Sensitivity of mRNA Translation

Modeling mRNA Translation With Ribosome Abortions

We derive a deterministic mathematical model for the flow of ribosomes along a mRNA called the ribosome flow model with extended objects and abortions (RFMEOA). This model incorporates important cellular features such as every ribosome covers several codons and they may detach from various regions along the track due to more realistic biological situations including phenomena of ribosome-ribosome collisions. We prove that the ribosome density profile along the mRNA in the RFMEOA and in particular, the protein production rate converge to a unique steady-state. Simulations of the RFMEOA demonstrate a surprising result that an increase in the initiation rate may sometimes lead to a decrease in the production rate. We believe that this model could be helpful to provide insight into the effects of premature termination on the protein expression and be useful for understanding and re-engineering the translation process.

On the gain of entrainment in the n-dimensional ribosome flow model

The ribosome flow model (RFM) is a phenomenological model for the flow of particles along a one-dimensional chain of n sites. It has been extensively used to study ribosome flow along the mRNA molecule during translation. When the transition rates along the chain are time-varying and jointly T-periodic the RFM entrains, i.e. every trajectory of the RFM converges to a unique T-periodic solution that depends on the transition rates, but not on the initial condition. In general, entrainment to periodic excitations like the 24 h solar day or the 50 Hz frequency of the electric grid is important in numerous natural and artificial systems. An interesting question, called the gain of entrainment (GOE) in the RFM, is whether proper coordination of the periodic translation rates along the mRNA can lead to a larger average protein production rate. Analysing the GOE in the RFM is non-trivial and partial results exist only for the RFM with dimensions n = 1, 2. We use a new approach to derive several results on the GOE in the general n-dimensional RFM. Perhaps surprisingly, we rigorously characterize several cases where there is no GOE, so to maximize the average production rate in these cases, the best choice is to use constant transition rates all along the chain.

Translation in the cell under fierce competition for shared resources: a mathematical model

During translation, mRNAs 'compete' for shared resources. Under stress conditions, during viral infection and also in high-throughput heterologous gene expression, these resources may become scarce, e.g. the pool of free ribosomes is starved, and then the competition may have a dramatic effect on the global dynamics of translation in the cell. We model this scenario using a network that includes m ribosome flow models (RFMs) interconnected via a pool of free ribosomes. Each RFM models ribosome flow along an mRNA molecule, and the pool models the shared resource. We assume that the number of mRNAs is large, so many ribosomes are attached to the mRNAs, and the pool is starved. Our analysis shows that adding an mRNA has an intricate effect on the total protein production. The new mRNA produces new proteins, but the other mRNAs produce less proteins, as the pool that feeds these mRNAs now has a smaller abundance of ribosomes. As the number of mRNAs increases, the marginal utility of adding another mRNA diminishes, and the total protein production rate saturates to a limiting value. We demonstrate our approach using an example of insulin protein production in a cell-free system.

Determinants of efficient modulation of ribosomal traffic jams

mRNA translation is the process which consumes most of the cellular energy. Thus, this process is under strong evolutionary selection for its optimization and rational optimization or reduction of the translation efficiency can impact the cell growth rate. Algorithms for modulating cell growth rate can have various applications in biotechnology, medicine, and agriculture. In this study, we demonstrate that the analysis of these algorithms can also be used for understanding translation.

We specifically describe and analyze various generic algorithms, based on comprehensive computational models and whole cell simulations of translation, for introducing silent mutations that can either reduce or increase ribosomal traffic jams along the mRNA. As a result, more or less resources are available, for the cell, promoting improved or reduced cells growth-rate, respectively. We then explore the cost of these algorithms' performance, in terms of their computational time, the number of mutations they introduce, the modified genomic region, the effect on local translation rates, and the properties of the modified genes.

Among others, we show that mRNA levels of a gene are much stronger predictors for the effect of its engineering on the ribosomal pool than the ribosomal density of the gene. We also demonstrate that the mutations at the ends of the coding regions have a stronger effect on the ribosomal pool. Furthermore, we report two optimization algorithms that exhibit a tread-off between the number of mutations they introduce and their executing time.

