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

Weak Pinning and Long-Range Anticorrelated Motion of Phase Boundaries in Driven Diffusive Systems

We show that domain walls separating coexisting extremal current phases in driven diffusive systems exhibit complex stochastic dynamics with a subdiffusive temporal growth of position fluctuations due to long-range anticorrelated current fluctuations and a weak pinning at long times. This weak pinning manifests itself in a saturated width of the domain wall position fluctuations that increases sublinearly with the system size. As a function of time t and system size L, the width w(t,L) has a scaling behavior w(t,L)=L^{3/4}f(t/L^{9/4}), with f(u) constant for u≫1 and f(u)∼u^{1/3} for u≪1. An Orstein-Uhlenbeck process with long-range anticorrelated noise is shown to capture this scaling behavior. The exponent 9/4 is a new dynamical exponent for relaxation processes in driven diffusive systems.

Sample-to-sample fluctuations of transport coefficients in the totally asymmetric simple exclusion process with quenched disorder

We consider the totally asymmetric simple exclusion processes on quenched random energy landscapes. We show that the current and the diffusion coefficient differ from those for homogeneous environments. Using the mean-field approximation, we analytically obtain the site density when the particle density is low or high. As a result, the current and the diffusion coefficient are described by the dilute limit of particles or holes, respectively. However, in the intermediate regime, due to the many-body effect, the current and the diffusion coefficient differ from those for single-particle dynamics. The current is almost constant and becomes the maximal value in the intermediate regime. Moreover, the diffusion coefficient decreases with the particle density in the intermediate regime. We obtain analytical expressions for the maximal current and the diffusion coefficient based on the renewal theory. The deepest energy depth plays a central role in determining the maximal current and the diffusion coefficient. As a result, the maximal current and the diffusion coefficient depend crucially on the disorder, i.e., non-self-averaging. Based on the extreme value theory, we find that sample-to-sample fluctuations of the maximal current and diffusion coefficient are characterized by the Weibull distribution. We show that the disorder averages of the maximal current and the diffusion coefficient converge to zero as the system size is increased and quantify the degree of the non-self-averaging effect for the maximal current and the diffusion coefficient.

Non-self-averaging of current in a totally asymmetric simple exclusion process with quenched disorder

We investigate the current properties in the totally asymmetric simple exclusion process (TASEP) on a quenched random energy landscape. In low- and high-density regimes, the properties are characterized by single-particle dynamics. In the intermediate one, the current becomes constant and is maximized. Based on the renewal theory, we derive accurate results for the maximum current. The maximum current significantly depends on a disorder realization, i.e., non-self-averaging (SA). We demonstrate that the disorder average of the maximum current decreases with the system size, and the sample-to-sample fluctuations of the maximum current exceed those of current in the low- and high-density regimes. We find a significant difference between single-particle dynamics and the TASEP. In particular, the non-SA behavior of the maximum current is always observed, whereas the transition from non-SA to SA for current in single-particle dynamics exists.

A model of minimal entropy generation for cytoskeletal transport systems with multiple interacting motors

We study the steady-state rate of entropy generation for multiple interacting particles. The description used is based on the partially asymmetric exclusion process in a lattice with periodic boundary conditions. Our methodology shows that in the steady-state, the rate of entropy generation is directly proportional to the bulk drift and the applied driving force. Since in many cases the driving force is unknown or hard to determine. We circumvent this by deriving a lower bound for the entropy, resulting in an extended thermodynamic uncertainty relation for the asymmetric simple exclusion process. We systematically compared this bound with the actual entropy generation. Thus, we identify the force regimes, and particles' density conditions where the entropy bound derived from this extended thermodynamic uncertainty relation is meaningful.

