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

Genome scale analysis of Escherichia coli with a comprehensive prokaryotic sequence-based biophysical model of translation initiation and elongation

Translation initiation in prokaryotes is affected by the mRNA folding and interaction of the ribosome binding site with the ribosomal RNA. The elongation rate is affected, among other factors, by the local biophysical properties of the coding regions, the decoding rates of different codons, and the interactions among ribosomes. Currently, there is no comprehensive biophysical model of translation that enables the prediction of mRNA translation dynamics based only on the transcript sequence and while considering all of these fundamental aspects of translation. In this study, we provide, for the first time, a computational simulative biophysical model of both translation initiation and elongation with all aspects mentioned above. We demonstrate our model performance and advantages focusing on Escherichia coli genes. We further show that the model enables prediction of translation rate, protein levels, and ribosome densities. In addition, our model enables quantifying the effect of silent mutations on translation rate in different parts of the transcript, the relative effect of mutations on translation initiation and elongation, and the effect of mutations on ribosome traffic jams. Thus, unlike previous models, the proposed one provides comprehensive information, facilitating future research in disciplines such as molecular evolution, synthetic biology, and functional genomics. A toolkit to estimate translation dynamics of transcripts is available at: https://www.cs.tau.ac.il/∼tamirtul/transim.

Entrainment in the master equation

The master equation plays an important role in many scientific fields including physics, chemistry, systems biology, physical finance and sociodynamics. We consider the master equation with periodic transition rates. This may represent an external periodic excitation like the 24 h solar day in biological systems or periodic traffic lights in a model of vehicular traffic. Using tools from systems and control theory, we prove that under mild technical conditions every solution of the master equation converges to a periodic solution with the same period as the rates. In other words, the master equation entrains (or phase locks) to periodic excitations. We describe two applications of our theoretical results to important models from statistical mechanics and epidemiology.

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

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

Determinants of translation speed are randomly distributed across transcripts resulting in a universal scaling of protein synthesis times

Ribosome profiling experiments have found greater than 100-fold variation in ribosome density along mRNA transcripts, indicating that individual codon elongation rates can vary to a similar degree. This wide range of elongation times, coupled with differences in codon usage between transcripts, suggests that the average codon translation-rate per gene can vary widely. Yet, ribosome run-off experiments have found that the average codon translation rate for different groups of transcripts in mouse stem cells is constant at 5.6 AA/s. How these seemingly contradictory results can be reconciled is the focus of this study. Here, we combine knowledge of the molecular factors shown to influence translation speed with genomic information from Escherichia coli, Saccharomyces cerevisiae and Homo sapiens to simulate the synthesis of cytosolic proteins in these organisms. The model recapitulates a near constant average translation rate, which we demonstrate arises because the molecular determinants of translation speed are distributed nearly randomly amongst most of the transcripts. Consequently, codon translation rates are also randomly distributed and fast-translating segments of a transcript are likely to be offset by equally probable slow-translating segments, resulting in similar average elongation rates for most transcripts. We also show that the codon usage bias does not significantly affect the near random distribution of codon translation rates because only about 10% of the total transcripts in an organism have high codon usage bias while the rest have little to no bias. Analysis of Ribo-Seq data and an in vivo fluorescent assay supports these conclusions.

Influence of multiplicative stochastic variation on translational elongation rates

Experimental data indicate that stochastic effects exerted at the level of translation contribute substantially to the variation in abundance of proteins expressed at moderate to high levels. This study analyzes the theoretical consequences of fluctuations in residue-specific elongation rates during translation. A simple analytical framework shows that rate variation during elongation gives rise to protein production rates that consist of sums of products of random variables. Simulations show that because the contribution to total variation of products of random variables greatly exceeds that of sums of random variables, the overall distribution exhibits approximately log-normal behavior. Empirical fits of the data can be satisfied by either sums of log-normal distributions, or sums of log-normal and log-logistic distributions. Elongation rate stochastic variation offers an accounting for a major component of biological variation. The analysis provided here highlights a probability distribution that is a natural extension of the Poisson and has broad applicability to many types of multiplicative noise processes.

Gene length as a regulator for ribosome recruitment and protein synthesis: theoretical insights

Protein synthesis rates are determined, at the translational level, by properties of the transcript’s sequence. The efficiency of an mRNA can be tuned by varying the ribosome binding sites controlling the recruitment of the ribosomes, or the codon usage establishing the speed of protein elongation. In this work we propose transcript length as a further key determinant of translation efficiency. Based on a physical model that considers the kinetics of ribosomes advancing on the mRNA and diffusing in its surrounding, as well as mRNA circularisation and ribosome drop-off, we explain how the transcript length may play a central role in establishing ribosome recruitment and the overall translation rate of an mRNA. According to our results, the proximity of the 3′ end to the ribosomal recruitment site of the mRNA could induce a feedback in the translation process that would favour the recycling of ribosomes. We also demonstrate how this process may be involved in shaping the experimental ribosome density-gene length dependence. Finally, we argue that cells could exploit this mechanism to adjust and balance the usage of its ribosomal resources.

