow site can lead to a first-order phase transition, and derive analytical expressions for the configuration of slow sites for this to happen. We apply this stochastic model to describe the translation of mRNAs. We show that the first-order phase transition, uncovered in this work, is the process responsible for the classification of the proteins having different biological functions.

Effects of multiple occupancy and interparticle interactions on selective transport through narrow channels: theory versus experiment

Many biological and artificial transport channels function without direct input of metabolic energy during a transport event and without structural rearrangements involving transitions from a closed to an open state. Nevertheless, such channels are able to maintain efficient and selective transport. It has been proposed that attractive interactions between the transported molecules and the channel can increase the transport efficiency and that the selectivity of such channels can be based on the strength of the interaction of the specifically transported molecules with the channel. Herein, we study the transport through narrow channels in a framework of a general kinetic theory, which naturally incorporates multiparticle occupancy of the channel and non-single-file transport. We study how the transport efficiency and the probability of translocation through the channel are affected by interparticle interactions in the confined space inside the channel, and establish conditions for selective transport. We compare the predictions of the model with the available experimental data and find good semiquantitative agreement. Finally, we discuss applications of the theory to the design of artificial nanomolecular sieves.

Fluctuations in protein synthesis from a single RNA template: stochastic kinetics of ribosomes

Proteins are polymerized by cyclic machines called ribosomes, which use their messenger RNA (mRNA) track also as the corresponding template, and the process is called translation. We explore, in depth and detail, the stochastic nature of the translation. We compute various distributions associated with the translation process; one of them--namely, the dwell time distribution--has been measured in recent single-ribosome experiments. The form of the distribution, which fits best with our simulation data, is consistent with that extracted from the experimental data. For our computations, we use a model that captures both the mechanochemistry of each individual ribosome and their steric interactions. We also demonstrate the effects of the sequence inhomogeneities of real genes on the fluctuations and noise in translation. Finally, inspired by recent advances in the experimental techniques of manipulating single ribosomes, we make theoretical predictions on the force-velocity relation for individual ribosomes. In principle, all our predictions can be tested by carrying out in vitro experiments.

Asymmetric exclusion processes with disorder: effect of correlations

Multiparticle dynamics in one-dimensional asymmetric exclusion processes with disorder is investigated theoretically by computational and analytical methods. It is argued that the general phase diagram consists of three nonequilibrium phases that are determined by the dynamic behavior at the entrance, at the exit and at the slowest defect bond in the bulk of the system. Specifically, we consider dynamics of asymmetric exclusion process with two identical defect bonds as a function of distance between them. Two approximate theoretical methods that treat the system as a sequence of segments with exact description of dynamics inside the segments and neglect correlations between them, are presented. In addition, a numerical iterative procedure for calculating dynamic properties of asymmetric exclusion systems is developed. Our theoretical predictions are compared with extensive Monte Carlo computer simulations. It is shown that correlations play an important role in the particle dynamics. When two defect bonds are far away from each other the strongest correlations are found at these bonds. However, bringing defect bonds closer leads to the shift of correlations to the region between them. Our analysis indicates that it is possible to develop a successful theoretical description of asymmetric exclusion processes with disorder by properly taking into account the correlations.

Stochasticity and traffic jams in the transcription of ribosomal RNA: Intriguing role of termination and antitermination

In fast-growing bacteria, ribosomal RNA (rRNA) is required to be transcribed at very high rates to sustain the high cellular demand on ribosome synthesis. This results in dense traffic of RNA polymerases (RNAP). We developed a stochastic model, integrating results of single-molecule and quantitative in vivo studies of Escherichia coli, to evaluate the quantitative effect of pausing, termination, and antitermination (AT) on rRNA transcription. Our calculations reveal that in dense RNAP traffic, spontaneous pausing of RNAP can lead to severe "traffic jams," as manifested in the broad distribution of inter-RNAP distances and can be a major factor limiting transcription and hence growth. Our results suggest the suppression of these pauses by the ribosomal AT complex to be essential at fast growth. Moreover, unsuppressed pausing by even a few nonantiterminated RNAPs can already reduce transcription drastically under dense traffic. However, the termination factor Rho can remove the nonantiterminated RNAPs and restore fast transcription. The results thus suggest an intriguing role by Rho to enhance rather than attenuate rRNA transcription.