A deterministic model for non-monotone relationship between translation of upstream and downstream open reading frames

Totally asymmetric simple exclusion process (TASEP) modelling was shown to offer a parsimonious explanation for the experimentally confirmed ability of a single upstream open reading frames (uORFs) to upregulate downstream translation during the integrated stress response. As revealed by numerical simulations, the model predicts that reducing the density of scanning ribosomes upstream of certain uORFs increases the flow of ribosomes downstream. To gain a better insight into the mechanism which ensures the non-monotone relation between the upstream and downstream flows, in this work, we propose a phenomenological deterministic model approximating the TASEP model of the translation process. We establish the existence of a stationary solution featuring the decreasing density along the uORF for the deterministic model. Further, we find an explicit non-monotone relation between the upstream ribosome density and the downstream flow for the stationary solution in the limit of increasing uORF length and increasingly leaky initiation. The stationary distribution of the TASEP model, the stationary solution of the deterministic model and the explicit limit are compared numerically.

Current-density relation in the exclusion process with dynamic obstacles

We investigate the totally asymmetric simple exclusion process (TASEP) in the presence of obstacles that dynamically bind and unbind from the lattice. The model is motivated by biological processes such as transcription in the presence of DNA-binding proteins. Similar models have been studied before using the mean-field approximation, but the exact relation between the particle current and density remains elusive. Here, we first show using extensive Monte Carlo simulations that the current-density relation in this model assumes a quasiparabolic form similar to that of the ordinary TASEP without obstacles. We then attempt to explain this relation using exact calculations in the limit of low and high density of particles. Our results suggest that the symmetric, quasiparabolic current-density relation arises through a nontrivial cancellation of higher-order terms, similarly as in the standard TASEP.

Dynamics of ribosomes in mRNA translation under steady- and nonsteady-state conditions

Recent advances in DNA sequencing and fluorescence imaging have made it possible to monitor the dynamics of ribosomes actively engaged in messenger RNA (mRNA) translation. Here, we model these experiments within the inhomogeneous totally asymmetric simple exclusion process (TASEP) using realistic kinetic parameters. In particular, we present analytic expressions to describe the following three cases: (a) translation of a newly transcribed mRNA, (b) translation in the steady state and, specifically, the dynamics of individual (tagged) ribosomes, and (c) runoff translation after inhibition of translation initiation. In cases (b) and (c) we develop an effective medium approximation to describe many-ribosome dynamics in terms of a single tagged ribosome in an effective medium. The predictions are in good agreement with stochastic simulations.

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

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

Biologically motivated three-species exclusion model: Effects of leaky scanning and overlapping genes on initiation of protein synthesis

The totally asymmetric simple exclusion process was originally introduced as a model for the trafficlike collective movement of ribosomes on a messenger RNA (mRNA) that serves as the track for the motorlike forward stepping of individual ribosomes. In each step, a ribosome elongates a protein by a single unit using the track also as a template for protein synthesis. But, prefabricated functionally competent ribosomes are not available to begin synthesis of protein; a subunit directionally scans the mRNA in search of the predesignated site where it is supposed to bind with the other subunit and begin the synthesis of the corresponding protein. However, because of "leaky" scanning, a fraction of the scanning subunits miss the target site and continue their search beyond the first target. Sometimes such scanners successfully identify the site that marks the site for initiation of the synthesis of a different protein. In this paper, we develop an exclusion model with three interconvertible species of hard rods to capture some of the key features of these biological phenomena and study the effects of the interference of the flow of the different species of rods on the same lattice. More specifically, we identify the mean time for the initiation of protein synthesis as appropriate mean first-passage time that we calculate analytically using the formalism of backward master equations. Despite the approximations made, our analytical predictions are in reasonably good agreement with the numerical data that we obtain by performing Monte Carlo simulations.

Power series solution of the inhomogeneous exclusion process

We develop a power series method for the nonequilibrium steady state of the inhomogeneous one-dimensional totally asymmetric simple exclusion process (TASEP) in contact with two particle reservoirs and with site-dependent hopping rates in the bulk. The power series is performed in the entrance or exit rates governing particle exchange with the reservoirs, and the corresponding particle current is computed analytically up to the cubic term in the entry or exit rate, respectively. We also show how to compute higher-order terms using combinatorial objects known as Young tableaux. Our results address the long outstanding problem of finding the exact nonequilibrium steady state of the inhomogeneous TASEP. The findings are particularly relevant to the modeling of mRNA translation in which the rate of translation initiation, corresponding to the entrance rate in the TASEP, is typically small.