Dynamics of translation can determine the spatial organization of membrane-bound proteins and their mRNA

Unlike most macromolecules that are homogeneously distributed in the bacterial cell, mRNAs that encode inner-membrane proteins can be concentrated near the inner membrane. Cotranslational insertion of the nascent peptide into the membrane brings the translating ribosome and the mRNA close to the membrane. This suggests that kinetic properties of translation can determine the spatial organization of these mRNAs and proteins, which can be modulated through posttranscriptional regulation. Here we use a simple stochastic model of translation to characterize the effect of mRNA properties on the dynamics and statistics of its spatial distribution. We show that a combination of the rate of translation initiation, the availability of secretory apparatuses, and the composition of the coding region determines the abundance of mRNAs near the membrane, as well as their residence time. We propose that the spatiotemporal dynamics of mRNAs can give rise to protein clusters on the membrane and determine their size distribution.

Ribosome flow model with extended objects

We study a deterministic mechanistic model for the flow of ribosomes along the mRNA molecule, called the ribosome flow model with extended objects (RFMEO). This model encapsulates many realistic features of translation including non-homogeneous transition rates along mRNA, the fact that every ribosome covers several codons, and the fact that ribosomes cannot overtake one another. The RFMEO is a mean-field approximation of an important model from statistical mechanics called the totally asymmetric simple exclusion process with extended objects (TASEPEO). We demonstrate that the RFMEO describes biophysical aspects of translation better than previous mean-field approximations, and that its predictions correlate well with those of TASEPEO. However, unlike TASEPEO, the RFMEO is amenable to rigorous analysis using tools from systems and control theory. We show that the ribosome density profile along the mRNA in the RFMEO converges to a unique steady-state density that depends on the length of the mRNA, the transition rates along it, and the number of codons covered by every ribosome, but not on the initial density of ribosomes along the mRNA. In particular, the protein production rate also converges to a unique steady state. Furthermore, if the transition rates along the mRNA are periodic with a common period T then the ribosome density along the mRNA and the protein production rate converge to a unique periodic pattern with period T, that is, the model entrains to periodic excitations in the transition rates. Analysis and simulations of the RFMEO demonstrate several counterintuitive results. For example, increasing the ribosome footprint may sometimes lead to an increase in the production rate. Also, for large values of the footprint the steady-state density along the mRNA may be quite complex (e.g. with quasi-periodic patterns) even for relatively simple (and non-periodic) transition rates along the mRNA. This implies that inferring the transition rates from the ribosome density may be non-trivial. We believe that the RFMEO could be useful for modelling, understanding and re-engineering translation as well as other important biological processes.

Optimal Translation Along a Circular mRNA

The ribosome flow model on a ring (RFMR) is a deterministic model for ribosome flow along a circularized mRNA. We derive a new spectral representation for the optimal steady-state production rate and the corresponding optimal steady-state ribosomal density in the RFMR. This representation has several important advantages. First, it provides a simple and numerically stable algorithm for determining the optimal values even in very long rings. Second, it enables efficient computation of the sensitivity of the optimal production rate to small changes in the transition rates along the mRNA. Third, it implies that the optimal steady-state production rate is a strictly concave function of the transition rates. Maximizing the optimal steady-state production rate with respect to the rates under an affine constraint on the rates thus becomes a convex optimization problem that admits a unique solution. This solution can be determined numerically using highly efficient algorithms. This optimization problem is important, for example, when re-engineering heterologous genes in a host organism. We describe the implications of our results to this and other aspects of translation.

A deterministic mathematical model for bidirectional excluded flow with Langmuir kinetics

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

A deterministic model for one-dimensional excluded flow with local interactions

Natural phenomena frequently involve a very large number of interacting molecules moving in confined regions of space. Cellular transport by motor proteins is an example of such collective behavior. We derive a deterministic compartmental model for the unidirectional flow of particles along a one-dimensional lattice of sites with nearest-neighbor interactions between the particles. The flow between consecutive sites is governed by a “soft” simple exclusion principle and by attracting or repelling forces between neighboring particles. Using tools from contraction theory, we prove that the model admits a unique steady-state and that every trajectory converges to this steady-state. Analysis and simulations of the effect of the attracting and repelling forces on this steady-state highlight the crucial role that these forces may play in increasing the steady-state flow, and reveal that this increase stems from the alleviation of traffic jams along the lattice. Our theoretical analysis clarifies microscopic aspects of complex multi-particle dynamic processes.

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.

