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

How total mRNA influences cell growth

Experimental observations tracing back to the 1960s imply that ribosome quantities play a prominent role in determining a cell's growth. Nevertheless, in biologically relevant scenarios, growth can also be influenced by the levels of mRNA and RNA polymerase. Here, we construct a quantitative model of biosynthesis providing testable scenarios for these situations. The model explores a theoretically motivated regime where RNA polymerases compete for genes and ribosomes for transcripts and gives general expressions relating growth rate, mRNA concentrations, ribosome, and RNA polymerase levels. On general grounds, the model predicts how the fraction of ribosomes in the proteome depends on total mRNA concentration and inspects an underexplored regime in which the trade-off between transcript levels and ribosome abundances sets the cellular growth rate. In particular, we show that the model predicts and clarifies three important experimental observations, in budding yeast and Escherichia coli bacteria: i) that the growth-rate cost of unneeded protein expression can be affected by mRNA levels, ii) that resource optimization leads to decreasing trends in mRNA levels at slow growth, and iii) that ribosome allocation may increase, stay constant, or decrease, in response to transcription-inhibiting antibiotics. Since the data indicate that a regime of joint limitation may apply in physiological conditions and not only to perturbations, we speculate that this regime is likely self-imposed.

Competition for resources in an exclusion model with biased lane-changing mechanism

The motivation for the proposed work is drawn from the attachment-detachment observed in biological and physical transport processes that entail finite resources. We investigate the influence of limited particle availability on particle dynamics within two parallel totally asymmetric simple exclusion lanes, with one lane incorporating only particle detachment and the other considering particle attachment. We establish a theoretical framework by employing vertical mean-field theory in conjunction with singular perturbation technique. The analytical findings are supported by numerical and stochastic validation using a finite-difference scheme and the Gillespie algorithm. By utilizing these approaches, we scrutinize various stationary properties, including particle densities, phase boundaries, and particle currents for both lanes. Our analysis reveals that the complexity of the phase diagram exhibits a nonmonotonic trend in the number of stationary phases as the particle count increases. Each phase diagram is constructed with respect to the intrinsic boundary parameters, illustrating both bulk and surface transitions occurring within the lanes. The interplay between finite resources and coupling mechanisms gives rise to two phases involving upward shock in one of the lanes, while two phases exhibit synchronized downward shock in both lanes. Finally, we delve into shock dynamics to comprehend critical phase transitions occurring in the system.

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

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

Exclusion processes on a roundabout traffic model with constrained resources

Motivated by the vehicular traffic phenomenon at roundabouts, we examine how the limited availability of resources affects the movement of two distinct types of particles on bidirectional lanes connected by two bridges, with each bridge specifically designated for the transportation of one species. To provide a theoretical ground for our findings, we employ a mean-field framework and successfully validate them through dynamic Monte Carlo simulations. Based on the theoretical analysis, we analytically derive various stationary properties, such as the particle densities, phase boundaries, and particle currents, for all the possible symmetric as well as asymmetric phases. The qualitative as well as quantitative behavior of the system is significantly affected by the constraint on the number of resources. The complexity of the phase diagram shows a nonmonotonic behavior with an increasing number of particles in the system. Analytical arguments enable the identification of several critical values for the total number of particles, leading to a qualitative change in the phase diagrams. The interplay of the finite resources and the bidirectional transport yields unanticipated and unusual features such as back-and-forth transition, the presence of two congested phases where particle movement is halted, as well as shock phases induced by boundaries and the bulk of the system. Also, it is found that spontaneous symmetry-breaking phenomena are induced even for very few particles in the system. Moreover, we thoroughly examine the location of shocks by varying the parameters controlling the system's boundaries, providing insights into possible phase transitions.

Multiple reentrance transitions in exclusion process with finite reservoir

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

Interplay of reservoirs in a bidirectional system

Motivated by the interplay of multiple species in several real world transport processes, we propose a bidirectional totally asymmetric simple exclusion process with two finite particle reservoirs regulating the inflow of oppositely directed particles corresponding to two different species. The system's stationary characteristics, such as densities, currents, etc., are investigated using a theoretical framework based on mean-field approximation and are supported by extensive Monte Carlo simulations. The impact of individual species populations, quantified by filling factor, has been comprehensively analyzed considering both equal and unequal conditions. For the equal case, the system exhibits the spontaneous symmetry-breaking phenomena and admits both symmetric as well as asymmetric phases. Moreover, the phase diagram exhibits a different asymmetric phase and displays a nonmonotonic variation in the number of phases with respect to the filling factor. For unequal filling factors, the phase schema can display at most five phases including a phase that shows maximal current for one of the species.