TASEP modelling provides a parsimonious explanation for the ability of a single uORF to derepress translation during the integrated stress response

Translation initiation is the rate-limiting step of protein synthesis that is downregulated during the Integrated Stress Response (ISR). Previously, we demonstrated that most human mRNAs that are resistant to this inhibition possess translated upstream open reading frames (uORFs), and that in some cases a single uORF is sufficient for the resistance. Here we developed a computational model of Initiation Complexes Interference with Elongating Ribosomes (ICIER) to gain insight into the mechanism. We explored the relationship between the flux of scanning ribosomes upstream and downstream of a single uORF depending on uORF features. Paradoxically, our analysis predicts that reducing ribosome flux upstream of certain uORFs increases initiation downstream. The model supports the derepression of downstream translation as a general mechanism of uORF-mediated stress resistance. It predicts that stress resistance can be achieved with long slowly decoded uORFs that do not favor translation reinitiation and that start with initiators of low leakiness.

Deciphering mRNA Sequence Determinants of Protein Production Rate

One of the greatest challenges in biophysical models of translation is to identify coding sequence features that affect the rate of translation and therefore the overall protein production in the cell. We propose an analytic method to solve a translation model based on the inhomogeneous totally asymmetric simple exclusion process, which allows us to unveil simple design principles of nucleotide sequences determining protein production rates. Our solution shows an excellent agreement when compared to numerical genome-wide simulations of S. cerevisiae transcript sequences and predicts that the first 10 codons, which is the ribosome footprint length on the mRNA, together with the value of the initiation rate, are the main determinants of protein production rate under physiological conditions. Finally, we interpret the obtained analytic results based on the evolutionary role of the codons' choice for regulating translation rates and ribosome densities.

Interference of two codirectional exclusion processes in the presence of a static bottleneck: A biologically motivated model

We develop a two-species exclusion process with a distinct pair of entry and exit sites for each species of rigid rods. The relatively slower forward stepping of the rods in an extended bottleneck region, located in between the two entry sites, controls the extent of interference of the codirectional flow of the two species of rods. The relative positions of the sites of entry of the two species of rods with respect to the location of the bottleneck are motivated by a biological phenomenon. However, the primary focus of the study here is to explore the effects of the interference of the flow of the two species of rods on their spatiotemporal organization and the regulations of this interference by the extended bottleneck. By a combination of mean-field theory and computer simulation, we calculate the flux of both species of rods and their density profiles as well as the composite phase diagrams of the system. If the bottleneck is sufficiently stringent, then some of the phases become practically unrealizable, although not ruled out on the basis of any fundamental physical principle. Moreover, the extent of suppression of flow of the downstream entrants by the flow of the upstream entrants can also be regulated by the strength of the bottleneck. We speculate on the possible implications of the results in the context of the biological phenomenon that motivated the formulation of the theoretical model.

Matrix-product ansatz for the totally asymmetric simple exclusion process with a generalized update on a ring

We apply the matrix-product ansatz to study the totally asymmetric simple exclusion process on a ring with a generalized discrete-time dynamics depending on two hopping probabilities, p and p[over ̃]. The model contains as special cases the TASEP with parallel update, when p[over ̃]=0, and with sequential backward-ordered update, when p[over ̃]=p. We construct a quadratic algebra and its two-dimensional matrix-product representation to obtain exact finite-size expressions for the partition function, the current of particles, and the two-point correlation function. Our main new result is the derivation of the finite-size pair correlation function. Its behavior is analyzed in different regimes of effective attraction and repulsion between the particles, depending on whether p[over ̃]>p or p[over ̃]<p. In particular, we explicitly obtain an analytic expression for the pair correlation function in the limit of irreversible aggregation p[over ̃]→1, when the stationary configurations contain just one cluster.