Increasing Protein Production Rates Can Decrease the Rate at Which Functional Protein Is Produced and Their Steady-State Levels

The rate at which soluble, functional protein is produced by the ribosome has recently been found to vary in complex and unexplained ways as various translation-associated rates are altered through synonymous codon substitutions. To understand this phenomenon, here, we combine a well-established ribosome-traffic model with a master-equation model of cotranslational domain folding to explore the scenarios that are possible for the protein production rate, J, and the functional-nascent protein production rate, F, as the rates of various translation processes are altered for five different E. coli proteins. We find that while J monotonically increases as the rates of translation-initiation, -elongation, and -termination increase, F can either increase or decrease. We show that F's nonmonotonic behavior arises within the model from two opposing trends: the tendency for increased translation rates to produce more total protein but less cotranslationally folded protein. We further demonstrate that under certain conditions these nonmonotonic changes in F can result in nonmonotonic variations in post-translational, steady-state levels of functional protein. These results provide a potential explanation for recent experimental observations in which the specific activity of enzymatic proteins decreased with increased synthesis rates. Additionally our model has the potential to be used to rationally design transcripts to maximize the production of functional nascent protein by simultaneously optimizing translation initiation, elongation, and termination rates.

A resource dependent protein synthesis model for evaluating synthetic circuits

Reliable in silico design of synthetic gene networks necessitates novel approaches to model the process of protein synthesis under the influence of limited resources. We present such a novel protein synthesis model which originates from the Ribosome Flow Model and among other things describes the movement of RNA-polymerase and ribosomes on mRNA and DNA templates, respectively. By analyzing the convergence properties of this model based upon geometric considerations, we present additional insights into the dynamic mechanisms of the process of protein synthesis. Further, we demonstrate how this model can be used to evaluate the performance of synthetic gene circuits under different loading scenarios.

Novel mRNA-specific effects of ribosome drop-off on translation rate and polysome profile

The well established phenomenon of ribosome drop-off plays crucial roles in translational accuracy and nutrient starvation responses during protein translation. When cells are under stress conditions, such as amino acid starvation or aminoacyl-tRNA depletion due to a high level of recombinant protein expression, ribosome drop-off can substantially affect the efficiency of protein expression. Here we introduce a mathematical model that describes the effects of ribosome drop-off on the ribosome density along the mRNA and on the concomitant protein synthesis rate. Our results show that ribosome premature termination may lead to non-intuitive ribosome density profiles, such as a ribosome density which increases from the 5' to the 3' end. Importantly, the model predicts that the effects of ribosome drop-off on the translation rate are mRNA-specific, and we quantify their resilience to drop-off, showing that the mRNAs which present ribosome queues are much less affected by ribosome drop-off than those which do not. Moreover, among those mRNAs that do not present ribosome queues, resilience to drop-off correlates positively with the elongation rate, so that sequences using fast codons are expected to be less affected by ribosome drop-off. This result is consistent with a genome-wide analysis of S. cerevisiae, which reveals that under favourable growth conditions mRNAs coding for proteins involved in the translation machinery, known to be highly codon biased and using preferentially fast codons, are highly resilient to ribosome drop-off. Moreover, in physiological conditions, the translation rate of mRNAs coding for regulatory, stress-related proteins, is less resilient to ribosome drop-off. This model therefore allows analysis of variations in the translational efficiency of individual mRNAs by accounting for the full range of known ribosome behaviours, as well as explaining mRNA-specific variations in ribosome density emerging from ribosome profiling studies.

Optimal Down Regulation of mRNA Translation

Down regulation of mRNA translation is an important problem in various bio-medical domains ranging from developing effective medicines for tumors and for viral diseases to developing attenuated virus strains that can be used for vaccination. Here, we study the problem of down regulation of mRNA translation using a mathematical model called the ribosome flow model (RFM). In the RFM, the mRNA molecule is modeled as a chain of n sites. The flow of ribosomes between consecutive sites is regulated by n + 1 transition rates. Given a set of feasible transition rates, that models the outcome of all possible mutations, we consider the problem of maximally down regulating protein production by altering the rates within this set of feasible rates. Under certain conditions on the feasible set, we show that an optimal solution can be determined efficiently. We also rigorously analyze two special cases of the down regulation optimization problem. Our results suggest that one must focus on the position along the mRNA molecule where the transition rate has the strongest effect on the protein production rate. However, this rate is not necessarily the slowest transition rate along the mRNA molecule. We discuss some of the biological implications of these results.