Reservoir crowding in a resource-constrained exclusion process with a dynamic defect

To understand the complicated transport processes that occur in biological and physical systems, we investigate a constrained totally asymmetric simple exclusion process with a stochastic defect particle. The defect particle might randomly emerge or vanish, resulting in a dynamic defect, and slows down the flow of moving particles when attached to the lattice. Using a mean-field technique, we examine the steady-state characteristics and boundary-layer analysis is provided to comprehend the properties of finite system. In a simplification, our theoretical method unifies three different parameter used to define the defect dynamics into one parameter termed the obstruction factor. It is found that the defect kinetics lead to emergence of phases where the current is defect restricted. The system shows nine phases overall, including bulk-induced and boundary-induced shock phases, with the phase schema showing no more than eight phases depending on the dynamics. We found that variation of obstruction does not lead to qualitative transition in the system, whereas the change in constraint on total particles affect the system qualitatively. All the theoretical outcomes have been validated using extensive Monte Carlo simulations.

Protein degradation sets the fraction of active ribosomes at vanishing growth

Growing cells adopt common basic strategies to achieve optimal resource allocation under limited resource availability. Our current understanding of such “growth laws” neglects degradation, assuming that it occurs slowly compared to the cell cycle duration. Here we argue that this assumption cannot hold at slow growth, leading to important consequences. We propose a simple framework showing that at slow growth protein degradation is balanced by a fraction of “maintenance” ribosomes. Consequently, active ribosomes do not drop to zero at vanishing growth, but as growth rate diminishes, an increasing fraction of active ribosomes performs maintenance. Through a detailed analysis of compiled data, we show that the predictions of this model agree with data from E. coli and S. cerevisiae. Intriguingly, we also find that protein degradation increases at slow growth, which we interpret as a consequence of active waste management and/or recycling. Our results highlight protein turnover as an underrated factor for our understanding of growth laws across kingdoms.

The idea that simple quantitative relationships relate cell physiology to cellular composition dates back to the 1950s, but the recent years saw a leap in our understanding of such “growth laws”, with relevant implications regarding the interdependence between growth, metabolism and biochemical networks. However, recent works on nutrient-limited growth mainly focused on laboratory conditions that are favourable to growth. Thus, our current mathematical understanding of the growth laws neglects protein degradation, under the argument that it occurs slowly compared to the timescale of the cell cycle. Instead, at slow growth the timescales of mass loss from protein degradation and dilution become comparable. In this work, we propose that protein degradation shapes the quantitative relationships between ribosome allocation and growth rate, and determines a fraction of ribosomes that do not contribute to growth and need to remain active to balance degradation.

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

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

Ribosome exit tunnel electrostatics

The impact of ribosome exit tunnel electrostatics on the protein elongation rate or on forces acting upon the nascent polypeptide chain are currently not fully elucidated. In the past, researchers have measured the electrostatic potential inside the ribosome polypeptide exit tunnel at a limited number of spatial points, at least in rabbit reticulocytes. Here we present a basic electrostatic model of the exit tunnel of the ribosome, providing a quantitative physical description of the tunnel interaction with the nascent proteins at all centro-axial points inside the tunnel. We show that a strong electrostatic screening is due to water molecules (not mobile ions) attracted to the ribosomal nucleic acid phosphate moieties buried in the immediate vicinity of the tunnel wall. We also show how the tunnel wall components and local ribosomal protein protrusions impact on the electrostatic potential profile and impede charged amino acid residues from progressing through the tunnel, affecting the elongation rate in a range of -40% to +85% when compared to the average elongation rate. The time spent by the ribosome to decode the genetic encrypted message is constrained accordingly. We quantitatively derive, at single-residue resolution, the axial forces acting on the nascent peptide from its particular sequence embedded in the tunnel. The model sheds light on how the experimental data point measurements of the potential are linked to the local structural chemistry of the inner wall, shape, and size of the tunnel. The model consistently connects experimental observations coming from different fields in molecular biology, x-ray crystallography, physical chemistry, biomechanics, and synthetic and multiomics biology. Our model should be a valuable tool to gain insight into protein synthesis dynamics, translational control, and the role of the ribosome's mechanochemistry in the cotranslational protein folding.