Hybrid agent-based model for quantitative in-silico cell-free protein synthesis

An advanced vision of the mRNA translation is presented through a hybrid modeling approach. The dynamics of the polysome formation was investigated by computer simulation that combined agent-based model and fine-grained Markov chain representation of the chemical kinetics. This approach allowed for the investigation of the polysome dynamics under non-steady-state and non-continuum conditions. The model is validated by the quantitative comparison of the simulation results and Luciferase protein production in cell-free system, as well as by testing of the hypothesis regarding the two possible mechanisms of the Edeine antibiotic. Calculation of the Hurst exponent demonstrated a relationship between the microscopic properties of amino acid elongation and the fractal dimension of the translation duration time series. The temporal properties of the amino acid elongation have indicated an anti-persistent behavior under low mRNA occupancy and evinced the appearance of long range interactions within the mRNA-ribosome system for high ribosome density. The dynamic and temporal characteristics of the polysomal system presented here can have a direct impact on the studies of the co-translation protein folding and provide a validated platform for cell-free system studies.

A Mechanistic Model for Cooperative Behavior of Co-transcribing RNA Polymerases

In fast-transcribing prokaryotic genes, such as an rrn gene in Escherichia coli, many RNA polymerases (RNAPs) transcribe the DNA simultaneously. Active elongation of RNAPs is often interrupted by pauses, which has been observed to cause RNAP traffic jams; yet some studies indicate that elongation seems to be faster in the presence of multiple RNAPs than elongation by a single RNAP. We propose that an interaction between RNAPs via the torque produced by RNAP motion on helically twisted DNA can explain this apparent paradox. We have incorporated the torque mechanism into a stochastic model and simulated transcription both with and without torque. Simulation results illustrate that the torque causes shorter pause durations and fewer collisions between polymerases. Our results suggest that the torsional interaction of RNAPs is an important mechanism in maintaining fast transcription times, and that transcription should be viewed as a cooperative group effort by multiple polymerases.

Identification of the mRNA targets of tRNA-specific regulation using genome-wide simulation of translation

tRNA gene copy number is a primary determinant of tRNA abundance and therefore the rate at which each tRNA delivers amino acids to the ribosome during translation. Low-abundance tRNAs decode rare codons slowly, but it is unclear which genes might be subject to tRNA-mediated regulation of expression. Here, those mRNA targets were identified via global simulation of translation. In-silico mRNA translation rates were compared for each mRNA in both wild-type and a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}${\rm{tRNA}}_{{\rm{CUG}}}^{{\rm{Gln}}}$\end{document}sup70-65 mutant, which exhibits a pseudohyphal growth phenotype and a 75% slower CAG codon translation rate. Of 4900 CAG-containing mRNAs, 300 showed significantly reduced in silico translation rates in a simulated tRNA mutant. Quantitative immunoassay confirmed that the reduced translation rates of sensitive mRNAs were \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}${\rm{tRNA}}_{{\rm{CUG}}}^{{\rm{Gln}}}$\end{document} concentration-dependent. Translation simulations showed that reduced \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}${\rm{tRNA}}_{{\rm{CUG}}}^{{\rm{Gln}}}$\end{document} concentrations triggered ribosome queues, which dissipated at reduced translation initiation rates. To validate this prediction experimentally, constitutive gcn2 kinase mutants were used to reduce in vivo translation initiation rates. This repaired the relative translational rate defect of target mRNAs in the sup70-65 background, and ameliorated sup70-65 pseudohyphal growth phenotypes. We thus validate global simulation of translation as a new tool to identify mRNA targets of tRNA-specific gene regulation.