A model for competition for ribosomes in the cell

A single mammalian cell includes an order of 10(4)-10(5) mRNA molecules and as many as 10(5)-10(6) ribosomes. Large-scale simultaneous mRNA translation induces correlations between the mRNA molecules, as they all compete for the finite pool of available ribosomes. This has important implications for the cell's functioning and evolution. Developing a better understanding of the intricate correlations between these simultaneous processes, rather than focusing on the translation of a single isolated transcript, should help in gaining a better understanding of mRNA translation regulation and the way elongation rates affect organismal fitness. A model of simultaneous translation is specifically important when dealing with highly expressed genes, as these consume more resources. In addition, such a model can lead to more accurate predictions that are needed in the interconnection of translational modules in synthetic biology. We develop and analyse a general dynamical model for large-scale simultaneous mRNA translation and competition for ribosomes. This is based on combining several ribosome flow models (RFMs) interconnected via a pool of free ribosomes. We use this model to explore the interactions between the various mRNA molecules and ribosomes at steady state. We show that the compound system always converges to a steady state and that it always entrains or phase locks to periodically time-varying transition rates in any of the mRNA molecules. We then study the effect of changing the transition rates in one mRNA molecule on the steady-state translation rates of the other mRNAs that results from the competition for ribosomes. We show that increasing any of the codon translation rates in a specific mRNA molecule yields a local effect, an increase in the translation rate of this mRNA, and also a global effect, the translation rates in the other mRNA molecules all increase or all decrease. These results suggest that the effect of codon decoding rates of endogenous and heterologous mRNAs on protein production is more complicated than previously thought. In addition, we show that increasing the length of an mRNA molecule decreases the production rate of all the mRNAs.

On the Ribosomal Density that Maximizes Protein Translation Rate

During mRNA translation, several ribosomes attach to the same mRNA molecule simultaneously translating it into a protein. This pipelining increases the protein translation rate. A natural and important question is what ribosomal density maximizes the protein translation rate. Using mathematical models of ribosome flow along both a linear and a circular mRNA molecules we prove that typically the steady-state protein translation rate is maximized when the ribosomal density is one half of the maximal possible density. We discuss the implications of our results to endogenous genes under natural cellular conditions and also to synthetic biology.

Predictive biophysical modeling and understanding of the dynamics of mRNA translation and its evolution

mRNA translation is the fundamental process of decoding the information encoded in mRNA molecules by the ribosome for the synthesis of proteins. The centrality of this process in various biomedical disciplines such as cell biology, evolution and biotechnology, encouraged the development of dozens of mathematical and computational models of translation in recent years. These models aimed at capturing various biophysical aspects of the process. The objective of this review is to survey these models, focusing on those based and/or validated on real large-scale genomic data. We consider aspects such as the complexity of the models, the biophysical aspects they regard and the predictions they may provide. Furthermore, we survey the central systems biology discoveries reported on their basis. This review demonstrates the fundamental advantages of employing computational biophysical translation models in general, and discusses the relative advantages of the different approaches and the challenges in the field.

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.

Phase coexistence and spatial correlations in reconstituting k-mer models

In reconstituting k-mer models, extended objects that occupy several sites on a one-dimensional lattice undergo directed or undirected diffusion, and reconstitute-when in contact-by transferring a single monomer unit from one k-mer to the other; the rates depend on the size of participating k-mers. This polydispersed system has two conserved quantities, the number of k-mers and the packing fraction. We provide a matrix product method to write the steady state of this model and to calculate the spatial correlation functions analytically. We show that for a constant reconstitution rate, the spatial correlation exhibits damped oscillations in some density regions separated, from other regions with exponential decay, by a disorder surface. In a specific limit, this constant-rate reconstitution model is equivalent to a single dimer model and exhibits a phase coexistence similar to the one observed earlier in totally asymmetric simple exclusion process on a ring with a defect.

Probabilistic Boolean Network Modelling and Analysis Framework for mRNA Translation

mRNA translation is a complex process involving the progression of ribosomes on the mRNA, resulting in the synthesis of proteins, and is subject to multiple layers of regulation. This process has been modelled using different formalisms, both stochastic and deterministic. Recently, we introduced a Probabilistic Boolean modelling framework for mRNA translation, which possesses the advantage of tools for numerically exact computation of steady state probability distribution, without requiring simulation. Here, we extend this model to incorporate both random sequential and parallel update rules, and demonstrate its effectiveness in various settings, including its flexibility in accommodating additional static and dynamic biological complexities and its role in parameter sensitivity analysis. In these applications, the results from the model analysis match those of TASEP model simulations. Importantly, the proposed modelling framework maintains the stochastic aspects of mRNA translation and provides a way to exactly calculate probability distributions, providing additional tools of analysis in this context. Finally, the proposed modelling methodology provides an alternative approach to the understanding of the mRNA translation process, by bridging the gap between existing approaches, providing new analysis tools, and contributing to a more robust platform for modelling and understanding translation.

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.

Community control in cellular protein production: consequences for amino acid starvation

Deprivation of essential nutrients can have stark consequences for many processes in a cell. We consider amino acid starvation, which can result in bottlenecks in mRNA translation when ribosomes stall due to lack of resources, i.e. tRNAs charged with the missing amino acid. Recent experiments also show less obvious effects such as increased charging of other (non-starved) tRNA species and selective charging of isoaccepting tRNAs. We present a mechanism which accounts for these observations and shows that production of some proteins can actually increase under starvation. One might assume that such responses could only be a result of sophisticated control pathways, but here we show that these effects can occur naturally due to changes in the supply and demand for different resources, and that control can be accomplished through selective use of rare codons. We develop a model for translation which includes the dynamics of the charging and use of aminoacylated tRNAs, explicitly taking into account the effect of specific codon sequences. This constitutes a new control mechanism in gene regulation which emerges at the community level, i.e. via resources used by all ribosomes.