Ribosome collisions and translation efficiency: optimization by codon usage and mRNA destabilization

Individual mRNAs are translated by multiple ribosomes that initiate translation with an interval of a few seconds. The ribosome speed is codon dependent, and ribosome queuing has been suggested to explain specific data for translation of some mRNAs in vivo. By modeling the stochastic translation process as a traffic problem, we here analyze conditions and consequences of collisions and queuing. The model allowed us to determine the on-rate (0.8 to 1.1 initiations/s) and the time (1 s) the preceding ribosome occludes initiation for Escherichia coli lacZ mRNA in vivo. We find that ribosome collisions and queues are inevitable consequences of a stochastic translation mechanism that reduce the translation efficiency substantially on natural mRNAs. The cells minimize collisions by having its mRNAs being unstable and by a highly selected codon usage in the start of the mRNA. The cost of mRNA breakdown is offset by the concomitant increase in translation efficiency.

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.

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.

Inhomogeneous exclusion processes with extended objects: the effect of defect locations

We study the effects of local inhomogeneities, i.e., slow sites of hopping rate q<1, in a totally asymmetric simple exclusion process for particles of size l>or=1 (in units of the lattice spacing). We compare the simulation results of l=1 and l>1 and notice that the existence of local defects has qualitatively similar effects on the steady state. We focus on the stationary current as well as the density profiles. If there is only a single slow site in the system, we observe a significant dependence of the current on the location of the slow site for both l=1 and l>1 cases. When two slow sites are introduced, more intriguing phenomena emerge, e.g., dramatic decreases in the current when the two are close together. In addition, we study the asymptotic behavior when q-->0. We also explore the associated density profiles and compare our findings to an earlier study using a simple mean-field theory. We then outline the biological significance of these effects.

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.

Molecular Spiders in One Dimension

Molecular spiders are synthetic bio-molecular systems which have "legs" made of short single-stranded segments of DNA. Spiders move on a surface covered with single-stranded DNA segments complementary to legs. Different mappings are established between various models of spiders and simple exclusion processes. For spiders with simple gait and varying number of legs we compute the diffusion coefficient; when the hopping is biased we also compute their velocity.

Power spectra of the total occupancy in the totally asymmetric simple exclusion process

As a solvable and broadly applicable model system, the totally asymmetric exclusion process enjoys iconic status in the theory of nonequilibrium phase transitions. Here, we focus on the time dependence of the total number of particles on a 1-dimensional open lattice and its power spectrum. Using both Monte Carlo simulations and analytic methods, we explore its behavior in different characteristic regimes. In the maximal current phase and on the coexistence line (between high and low density phases), the power spectrum displays algebraic decay, with exponents -1.62 and -2.00, respectively. Deep within the high and low density phases, we find pronounced oscillations, which damp into power laws. This behavior can be understood in terms of driven biased diffusion with conserved noise in the bulk.

Traffic of interacting ribosomes: effects of single-machine mechanochemistry on protein synthesis

Many ribosomes simultaneously move on the same messenger RNA (mRNA), each synthesizing separately a copy of the same protein. In contrast to the earlier models, here we develop a "unified" theoretical model that not only incorporates the mutual exclusions of the interacting ribosomes, but also describes explicitly the mechanochemistry of each of these macromolecular machines during protein synthesis. Using analytical and numerical techniques of nonequilibrium statistical mechanics, we analyze the rates of protein synthesis and the spatiotemporal organization of the ribosomes in this model. We also predict how these properties would change with the changes in the rates of the various chemomechanical processes in each ribosome. Finally, we illustrate the power of this model by making experimentally testable predictions on the rates of protein synthesis and the density profiles of the ribosomes on some mRNAs in E-coli.

Bottleneck-induced transitions in a minimal model for intracellular transport

We consider the influence of disorder on the nonequilibrium steady state of a minimal model for intracellular transport. In this model particles move unidirectionally according to the totally asymmetric exclusion process (TASEP) and are coupled to a bulk reservoir by Langmuir kinetics. Our discussion focuses on localized point defects acting as a bottleneck for the particle transport. Combining analytic methods and numerical simulations, we identify a rich phase behavior as a function of the defect strength. Our analytical approach relies on an effective mean-field theory obtained by splitting the lattice into two subsystems, which are effectively connected exploiting the local current conservation. Introducing the key concept of a carrying capacity, the maximal current that can flow through the bulk of the system (including the defect), we discriminate between the cases where the defect is irrelevant and those where it acts as a bottleneck and induces various novel phases (called bottleneck phases). Contrary to the simple TASEP in the presence of inhomogeneities, many scenarios emerge and translate into rich underlying phase diagrams, the topological properties of which are discussed.