Rapid Curtailing of the Stringent Response by Toxin-Antitoxin Module-Encoded mRNases

Escherichia coli regulates its metabolism to adapt to changes in the environment, in particular to stressful downshifts in nutrient quality. Such shifts elicit the so-called stringent response, coordinated by the alarmone guanosine tetra- and pentaphosphate [(p)ppGpp]. On sudden amino acid (aa) starvation, RelA [(p)ppGpp synthetase I] activity is stimulated by binding of uncharged tRNAs to a vacant ribosomal site; the (p)ppGpp level increases dramatically and peaks within the time scale of a few minutes. The decrease of the (p)ppGpp level after the peak is mediated by the decreased production of mRNA by (p)ppGpp-associated transcriptional regulation, which reduces the vacant ribosomal A site and thus constitutes negative feedback to the RelA-dependent (p)ppGpp synthesis. Here we showed that on sudden isoleucine starvation, this peak was higher in an E. coli strain that lacks the 10 known mRNase-encoding toxin-antitoxin (TA) modules present in the wild-type (wt) strain. This observation suggested that toxins are part of the negative-feedback mechanism to control the (p)ppGpp level during the early stringent response. We built a ribosome trafficking model to evaluate the fold increase in RelA activity just after the onset of aa starvation. Combining this with a feedback model between the (p)ppGpp level and the mRNA level, we obtained reasonable fits to the experimental data for both strains. The analysis revealed that toxins are activated rapidly, within a minute after the onset of starvation, reducing the mRNA half-life by ∼30%.

IMPORTANCE

The early stringent response elicited by amino acid starvation is controlled by a sharp increase of the cellular (p)ppGpp level. Toxin-antitoxin module-encoded mRNases are activated by (p)ppGpp through enhanced degradation of antitoxins. The present work shows that this activation happens over a very short time scale and that the activated mRNases negatively affect the (p)ppGpp level. The proposed mathematical model of (p)ppGpp regulation through the mRNA level highlights the importance of several feedback loops in early (p)ppGpp regulation.

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.

Stepping and crowding of molecular motors: statistical kinetics from an exclusion process perspective

Motor enzymes are remarkable molecular machines that use the energy derived from the hydrolysis of a nucleoside triphosphate to generate mechanical movement, achieved through different steps that constitute their kinetic cycle. These macromolecules, nowadays investigated with advanced experimental techniques to unveil their molecular mechanisms and the properties of their kinetic cycles, are implicated in many biological processes, ranging from biopolymerization (e.g., RNA polymerases and ribosomes) to intracellular transport (motor proteins such as kinesins or dyneins). Although the kinetics of individual motors is well studied on both theoretical and experimental grounds, the repercussions of their stepping cycle on the collective dynamics still remains unclear. Advances in this direction will improve our comprehension of transport process in the natural intracellular medium, where processive motor enzymes might operate in crowded conditions. In this work, we therefore extend contemporary statistical kinetic analysis to study collective transport phenomena of motors in terms of lattice gas models belonging to the exclusion process class. Via numerical simulations, we show how to interpret and use the randomness calculated from single particle trajectories in crowded conditions. Importantly, we also show that time fluctuations and non-Poissonian behavior are intrinsically related to spatial correlations and the emergence of large, but finite, clusters of comoving motors. The properties unveiled by our analysis have important biological implications on the collective transport characteristics of processive motor enzymes in crowded conditions.

Evolutionary Pressures on the Yeast Transcriptome

Codon usage bias (CUB) is the well known phenomenon that the frequency of synonymous codons is unequal. This is presumably the result of adaptive pressures favouring some codons over others. The underlying reason for this pressure is unknown, although a large number of possible driver mechanisms have been proposed; one of them is the decoding time. The standard model to calculate decoding time is the Gromadski-Rodnina model. Yet, recently, there have been a number of studies arguing to the effect that this conventional speed-model is not relevant to understand the dynamics of translation. However, results remain inconclusive so far. This contribution takes a novel approach to address this issue based on comparing mRNA with random synonymous variants to estimate the evolutionary pressures that have acted on the transcriptome. It emerges that over 70 percent of ORFs have been subject to a strong selection pressure for translation speed and that there is also a strong selection pressure for the avoidance of traffic jams. Finally, it is also shown that both homogeneous and very heterogeneous transcripts are over-represented. These results corroborate the validity of the Gromadski-Rodnina model.