Maximizing protein translation rate in the non-homogeneous ribosome flow model: a convex optimization approach

Translation is an important stage in gene expression. During this stage, macro-molecules called ribosomes travel along the mRNA strand linking amino acids together in a specific order to create a functioning protein. An important question, related to many biomedical disciplines, is how to maximize protein production. Indeed, translation is known to be one of the most energy-consuming processes in the cell, and it is natural to assume that evolution shaped this process so that it maximizes the protein production rate. If this is indeed so then one can estimate various parameters of the translation machinery by solving an appropriate mathematical optimization problem. The same problem also arises in the context of synthetic biology, namely, re-engineer heterologous genes in order to maximize their translation rate in a host organism. We consider the problem of maximizing the protein production rate using a computational model for translation-elongation called the ribosome flow model (RFM). This model describes the flow of the ribosomes along an mRNA chain of length n using a set of n first-order nonlinear ordinary differential equations. It also includes n + 1 positive parameters: the ribosomal initiation rate into the mRNA chain, and n elongation rates along the chain sites. We show that the steady-state translation rate in the RFM is a strictly concave function of its parameters. This means that the problem of maximizing the translation rate under a suitable constraint always admits a unique solution, and that this solution can be determined using highly efficient algorithms for solving convex optimization problems even for large values of n. Furthermore, our analysis shows that the optimal translation rate can be computed based only on the optimal initiation rate and the elongation rate of the codons near the beginning of the ORF. We discuss some applications of the theoretical results to synthetic biology, molecular evolution, and functional genomics.

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.

Synonymous codons, ribosome speed, and eukaryotic gene expression regulation

Quantitative control of gene expression occurs at multiple levels, including the level of translation. Within the overall process of translation, most identified regulatory processes impinge on the initiation phase. However, recent studies have revealed that the elongation phase can also regulate translation if elongation and initiation occur with specific, not mutually compatible rate parameters. Translation elongation then limits the overall amount of protein that can be made from an mRNA. Several recently discovered control mechanisms of biological pathways are based on such elongation control. Here, we review the molecular mechanisms that determine ribosome speed in eukaryotic organisms, and discuss under which conditions ribosome speed can become the controlling parameter of gene expression levels.

Maximizing Protein Translation Rate in the Ribosome Flow Model: The Homogeneous Case

Gene translation is the process in which intracellular macro-molecules, called ribosomes, decode genetic information in the mRNA chain into the corresponding proteins. Gene translation includes several steps. During the elongation step, ribosomes move along the mRNA in a sequential manner and link amino-acids together in the corresponding order to produce the proteins. The homogeneous ribosome flow model (HRFM) is a deterministic computational model for translation-elongation under the assumption of constant elongation rates along the mRNA chain. The HRFM is described by a set of n first-order nonlinear ordinary differential equations, where n represents the number of sites along the mRNA chain. The HRFM also includes two positive parameters: ribosomal initiation rate and the (constant) elongation rate. In this paper, we show that the steady-state translation rate in the HRFM is a concave function of its parameters. This means that the problem of determining the parameter values that maximize the translation rate is relatively simple. Our results may contribute to a better understanding of the mechanisms and evolution of translation-elongation. We demonstrate this by using the theoretical results to estimate the initiation rate in M. musculus embryonic stem cell. The underlying assumption is that evolution optimized the translation mechanism. For the infinite-dimensional HRFM, we derive a closed-form solution to the problem of determining the initiation and transition rates that maximize the protein translation rate. We show that these expressions provide good approximations for the optimal values in the n-dimensional HRFM already for relatively small values of n. These results may have applications for synthetic biology where an important problem is to re-engineer genomic systems in order to maximize the protein production rate.

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.

Properties and determinants of codon decoding time distributions

Background

Codon decoding time is a fundamental property of mRNA translation believed to affect the abundance, function, and properties of proteins. Recently, a novel experimental technology--ribosome profiling--was developed to measure the density, and thus the speed, of ribosomes at codon resolution. Specifically, this method is based on next-generation sequencing, which theoretically can provide footprint counts that correspond to the probability of observing a ribosome in this position for each nucleotide in each transcript.

Results

In this study, we report for the first time various novel properties of the distribution of codon footprint counts in five organisms, based on large-scale analysis of ribosomal profiling data. We show that codons have distinctive footprint count distributions. These tend to be preserved along the inner part of the ORF, but differ at the 5' and 3' ends of the ORF, suggesting that the translation-elongation stage actually includes three biophysical sub-steps. In addition, we study various basic properties of the codon footprint count distributions and show that some of them correlate with the abundance of the tRNA molecule types recognizing them.