Accelerating Whole-Cell Simulations of mRNA Translation Using a Dedicated Hardware

In recent years, intracellular biophysical simulations have been used with increasing frequency not only for answering basic scientific questions but also in the field of synthetic biology. However, since these models include networks of interaction between millions of components, they are extremely time-consuming and cannot run easily on parallel computers. In this study, we demonstrate for the first time a novel approach addressing this challenge by using a dedicated hardware designed specifically to simulate such processes. As a proof of concept, we specifically focus on mRNA translation, which is the process consuming most of the energy in the cell. We design a hardware that simulates translation in Escherichia coli and Saccharomyces cerevisiae for thousands of mRNAs and ribosomes, which is in orders of magnitude faster than a similar software solution. With the sharp increase in the amount of genomic data available today and the complexity of the corresponding models inferred from them, we believe that the strategy suggested here will become common and can be used among others for simulating entire cells with all gene expression steps.

Asymmetric exclusion processes with fixed resources: Reservoir crowding and steady states

We study the reservoir crowding effect by considering the nonequilibrium steady states of an asymmetric exclusion process (TASEP) coupled to a reservoir with fixed available resources and dynamically coupled entry and exit rate. We elucidate how the steady states are controlled by the interplay between the coupled entry and exit rates, both being dynamically controlled by the reservoir population, and the fixed total particle number in the system. The TASEP can be in the low-density, high-density, maximal current, and shock phases. We show that such a TASEP is different from an open TASEP for all values of available resources: here the TASEP can support only localized domain walls for any (finite) amount of resources that do not tend to delocalize even for large resources, a feature attributed to the form of the dynamic coupling between the entry and exit rates. Furthermore, in the limit of infinite resources, in contrast to an open TASEP, the TASEP can be found in its high-density phase only for any finite values of the control parameters, again as a consequence of the coupling between the entry and exit rates.

Exclusion process on two intersecting lanes with constrained resources: Symmetry breaking and shock dynamics

We present a study of the exclusion process on a peculiar topology of network with two intersecting lanes, competing for the particles in a reservoir with finite capacity. To provide a theoretical ground for our findings, we exploit mean-field approximation along with domain-wall theory. The stationary properties of the system, including phase transitions, density profiles, and position of the domain wall are derived analytically. Under the similar dynamical rules, the particles of both lanes interact only at the intersected site. The symmetry of the system is maintained until the number of particles do not exceed the total number of sites. However, beyond this, the symmetry breaking phenomenon occurs, resulting in the appearance of asymmetric phases and continues to persist even for an infinite number of particles. The complexity of the phase diagram shows a nonmonotonic behavior with an increasing number of particles in the system. A bulk induced shock appears in a symmetric phase, whereas, a boundary induced shock is observed in the symmetric as well as the asymmetric phase. Monitoring the location of localized shock with increasing entry of particles, we explain the possible phase transitions. The theoretical results are supported by extensive Monte Carlo simulations and explained using simple physical arguments.

Whole cell biophysical modeling of codon-tRNA competition reveals novel insights related to translation dynamics

The importance of mRNA translation models has been demonstrated across many fields of science and biotechnology. However, a whole cell model with codon resolution and biophysical dynamics is still lacking. We describe a whole cell model of translation for E. coli. The model simulates all major translation components in the cell: ribosomes, mRNAs and tRNAs. It also includes, for the first time, fundamental aspects of translation, such as competition for ribosomes and tRNAs at a codon resolution while considering tRNAs wobble interactions and tRNA recycling. The model uses parameters that are tightly inferred from large scale measurements of translation. Furthermore, we demonstrate a robust modelling approach which relies on state-of-the-art practices of translation modelling and also provides a framework for easy generalizations. This novel approach allows simulation of thousands of mRNAs that undergo translation in the same cell with common resources such as ribosomes and tRNAs in feasible time. Based on this model, we demonstrate, for the first time, the direct importance of competition for resources on translation and its accurate modelling. An effective supply-demand ratio (ESDR) measure, which is related to translation factors such as tRNAs, has been devised and utilized to show superior predictive power in complex scenarios of heterologous gene expression. The devised model is not only more accurate than the existing models, but, more importantly, provides a framework for analyzing complex whole cell translation problems and variables that haven't been explored before, making it important in various biomedical fields.