Driven lattice gas of dimers coupled to a bulk reservoir

We investigate the nonequilibrium steady state of a one-dimensional (1D) lattice gas of dimers. The dynamics is described by a totally asymmetric exclusion process (TASEP) supplemented by attachment and detachment processes, mimicking chemical equilibrium of the 1D driven transport with the bulk reservoir. The steady-state phase diagram and current and density profiles are calculated using both a refined mean-field theory and extensive stochastic simulations. As a consequence of the on-off kinetics, a phase coexistence region arises intervening between low and high density phases such that the discontinuous transition line of the TASEP splits into two continuous ones. The results of the mean-field theory and simulations are found to coincide. We show that the physical picture obtained in the corresponding lattice gas model with monomers is robust, in the sense that the phase diagram changes quantitatively, but the topology remains unaltered. The mechanism for phase separation is identified as generic for a wide class of driven 1D lattice gases.

Experimental and Computational Analysis of Translation Products in Apomyoglobin Expression

This work focuses on the experimental analysis of the time-course of protein expression in a cell-free system, in conjunction with the development of a computational model, denoted as progressive chain buildup (PCB), able to simulate translation kinetics and product formation as a function of starting reactant concentrations. Translation of the gene encoding the apomyoglobin (apoMb) model protein was monitored in an Escherichia coli cell-free system under different experimental conditions. Experimentally observed protein expression yields, product accumulation time-course and expression completion times match with the predictions by the PCB model. This algorithm regards elementary single-residue elongations as apparent second-order events and it accounts for aminoacyl-tRNA regeneration during translation. We have used this computational approach to model full-length protein expression and to explore the kinetic behavior of incomplete chains generated during protein biosynthesis. Most of the observed incomplete chains are non-obligatory dead-end species, in that their formation is not mandatory for full-length protein expression, and that they are unable to convert to the expected final translation product. These truncated polypeptides do not arise from post-translational degradation of full-length protein, but from a distinct subpopulation of chains which expresses intrinsically more slowly than the population leading to full-length product. The PCB model is a valuable tool to predict full-length and incomplete chain populations and formulate experimentally testable hypotheses on their origin. PCB simulations are applicable to E.coli cell-free expression systems (both in batch and dialysis mode) under the control of T7 RNA polymerase and to other environments where transcription and translation can be regarded as kinetically decoupled.

Numerical method for accessing the universal scaling function for a multiparticle discrete time asymmetric exclusion process

In the universality class of the one-dimensional Kardar-Parisi-Zhang (KPZ) surface growth, Derrida and Lebowitz conjectured the universality of not only the scaling exponents, but of an entire scaling function. Since and Derrida and Lebowitz's original publication [Phys. Rev. Lett. 80, 209 (1998)] this universality has been verified for a variety of continuous-time, periodic-boundary systems in the KPZ universality class. Here, we present a numerical method for directly examining the entire particle flux of the asymmetric exclusion process (ASEP), thus providing an alternative to more difficult cumulant ratios studies. Using this method, we find that the Derrida-Lebowitz scaling function (DLSF) properly characterizes the large-system-size limit (N--> infinity) of a single-particle discrete time system, even in the case of very small system sizes (N< or =22). This fact allows us to not only verify that the DLSF properly characterizes multiple-particle discrete-time asymmetric exclusion processes, but also provides a way to numerically solve for quantities of interest, such as the particle hopping flux. This method can thus serve to further increase the ease and accessibility of studies involving even more challenging dynamics, such as the open-boundary ASEP.

Anomalous nucleation far from equilibrium

We present precision Monte Carlo data and analytic arguments for an asymmetric exclusion process, involving two species of particles driven in opposite directions on a 2xL lattice. To resolve a stark discrepancy between earlier simulation data and an analytic conjecture, we argue that the presence of a single macroscopic cluster is an intermediate stage of a complex nucleation process: in smaller systems, this cluster is destabilized while larger systems form multiple clusters. Both limits lead to exponential cluster size distributions, controlled by very different length scales.

Exact probability function for bulk density and current in the asymmetric exclusion process

We examine the asymmetric simple exclusion process with open boundaries, a paradigm of driven diffusive systems, having a nonequilibrium steady-state transition. We provide a full derivation and expanded discussion and digression on results previously reported briefly in M. Depken and R. Stinchcombe, Phys. Rev. Lett. 93, 040602 (2004). In particular we derive an exact form for the joint probability function for the bulk density and current, both for finite systems, and also in the thermodynamic limit. The resulting distribution is non-Gaussian, and while the fluctuations in the current are continuous at the continuous phase transitions, the density fluctuations are discontinuous. The derivations are done by using the standard operator algebraic techniques and by introducing a modified version of the original operator algebra. As a by-product of these considerations we also arrive at a very simple way of calculating the normalization constant appearing in the standard treatment with the operator algebra. Like the partition function in equilibrium systems, this normalization constant is shown to completely characterize the fluctuations, albeit in a very different manner.