Conclusions

Our approach emphasizes the advantages of analyzing ribosome profiling and similar types of data via a comparative genomic codon-distribution-centric view. Thus, our methods can be used in future studies related to translation and even transcription elongation.

Detecting translational regulation by change point analysis of ribosome profiling data sets

Ribo-Seq maps the location of translating ribosomes on mature mRNA transcripts. While during normal translation, ribosome density is constant along the length of the mRNA coding region, this can be altered in response to translational regulatory events. In the present study, we developed a method to detect translational regulation of individual mRNAs from their ribosome profiles, utilizing changes in ribosome density. We used mathematical modeling to show that changes in ribosome density should occur along the mRNA at the point of regulation. We analyzed a Ribo-Seq data set obtained for mouse embryonic stem cells and showed that normalization by corresponding RNA-Seq can be used to improve the Ribo-Seq quality by removing bias introduced by deep-sequencing and alignment artifacts. After normalization, we applied a change point algorithm to detect changes in ribosome density present in individual mRNA ribosome profiles. Additional sequence and gene isoform information obtained from the UCSC Genome Browser allowed us to further categorize the detected changes into different mechanisms of regulation. In particular, we detected several mRNAs with known post-transcriptional regulation, e.g., premature termination for selenoprotein mRNAs and translational control of Atf4, but also several more mRNAs with hitherto unknown translational regulation. Additionally, our approach proved useful for identification of new transcript isoforms.

The effect of tRNA levels on decoding times of mRNA codons

The possible effect of transfer ribonucleic acid (tRNA) concentrations on codons decoding time is a fundamental biomedical research question; however, due to a large number of variables affecting this process and the non-direct relation between them, a conclusive answer to this question has eluded so far researchers in the field. In this study, we perform a novel analysis of the ribosome profiling data of four organisms which enables ranking the decoding times of different codons while filtering translational phenomena such as experimental biases, extreme ribosomal pauses and ribosome traffic jams. Based on this filtering, we show for the first time that there is a significant correlation between tRNA concentrations and the codons estimated decoding time both in prokaryotes and in eukaryotes in natural conditions (−0.38 to −0.66, all P values <0.006); in addition, we show that when considering tRNA concentrations, codons decoding times are not correlated with aminoacyl-tRNA levels. The reported results support the conjecture that translation efficiency is directly influenced by the tRNA levels in the cell. Thus, they should help to understand the evolution of synonymous aspects of coding sequences via the adaptation of their codons to the tRNA pool.

Exclusion processes with avalanches

In an exclusion process with avalanches, when a particle hops to a neighboring empty site which is adjacent to an island the particle on the other end of the island immediately hops, and if it joins another island this triggers another hop. There are no restrictions on the length of the islands and the duration of the avalanche. This process is well defined in the low-density region ρ < 1/2. We describe the nature of steady states (on a ring) and determine all correlation functions. For the asymmetric version of the process, we compute the steady state current, and we describe shock and rarefaction waves which arise in the evolution of the step-function initial profile. For the symmetric version, we determine the diffusion coefficient and examine the evolution of a tagged particle.

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

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

Entrainment to periodic initiation and transition rates in a computational model for gene translation

Periodic oscillations play an important role in many biomedical systems. Proper functioning of biological systems that respond to periodic signals requires the ability to synchronize with the periodic excitation. For example, the sleep/wake cycle is a manifestation of an internal timing system that synchronizes to the solar day. In the terminology of systems theory, the biological system must entrain or phase-lock to the periodic excitation. Entrainment is also important in synthetic biology. For example, connecting several artificial biological systems that entrain to a common clock may lead to a well-functioning modular system. The cell-cycle is a periodic program that regulates DNA synthesis and cell division. Recent biological studies suggest that cell-cycle related genes entrain to this periodic program at the gene translation level, leading to periodically-varying protein levels of these genes. The ribosome flow model (RFM) is a deterministic model obtained via a mean-field approximation of a stochastic model from statistical physics that has been used to model numerous processes including ribosome flow along the mRNA. Here we analyze the RFM under the assumption that the initiation and/or transition rates vary periodically with a common period . We show that the ribosome distribution profile in the RFM entrains to this periodic excitation. In particular, the protein synthesis pattern converges to a unique periodic solution with period . To the best of our knowledge, this is the first proof of entrainment in a mathematical model for translation that encapsulates aspects such as initiation and termination rates, ribosomal movement and interactions, and non-homogeneous elongation speeds along the mRNA. Our results support the conjecture that periodic oscillations in tRNA levels and other factors related to the translation process can induce periodic oscillations in protein levels, and may suggest a new approach for re-engineering genetic systems to obtain a desired, periodic, protein synthesis rate.