Ribosome recycling induces optimal translation rate at low ribosomal availability

During eukaryotic cellular protein synthesis, ribosomal translation is made more efficient through interaction between the two ends of the messenger RNA (mRNA). Ribosomes reaching the 3′ end of the mRNA can thus recycle and begin translation again on the same mRNA, the so-called ‘closed-loop’ model. Using a driven diffusion lattice model of translation, we study the effects of ribosome recycling on the dynamics of ribosome flow and density on the mRNA. We show that ribosome recycling induces a substantial increase in ribosome current. Furthermore, for sufficiently large values of the recycling rate, the lattice does not transition directly from low to high ribosome density, as seen in lattice models without recycling. Instead, a maximal current phase becomes accessible for much lower values of the initiation rate, and multiple phase transitions occur over a wide region of the phase plane. Crucially, we show that in the presence of ribosome recycling, mRNAs can exhibit a peak in protein production at low values of the initiation rate, beyond which translation rate decreases. This has important implications for translation of certain mRNAs, suggesting that there is an optimal concentration of ribosomes at which protein synthesis is maximal, and beyond which translational efficiency is impaired.

Minimal model of transcriptional elongation processes with pauses

Fundamental biological processes such as transcription and translation, where a genetic sequence is sequentially read by a macromolecule, have been well described by a classical model of nonequilibrium statistical physics, the totally asymmetric exclusion principle (TASEP). This model describes particles hopping between sites of a one-dimensional lattice, with the particle current determining the transcription or translation rate. An open problem is how to analyze a TASEP where particles can pause randomly, as has been observed during transcription. In this work, we report that surprisingly, a simple mean-field model predicts well the particle current for all values of the average pause duration, using a simple description of blocking behind paused particles.

Efficiency at maximum power of motor traffic on networks

We study motor traffic on Bethe networks subject to hard-core exclusion for both tightly coupled one-state machines and loosely coupled two-state machines that perform work against a constant load. In both cases we find an interaction-induced enhancement of the efficiency at maximum power (EMP) as compared to noninteracting motors. The EMP enhancement occurs for a wide range of network and single-motor parameters and is due to a change in the characteristic load-velocity relation caused by phase transitions in the system. Using a quantitative measure of the trade-off between the EMP enhancement and the corresponding loss in the maximum output power we identify parameter regimes where motor traffic systems operate efficiently at maximum power without a significant decrease in the maximum power output due to jamming effects.

mRNA translation and protein synthesis: an analysis of different modelling methodologies and a new PBN based approach

Background

mRNA translation involves simultaneous movement of multiple ribosomes on the mRNA and is also subject to regulatory mechanisms at different stages. Translation can be described by various codon-based models, including ODE, TASEP, and Petri net models. Although such models have been extensively used, the overlap and differences between these models and the implications of the assumptions of each model has not been systematically elucidated. The selection of the most appropriate modelling framework, and the most appropriate way to develop coarse-grained/fine-grained models in different contexts is not clear.

Results

We systematically analyze and compare how different modelling methodologies can be used to describe translation. We define various statistically equivalent codon-based simulation algorithms and analyze the importance of the update rule in determining the steady state, an aspect often neglected. Then a novel probabilistic Boolean network (PBN) model is proposed for modelling translation, which enjoys an exact numerical solution. This solution matches those of numerical simulation from other methods and acts as a complementary tool to analytical approximations and simulations. The advantages and limitations of various codon-based models are compared, and illustrated by examples with real biological complexities such as slow codons, premature termination and feedback regulation. Our studies reveal that while different models gives broadly similiar trends in many cases, important differences also arise and can be clearly seen, in the dependence of the translation rate on different parameters. Furthermore, the update rule affects the steady state solution.