mRNA translation is a fundamental process in all living organisms and the importance of its modeling has been demonstrated across many fields of science and biotechnology. Specifically, modeling a whole cell context with a high resolution has been a great challenge in the field, making many important problems un-addressable. In this study we devised a novel model, which allows, for the first time, simultaneous simulation of thousands of mRNAs, along with various bio-physical aspects that affect translation (such as codon-resolution dynamics and shared resources pool of both ribosomes and tRNAs). We demonstrated (using experimental data) that this model is more accurate than existing ones, and, more importantly, provides a framework for addressing complex translation problems (such as heterologous expression) at whole cell scale and in reasonable time. We demonstrated the model using E. coli data, but the model can be easily tailored to other organisms as well. Our model addresses an urgent unmet need for biophysically accurate whole cell translation model with resources coupling and has potential applications in many fields, including medicine and biotechnology.

Modelling and measuring intracellular competition for finite resources during gene expression

Dissecting the competition between genes for shared expressional resources is of fundamental importance for understanding the interplay between cellular components. Owing to the relationship between gene expression and cellular fitness, genomes are shaped by evolution to improve resource allocation. Whereas experimental approaches to investigate intracellular competition require technical resources and human expertise, computational models and in silico simulations allow vast numbers of experiments to be carried out and controlled easily, and with significantly reduced costs. Thus, modelling competition has a pivotal role in understanding the effects of competition on the biophysics of the cell. In this article, we review various computational models proposed to describe the different types of competition during gene expression. We also present relevant synthetic biology experiments and their biotechnological implications, and discuss the open questions in the field.

A stochastic model for simulating ribosome kinetics in vivo

Computational modelling of in vivo protein synthesis is highly complicated, as it requires the simulation of ribosomal movement over the entire transcriptome, as well as consideration of the concentration effects from 40+ different types of tRNAs and numerous other protein factors. Here I report on the development of a stochastic model for protein translation that is capable of simulating the dynamical process of in vivo protein synthesis in a prokaryotic cell containing several thousand unique mRNA sequences, with explicit nucleotide information for each, and report on a number of biological predictions which are beyond the scope of existing models. In particular, I show that, when the complex network of concentration dependent interactions between elongation factors, tRNAs, ribosomes, and other factors required for protein synthesis are included in full detail, several biological phenomena, such as the increasing peptide elongation rate with bacterial growth rate, are predicted as emergent properties of the model. The stochastic model presented here demonstrates the importance of considering the translational process at this level of detail, and provides a platform to interrogate various aspects of translation that are difficult to study in more coarse-grained models.

An equilibrium model for ribosome competition

The number of ribosomes in a cell is considered as limiting, and gene expression is thus largely determined by their cellular concentration. In this work we develop a toy model to study the trade-off between the ribosomal supply and the demand of the translation machinery, dictated by the composition of the transcript pool. Our equilibrium framework is useful to highlight qualitative behaviours and new means of gene expression regulation determined by the fine balance of this trade-off. We also speculate on the possible impact of these mechanisms on cellular physiology.

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.

Driven transport on a flexible polymer with particle recycling: A model inspired by transcription and translation

Many theoretical works have attempted to coarse grain gene expression at the level of transcription and translation via frameworks based on exclusion processes. Usually in these models the three-dimensional conformation of the substrates (DNA and mRNA) is neglected, and particles move on a static unidimensional lattice in contact to an infinite reservoir. In this work we generalize the paradigmatic exclusion process and study the transport of particles along a unidimensional polymerlike flexible lattice immersed in a three-dimensional particle reservoir. We study the recycling of particles in the reservoir, how the transport is influenced by the global conformation of the lattice, and, in turn, how particle density dictates the structure of the polymer.

Networks of ribosome flow models for modeling and analyzing intracellular traffic