Occupation probabilities and fluctuations in the asymmetric simple inclusion process

The asymmetric simple inclusion process (ASIP), a lattice-gas model of unidirectional transport and aggregation, was recently proposed as an "inclusion" counterpart of the asymmetric simple exclusion process. In this paper we present an exact closed-form expression for the probability that a given number of particles occupies a given set of consecutive lattice sites. Our results are expressed in terms of the entries of Catalan's trapezoids-number arrays which generalize Catalan's numbers and Catalan's triangle. We further prove that the ASIP is asymptotically governed by the following: (i) an inverse square-root law of occupation, (ii) a square-root law of fluctuation, and (iii) a Rayleigh law for the distribution of interexit times. The universality of these results is discussed.

Most associations between transcript features and gene expression are monotonic

Dozens of previous studies in the field have dealt with the relations between transcript features and their expression. Indeed, understanding the way gene expression is encoded in transcripts should not only contribute to disciplines, such as functional genomics and molecular evolution, but also to biotechnology and human health. Previous studies in the field mainly aimed at predicting protein levels of genes based on their transcript features. Most of the models employed in this context assume that the effect of each transcript feature on gene expression is monotonic. In the current study we aim to understand, for the first time, if indeed the relations between transcript features (i.e., the UTRs and ORF) and measurements related to the different stages of gene expression is monotonic. To this end, we analyze 5432 transcript features and perform gene expression measurements (mRNA levels, ribosomal densities, protein levels, etc.) of 4367 S. cerevisiae genes. We use the Maximal Information Coefficient (MIC) in order to identify potential relations that are not necessarily linear or monotonic. Our analyses demonstrate that the relation between most transcript features and the examined gene expression measurements is monotonic (only up to 1-5% of the variables, with significance levels of 0.001, are non-monotonic); in addition, in the cases of deviation from monotonicity the relation/deviation is very weak. These results should help in guiding the development of computational gene expression modeling and engineering, and improve the understanding of this process. Furthermore, the relatively simple relations between a transcript's nucleotide composition and its expression should contribute towards better understanding of transcript evolution at the molecular level.

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.

The RNA polymerase flow model of gene transcription

Gene expression is a fundamental cellular process by which proteins are synthesized based on the information coded in the genes. The two major steps of this process are the transcription of the DNA segment corresponding to a gene to mRNA molecules and the translation of the mRNA molecules to proteins by the ribosome. Thus, understanding, modeling and engineering the different stages of this process have both important biotechnological applications and contributions to basic life science. In previous studies we have introduced the Homogenous Ribosome Flow Model (HRFM) and demonstrated its advantages in analyses of the translation process. In this study we introduce the RNA Polymerase Flow Model (RPFM), a non trivial extension of the HRFM, which also includes a backward flow and can be used for modeling transcription and maybe other similar processes. We compare the HRFM and the RPFM in the three regimes of the transcription process: rate limiting initiation, rate limiting elongation and rate limiting termination via a simulative and analytical analysis. In addition, based on experimental data, we show that RPFM is a better choice for modeling transcription process.

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.

Improved understanding of gene expression regulation using systems biology

This article reviews the current state of systems biology approaches, including the experimental tools used to generate 'omic' data and computational frameworks to interpret this data. Through illustrative examples, systems biology approaches to understand gene expression and gene expression regulation are discussed. Some of the challenges facing this field and the future opportunities in the systems biology era are highlighted.

Interplay between finite resources and a local defect in an asymmetric simple exclusion process

When particle flux is regulated by multiple factors such as particle supply and varying transport rate, it is important to identify the respective dominant regimes. We extend the well-studied totally asymmetric simple exclusion model to investigate the interplay between a controlled entrance and a local defect site. The model mimics cellular transport phenomena where there is typically a finite particle pool and nonuniform moving rates due to biochemical kinetics. Our simulations reveal regions where, despite an increasing particle supply, the current remains constant while particles redistribute in the system. Exploiting a domain wall approach with mean-field approximation, we provide a theoretical ground for our findings. The results in steady-state current and density profiles provide quantitative insights into the regulation of the transcription and translation process in bacterial protein synthesis.

Transcript features alone enable accurate prediction and understanding of gene expression in S. cerevisiae

Background

Gene expression is a central process in all living organisms. Central questions in the field are related to the way the expression levels of genes are encoded in the transcripts and affect their evolution, and the potential to predict expression levels solely by transcript features. In this study we analyze S. cerevisiae, a model organism with the most abundant relevant cellular and genomic measurements, to evaluate the accuracy in which expression levels can be predicted by different parts of the transcript. To this end, we perform various types of regression analyses based on a total of 5323 features of the transcript. The main advantage of the proposed predictors over previous ones is related to the accurate and comprehensive definitions of the relevant transcript features, which are based on biophysical knowledge of the gene transcription and translation processes, their modeling and evolution.