Conclusions

The codon-based models are based on different levels of abstraction. Our analysis suggests that a multiple model approach to understanding translation allows one to ascertain which aspects of the conclusions are robust with respect to the choice of modelling methodology, and when (and why) important differences may arise. This approach also allows for an optimal use of analysis tools, which is especially important when additional complexities or regulatory mechanisms are included. This approach can provide a robust platform for dissecting translation, and results in an improved predictive framework for applications in systems and synthetic biology.

Controlling translation elongation efficiency: tRNA regulation of ribosome flux on the mRNA

Gene expression can be regulated by a wide variety of mechanisms. One example concerns the growing body of evidence that the protein-production rate can be regulated at the level of translation elongation by controlling ribosome flux across the mRNA. Variations in the abundance of tRNA molecules cause different rates of translation of their counterpart codons. This, in turn, produces a variable landscape of translational rate across each and every mRNA, with the dynamic formation and deformation of ribosomal queues being regulated by both tRNA availability and the rates of translation initiation and termination. In the present article, a range of examples of tRNA control of gene expression are reviewed, and the use of mathematical modelling to develop a predictive understanding of the consequences of that regulation is discussed and explained. These findings encourage a view that predicting the protein-synthesis rate of each mRNA requires a holistic understanding of how each stage of translation, including elongation, contributes to the overall protein-production 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.

Transport dynamics of molecular motors that switch between an active and inactive state

Molecular motors are involved in key transport processes in the cell. Many of these motors can switch from an active to a nonactive state, either spontaneously or depending on their interaction with other molecules. When active, the motors move processively along the filaments, while when inactive they are stationary. We treat here the simple case of spontaneously switching motors, between the active and inactive states, along an open linear track. We use our recent analogy with vehicular traffic, where we go beyond the mean-field description. We map the phase diagram of this system, and find that it clearly breaks the symmetry between the different phases, as compared to the standard total asymmetric exclusion process. We make several predictions that may be testable using molecular motors in vitro and in living cells.

Maximum power operation of interacting molecular motors

We study the mechanical and thermodynamic properties of different traffic models for kinesin which are relevant in biological and experimental contexts. We find that motor-motor interactions play a fundamental role by enhancing the thermodynamic efficiency at maximum power of the motors, as compared to the noninteracting system, in a wide range of biologically compatible scenarios. We furthermore consider the case where the motor-motor interaction directly affects the internal chemical cycle and investigate the effect on the system dynamics and thermodynamics.

Ribosome traffic on mRNAs maps to gene ontology: genome-wide quantification of translation initiation rates and polysome size regulation

To understand the complex relationship governing transcript abundance and the level of the encoded protein, we integrate genome-wide experimental data of ribosomal density on mRNAs with a novel stochastic model describing ribosome traffic dynamics during translation elongation. This analysis reveals that codon arrangement, rather than simply codon bias, has a key role in determining translational efficiency. It also reveals that translation output is governed both by initiation efficiency and elongation dynamics. By integrating genome-wide experimental data sets with simulation of ribosome traffic on all Saccharomyces cerevisiae ORFs, mRNA-specific translation initiation rates are for the first time estimated across the entire transcriptome. Our analysis identifies different classes of mRNAs characterised by their initiation rates, their ribosome traffic dynamics, and by their response to ribosome availability. Strikingly, this classification based on translational dynamics maps onto key gene ontological classifications, revealing evolutionary optimisation of translation responses to be strongly influenced by gene function.

Gene expression regulation is central to all living systems. Here we introduce a new framework and methodology to study the last stage of protein production in cells, where the genetic information encoded in the mRNAs is translated from the language of nucleotides into functional proteins. The process, on each mRNA, is carried out concurrently by several ribosomes; like cars on a small countryside road, they cannot overtake each other, and can form queues. By integrating experimental data with genome-wide simulations of our model, we analyse ribosome traffic across the entire Saccharomyces cerevisiae genome, and for the first time estimate mRNA-specific translation initiation rates for each transcript. Crucially, we identify different classes of mRNAs characterised by different ribosome traffic dynamics. Remarkably, this classification based on translational dynamics, and the evaluation of mRNA-specific initiation rates, map onto key gene ontological classifications, revealing evolutionary optimisation of translation responses to be strongly influenced by gene function.

