Slow sites in an exclusion process with limited resources

Availability versus carrying capacity: Phases of asymmetric exclusion processes competing for finite pools of resources

We address how the interplay between the finite availability and carrying capacity of particles at different parts of a spatially extended system can control the steady-state currents and density profiles in the one-dimensional current-carrying lanes connecting the different parts of the system. To study this, we set up a minimal model consisting of two particle reservoirs of the same finite carrying capacity connected by two equally sized antiparallel totally asymmetric simple exclusion processes (TASEPs). We focus on the steady-state currents and particle density profiles in the two TASEP lanes. The ensuing phases and the phase diagrams, which can be remarkably complex, are parametrized by the model parameters defining particle exchange between the TASEP lanes and the reservoirs and the filling fraction of the particles that determine the total resources available. These parameters may be tuned to make the densities of the two TASEP lanes globally uniform or piece-wise continuous in the form of a combination of a single localized domain wall and a spatially constant density or a pair of delocalized domain walls. Our model reveals that the two reservoirs can be preferentially populated or depopulated in the steady states.

Availability, storage capacity, and diffusion: Stationary states of an asymmetric exclusion process connected to two reservoirs

We explore how the interplay of finite availability, carrying capacity of particles at different parts of a spatially extended system, and particle diffusion between them control the steady-state currents and density profiles in a one-dimensional current-carrying channel connecting the different parts of the system. To study this, we construct a minimal model consisting of two particle reservoirs of finite carrying capacities connected by a totally asymmetric simple exclusion process (TASEP). In addition to particle transport via TASEP between the reservoirs, the latter can also directly exchange particles via Langmuir kinetics-like processes, modeling particle diffusion between them that can maintain a steady current in the system. We calculate the steady-state density profiles and the associated particle currents in the TASEP lane. The resulting phases and the phase diagrams are quite different from an open TASEP, and are characterized by the model parameters defining particle exchanges between the TASEP and the reservoirs, direct particle exchanges between the reservoirs, and the filling fraction of the particles that determines the total resources available. These parameters can be tuned to make the density on the TASEP lane globally uniform or piecewise continuous, and can make the two reservoirs preferentially populated or depopulated.

Ribosome Abundance Control in Prokaryotes

Cell growth is an essential phenotype of any unicellular organism and it crucially depends on precise control of protein synthesis. We construct a model of the feedback mechanisms that regulate abundance of ribosomes in E. coli, a prototypical prokaryotic organism. Since ribosomes are needed to produce more ribosomes, the model includes a positive feedback loop central to the control of cell growth. Our analysis of the model shows that there can be only two coexisting equilibrium states across all 23 parameters. This precludes the existence of hysteresis, suggesting that the ribosome abundance changes continuously with parameters. These states are related by a transcritical bifurcation, and we provide an analytic formula for parameters that admit either state.

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.

Trade-offs among transcription elongation rate, number, and duration of ubiquitous pauses on highly transcribed bacterial genes

In this paper, we study the limitations imposed on the transcription process by the presence of short ubiquitous pauses and crowding. These effects are especially pronounced in highly transcribed genes such as ribosomal genes (rrn) in fast growing bacteria. Our model indicates that the quantity and duration of pauses reported for protein-coding genes is incompatible with the average elongation rate observed in rrn genes. When maximal elongation rate is high, pause-induced traffic jams occur, increasing promoter occlusion, thereby lowering the initiation rate. This lowers average transcription rate and increases average transcription time. Increasing maximal elongation rate in the model is insufficient to match the experimentally observed average elongation rate in rrn genes. This suggests that there may be rrn-specific modifications to RNAP, which then experience fewer pauses, or pauses of shorter duration than those in protein-coding genes. We identify model parameter triples (maximal elongation rate, mean pause duration time, number of pauses) which are compatible with experimentally observed elongation rates. Average transcription time and average transcription rate are the model outputs investigated as proxies for cell fitness. These fitness functions are optimized for different parameter choices, opening up a possibility of differential control of these aspects of the elongation process, with potential evolutionary consequences. As an example, a gene's average transcription time may be crucial to fitness when the surrounding medium is prone to abrupt changes. This paper demonstrates that a functional relationship among the model parameters can be estimated using a standard statistical analysis, and this functional relationship describes the various trade-offs that must be made in order for the gene to control the elongation process and achieve a desired average transcription time. It also demonstrates the robustness of the system when a range of maximal elongation rates can be balanced with transcriptional pause data in order to maintain a desired fitness.

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.

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.

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.

Asymmetric exclusion process with global hopping

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

A max-plus model of ribosome dynamics during mRNA translation

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

The dynamics of supply and demand in mRNA translation

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

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