Results

Cross validation analyses of our predictors demonstrate that they achieve a correlation of 0.68/0.68/0.70/0.61/0.81 with mRNA levels, ribosomal density, protein levels, proteins per mRNA molecule (PPR), and ribosomal load (RL) respectively (all p-values <10-140). When we consider predictors that are based exclusively on the features related to different parts of the transcript (5'UTR, ORF, 3'UTR), the correlations with protein levels were 0.27/0.71/0.25 (all p-values <10-5), suggesting that the information in the UTRs is redundant, and features of the ORF alone yield similar predictions to the ones obtained based on the entire transcript.

Conclusions

The reported results demonstrate that in the analyzed model organism the expression levels of a gene are encoded in the transcript. Specifically, the prediction of a large fraction of the variance of the different gene expression steps based on transcript features alone is feasible in S. cerevisiae. We report dozens of novel transcript features related to expression levels predictions, demonstrating how such analyses can aid in understanding the gene expression process and its evolution, and how such predictors can be designed for other organisms in the future.

Control of ribosome traffic by position-dependent choice of synonymous codons

Messenger RNA (mRNA) encodes a sequence of amino acids by using codons. For most amino acids, there are multiple synonymous codons that can encode the amino acid. The translation speed can vary from one codon to another, thus there is room for changing the ribosome speed while keeping the amino acid sequence and hence the resulting protein. Recently, it has been noticed that the choice of the synonymous codon, via the resulting distribution of slow- and fast-translated codons, affects not only on the average speed of one ribosome translating the mRNA but also might have an effect on nearby ribosomes by affecting the appearance of 'traffic jams' where multiple ribosomes collide and form queues. To test this 'context effect' further, we here investigate the effect of the sequence of synonymous codons on the ribosome traffic by using a ribosome traffic model with codon-dependent rates, estimated from experiments. We compare the ribosome traffic on wild-type (WT) sequences and sequences where the synonymous codons were swapped randomly. By simulating translation of 87 genes, we demonstrate that the WT sequences, especially those with a high bias in codon usage, tend to have the ability to reduce ribosome collisions, hence optimizing the cellular investment in the translation apparatus. The magnitude of such reduction of the translation time might have a significant impact on the cellular growth rate and thereby have importance for the survival of the species.

Explicit expression for the steady-state translation rate in the infinite-dimensional homogeneous ribosome flow model

Gene translation is a central stage in the intracellular process of protein synthesis. Gene translation proceeds in three major stages: initiation, elongation, and termination. During the elongation step, ribosomes (intracellular macromolecules) link amino acids together in the order specified by messenger RNA (mRNA) molecules. The homogeneous ribosome flow model (HRFM) is a mathematical model of translation-elongation under the assumption of constant elongation rate along the mRNA sequence. The HRFM includes $(n)$ first-order nonlinear ordinary differential equations, where $(n)$ represents the length of the mRNA sequence, and two positive parameters: ribosomal initiation rate and the (constant) elongation rate. Here, we analyze the HRFM when $(n)$ goes to infinity and derive a simple expression for the steady-state protein synthesis rate. We also derive bounds that show that the behavior of the HRFM for finite, and relatively small, values of $(n)$ is already in good agreement with the closed-form result in the infinite-dimensional case. For example, for $(n=15)$, the relative error is already less than 4 percent. Our results can, thus, be used in practice for analyzing the behavior of finite-dimensional HRFMs that model translation. To demonstrate this, we apply our approach to estimate the mean initiation rate in M. musculus, finding it to be around 0.17 codons per second.

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.

Modeling cytoskeletal traffic: an interplay between passive diffusion and active transport

We introduce the totally asymmetric simple exclusion process with Langmuir kinetics on a network as a microscopic model for active motor protein transport on the cytoskeleton, immersed in the diffusive cytoplasm. We discuss how the interplay between active transport along a network and infinite diffusion in a bulk reservoir leads to a heterogeneous matter distribution on various scales: we find three regimes for steady state transport, corresponding to the scale of the network, of individual segments, or local to sites. At low exchange rates strong density heterogeneities develop between different segments in the network. In this regime one has to consider the topological complexity of the whole network to describe transport. In contrast, at moderate exchange rates the transport through the network decouples, and the physics is determined by single segments and the local topology. At last, for very high exchange rates the homogeneous Langmuir process dominates the stationary state. We introduce effective rate diagrams for the network to identify these different regimes. Based on this method we develop an intuitive but generic picture of how the stationary state of excluded volume processes on complex networks can be understood in terms of the single-segment phase diagram.

A traffic flow model for bio-polymerization processes

Bio-polymerization processes like transcription and translation are central to proper function of a cell. The speed at which the bio-polymer grows is affected both by the number of pauses of elongation machinery, as well the number of bio-polymers due to crowding effects. In order to quantify these effects in fast transcribing ribosome genes, we rigorously show that a classical traffic flow model is the limit of a mean occupancy ODE model. We compare the simulation of this model to a stochastic model and evaluate the combined effect of the polymerase density and the existence of pauses on the instantaneous transcription rate of ribosomal genes.