The ribosome flow model with input and output (RFMIO) is a deterministic dynamical system that has been used to study the flow of ribosomes during mRNA translation. The input of the RFMIO controls its initiation rate and the output represents the ribosome exit rate (and thus the protein production rate) at the 3′ end of the mRNA molecule. The RFMIO and its variants encapsulate important properties that are relevant to modeling ribosome flow such as the possible evolution of “traffic jams” and non-homogeneous elongation rates along the mRNA molecule, and can also be used for studying additional intracellular processes such as transcription, transport, and more. Here we consider networks of interconnected RFMIOs as a fundamental tool for modeling, analyzing and re-engineering the complex mechanisms of protein production. In these networks, the output of each RFMIO may be divided, using connection weights, between several inputs of other RFMIOs. We show that under quite general feedback connections the network has two important properties: (1) it admits a unique steady-state and every trajectory converges to this steady-state; and (2) the problem of how to determine the connection weights so that the network steady-state output is maximized is a convex optimization problem. These mathematical properties make these networks highly suitable as models of various phenomena: property (1) means that the behavior is predictable and ordered, and property (2) means that determining the optimal weights is numerically tractable even for large-scale networks. For the specific case of a feed-forward network of RFMIOs we prove an additional useful property, namely, that there exists a spectral representation for the network steady-state, and thus it can be determined without any numerical simulations of the dynamics. We describe the implications of these results to several fundamental biological phenomena and biotechnological objectives.

Self-organized system-size oscillation of a stochastic lattice-gas model

The totally asymmetric simple exclusion process (TASEP) is a paradigmatic stochastic model for nonequilibrium physics, and has been successfully applied to describe active transport of molecular motors along cytoskeletal filaments. Building on this simple model, we consider a two-lane lattice-gas model that couples directed transport (TASEP) to diffusive motion in a semiclosed geometry, and simultaneously accounts for spontaneous growth and particle-induced shrinkage of the system's size. This particular extension of the TASEP is motivated by the question of how active transport and diffusion might influence length regulation in confined systems. Surprisingly, we find that the size of our intrinsically stochastic system exhibits robust temporal patterns over a broad range of growth rates. More specifically, when particle diffusion is slow relative to the shrinkage dynamics, we observe quasiperiodic changes in length. We provide an intuitive explanation for the occurrence of these self-organized temporal patterns, which is based on the imbalance between the diffusion and shrinkage speed in the confined geometry. Finally, we formulate an effective theory for the oscillatory regime, which explains the origin of the oscillations and correctly predicts the dependence of key quantities, such as the oscillation frequency, on the growth rate.

Limited Resources Induce Bistability in Microtubule Length Regulation

The availability of protein is an important factor for the determination of the size of the mitotic spindle. Involved in spindle-size regulation is kinesin-8, a molecular motor and microtubule (MT) depolymerase, which is known to tightly control MT length. Here, we propose and analyze a theoretical model in which kinesin-induced MT depolymerization competes with spontaneous polymerization while supplies of both tubulin and kinesin are limited. In contrast to previous studies where resources were unconstrained, we find that, for a wide range of concentrations, MT length regulation is bistable. We test our predictions by conducting in vitro experiments and find that the bistable behavior manifests in a bimodal MT length distribution.

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.

Multispecies exclusion process with fusion and fission of rods: A model inspired by intraflagellar transport

We introduce a multispecies exclusion model where length-conserving probabilistic fusion and fission of the hard rods are allowed. Although all rods enter the system with the same initial length ℓ=1, their length can keep changing, because of fusion and fission, as they move in a step-by-step manner towards the exit. Two neighboring hard rods of lengths ℓ_{1} and ℓ_{2} can fuse into a single rod of longer length ℓ=ℓ_{1}+ℓ_{2} provided ℓ≤N. Similarly, length-conserving fission of a rod of length ℓ^{'}≤N results in two shorter daughter rods. Based on the extremum current hypothesis, we plot the phase diagram of the model under open boundary conditions utilizing the results derived for the same model under periodic boundary condition using mean-field approximation. The density profile and the flux profile of rods are in excellent agreement with computer simulations. Although the fusion and fission of the rods are motivated by similar phenomena observed in intraflagellar transport (IFT) in eukaryotic flagella, this exclusion model is too simple to account for the quantitative experimental data for any specific organism. Nevertheless, the concepts of "flux profile" and "transition zone" that emerge from the interplay of fusion and fission in this model are likely to have important implications for IFT and for other similar transport phenomena in long cell protrusions.

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.

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.

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.

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.

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.

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.

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.

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.