A yeast tRNA mutant that causes pseudohyphal growth exhibits reduced rates of CAG codon translation

In Saccharomyces cerevisiae, the SUP70 gene encodes the CAG-decoding tRNA(Gln)(CUG). A mutant allele, sup70-65, induces pseudohyphal growth on rich medium, an inappropriate nitrogen starvation response. This mutant tRNA is also a UAG nonsense suppressor via first base wobble. To investigate the basis of the pseudohyphal phenotype, 10 novel sup70 UAG suppressor alleles were identified, defining positions in the tRNA(Gln)(CUG) anticodon stem that restrict first base wobble. However, none conferred pseudohyphal growth, showing altered CUG anticodon presentation cannot itself induce pseudohyphal growth. Northern blot analysis revealed the sup70-65 tRNA(Gln)(CUG) is unstable, inefficiently charged, and 80% reduced in its effective concentration. A stochastic model simulation of translation predicted compromised expression of CAG-rich ORFs in the tRNA(Gln)(CUG)-depleted sup70-65 mutant. This prediction was validated by demonstrating that luciferase expression in the mutant was 60% reduced by introducing multiple tandem CAG (but not CAA) codons into this ORF. In addition, the sup70-65 pseudohyphal phenotype was partly complemented by overexpressing CAA-decoding tRNA(Gln)(UUG), an inefficient wobble-decoder of CAG. We thus show that introducing codons decoded by a rare tRNA near the 5' end of an ORF can reduce eukaryote translational expression, and that the mutant tRNA(CUG)(Gln) constitutive pseudohyphal differentiation phenotype correlates strongly with reduced CAG decoding efficiency.

Efficiency at maximum power of interacting molecular machines

We investigate the efficiency of systems of molecular motors operating at maximum power. We consider two models of kinesin motors on a microtubule: for both the simplified and the detailed model, we find that the many-body exclusion effect enhances the efficiency at maximum power of the many-motor system, with respect to the single motor case. Remarkably, we find that this effect occurs in a limited region of the system parameters, compatible with the biologically relevant range.

The architecture of eukaryotic translation

Translation in baker’s yeast involves the coordinated interaction of 200 000 ribosomes, 3 000 000 tRNAs and between 15 000 and 60 000 mRNAs. It is currently unknown whether this specific constellation of components has particular relevance for the requirements of the yeast proteome, or whether this is simply a frozen accident. Our study uses a computational simulation model of the genome-wide translational apparatus of yeast to explore quantitatively which combinations of mRNAs, ribosomes and tRNAs can produce viable proteomes. Surprisingly, we find that if we only consider total translational activity over time without regard to composition of the proteome, then there are many and widely differing combinations that can generate equivalent synthesis yields. In contrast, translational activity required for generating specific proteomes can only be achieved within a much more constrained parameter space. Furthermore, we find that strongly ribosome limited regimes are optimal for cells in that they are resource efficient and simplify the dynamics of the system.

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.

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.

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.

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.

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.

Quality control by a mobile molecular workshop: quality versus quantity

Ribosome is a molecular machine that moves on a messenger RNA (mRNA) track while, simultaneously, polymerizing a protein using the mRNA also as the corresponding template. We define, and analytically calculate, two different measures of the efficiency of this machine. However, we argue that its performance is evaluated better in terms of the translational fidelity and the speed with which it polymerizes a protein. We define both these quantities and calculate these analytically. Fidelity is a measure of the quality of the products, while the total quantity of products synthesized in a given interval depends on the speed of polymerization. We show that for synthesizing a large quantity of proteins, it is not necessary to sacrifice the quality. We also explore the effects of the quality control mechanism on the strength of mechanochemical coupling. We suggest experiments for testing some of the ideas presented here.