Molecular mechanisms for microtubule length regulation by kinesin-8 and XMAP215 proteins

The cytoskeleton is regulated by a plethora of enzymes that influence the stability and dynamics of cytoskeletal filaments. How microtubules (MTs) are controlled is of particular importance for mitosis, during which dynamic MTs are responsible for proper segregation of chromosomes. Molecular motors of the kinesin-8 protein family have been shown to depolymerize MTs in a length-dependent manner, and recent experimental and theoretical evidence suggests a possible role for kinesin-8 in the dynamic regulation of MTs. However, so far the detailed molecular mechanisms of how these molecular motors interact with the growing MT tip remain elusive. Here we show that two distinct scenarios for the interactions of kinesin-8 with the MT tip lead to qualitatively different MT dynamics, including accurate length control as well as intermittent dynamics. We give a comprehensive analysis of the regimes where length regulation is possible and characterize how the stationary length depends on the biochemical rates and the bulk concentrations of the various proteins. For a neutral scenario, where MTs grow irrespective of whether the MT tip is occupied by a molecular motor, length regulation is possible only for a narrow range of biochemical rates, and, in particular, limited to small polymerization rates. By contrast, for an inhibition scenario, where the presence of a motor at the MT tip inhibits MT growth, the regime where length regulation is possible is extremely broad and includes high growth rates. These results also apply to situations where a polymerizing enzyme like XMAP215 and kinesin-8 mutually exclude each other from the MT tip. Moreover, we characterize the differences in the stochastic length dynamics between the two scenarios. While for the neutral scenario length is tightly controlled, length dynamics is intermittent for the inhibition scenario and exhibits extended periods of MT growth and shrinkage. On a broader perspective, the set of models established in this work quite generally suggest that mutual exclusion of molecules at the ends of cytoskeletal filaments is an important factor for filament dynamics and regulation.

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.

Motor protein traffic regulation by supply-demand balance of resources

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

Nonequilibrium stochastic model for tRNA binding time statistics

Protein translation is one of the most important processes in cell life, but despite being well-understood biochemically, the implications of its intrinsic stochastic nature have not been fully elucidated. In this paper we develop a microscopic and stochastic model which describes a crucial step in protein translation, namely the binding of the tRNA to the ribosome. Our model explicitly takes into consideration tRNA recharging dynamics, spatial inhomogeneity, and stochastic fluctuations in the number of charged tRNAs around the ribosome. By analyzing this nonequilibrium system we are able to derive the statistical distribution of the times needed by the tRNAs to bind to the ribosome, and to show that it deviates from an exponential due to the coupling between the fluctuations of charged and uncharged populations of tRNA.

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.

Position dependence of the particle density in a double-chain section of a linear network in a totally asymmetric simple exclusion process

We report here results on the study of the totally asymmetric simple exclusion process, defined on an open network, consisting of head and tail simple-chain segments with a double-chain section inserted in between. Results of numerical simulations for relatively short chains reveal an interesting feature of the network. When the current through the system takes its maximum value, a simple translation of the double-chain section forward or backward along the network leads to a sharp change in the shape of the density profiles in the parallel chains, thus affecting the total number of particles in that part of the network. In the symmetric case of equal injection and ejection rates α=β>1/2 and equal lengths of the head and tail sections, the density profiles in the two parallel chains are almost linear, characteristic of the coexistence line (shock phase). Upon moving the section forward (backward), their shape changes to the one typical for the high- (low-) density phases of a simple chain. The total bulk density of particles in a section with a large number of parallel chains is evaluated too. The observed effect might have interesting implications for the traffic flow control as well as for biological transport processes in living cells. An explanation of this phenomenon is offered in terms of a finite-size dependence of the effective injection and ejection rates at the ends of the double-chain section.

Ribosome flow model with positive feedback

Eukaryotic mRNAs usually form a circular structure; thus, ribosomes that terminate translation at the 3' end can diffuse with increased probability to the 5' end of the transcript, initiating another cycle of translation. This phenomenon describes ribosomal flow with positive feedback--an increase in the flow of ribosomes terminating translating the open reading frame increases the ribosomal initiation rate. The aim of this paper is to model and rigorously analyse translation with feedback. We suggest a modified version of the ribosome flow model, called the ribosome flow model with input and output. In this model, the input is the initiation rate and the output is the translation rate. We analyse this model after closing the loop with a positive linear feedback. We show that the closed-loop system admits a unique globally asymptotically stable equilibrium point. From a biophysical point of view, this means that there exists a unique steady state of ribosome distributions along the mRNA, and thus a unique steady-state translation rate. The solution from any initial distribution will converge to this steady state. The steady-state distribution demonstrates a decrease in ribosome density along the coding sequence. For the case of constant elongation rates, we obtain expressions relating the model parameters to the equilibrium point. These results may perhaps be used to re-engineer the biological system in order to obtain a desired translation rate.

Entrainment and unit velocity: surprises in an accelerated exclusion process

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

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.