The reported results here are fundamental both for understanding the biophysics and evolution of translation, as well as for developing efficient approaches for its engineering.

Variability in mRNA translation: a random matrix theory approach

The rate of mRNA translation depends on the initiation, elongation, and termination rates of ribosomes along the mRNA. These rates depend on many "local" factors like the abundance of free ribosomes and tRNA molecules in the vicinity of the mRNA molecule. All these factors are stochastic and their experimental measurements are also noisy. An important question is how protein production in the cell is affected by this considerable variability. We develop a new theoretical framework for addressing this question by modeling the rates as identically and independently distributed random variables and using tools from random matrix theory to analyze the steady-state production rate. The analysis reveals a principle of universality: the average protein production rate depends only on the of the set of possible values that the random variable may attain. This explains how total protein production can be stabilized despite the overwhelming stochasticticity underlying cellular processes.

The Correlation Between DsRed mRNA Levels and Transient DsRed Protein Expression in Plants Depends on Leaf Age and the 5' Untranslated Region

The yield of recombinant proteins in plants determines their economic competitiveness as a production platform compared to microbes and mammalian cells. The promoter, untranslated regions (UTRs) and codon usage can all contribute to the yield, but potential interactions among these components have not been examined in detail. Here the effect of two promoters (35SS and nos) and four 5'UTRs on the spatiotemporal expression of DsRed mRNA and the accumulation of DsRed protein during transient expression in tobacco (Nicotiana tabacum) mediated by Agrobacterium tumefaciens is investigated. The authors found that the mRNA levels peaked 2-3 days post-infiltration (dpi), and rapidly declined thereafter, whereas DsRed protein was first detected after ≈3 days and concentrations continued to increase until at least 5 dpi. This temporal decoupling of mRNA and protein expression was strongest in the older leaves, which also produced the lowest DsRed yields. The accumulation of DsRed linearly correlated with mRNA levels in all but the youngest leaves, where more DsRed was synthesized per mRNA molecule. This was the case for both promoters, although the nos promoter had a higher protein/mRNA ratio than the 35SS promoter. Furthermore, the type of 5'UTR affected DsRed protein accumulation by 50% starting from similar levels of mRNA. The authors concluded that DsRed mRNA levels are not the limiting factor for DsRed protein expression in plants, but that translation-associated processes such as initiation, elongation, and release are bottlenecks that should be addressed in future studies.

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.

Genome-Scale Analysis of Perturbations in Translation Elongation Based on a Computational Model

Perturbations play an important role both in engineered systems and cellular processes. Thus, understanding their effect on protein synthesis should contribute to all biomedical disciplines. Here we describe the first genome-scale analysis of perturbations in translation-related factors in S. cerevisiae. To this end, we used simulations based on a computational model that takes into consideration the fundamental stochastic and bio-physical nature of translation. We found that the initiation rate has a key role in determining the sensitivity to perturbations. For low initiation rates, the first codons of the coding region dominate the sensitivity, which is highly correlated with the ratio between initiation rate and mean elongation rate (r = −0.95), with the open reading frame (ORF) length (r = 0.6) and with protein abundance (r = 0.45). For high initiation rates (that may rise, for example, due to cellular growth), the sensitivity of a gene is dominated by all internal codons and is correlated with the decoding rate. We found that various central intracellular functions are associated with the sensitivity: for example, both genes that are sensitive and genes that are robust to perturbations are over-represented in the group of genes related to translation regulation; this may suggest that robustness to perturbations is a trait that undergoes evolutionary selection in relation to the function of the encoded protein. We believe that the reported results, due to their quantitative value and genome-wide perspective, should contribute to disciplines such as synthetic biology, functional genomics, comparative genomics and molecular evolution.

Controllability Analysis and Control Synthesis for the Ribosome Flow Model

The ribosomal density along different parts of the coding regions of the mRNA molecule affects various fundamental intracellular phenomena including: protein production rates, global ribosome allocation and organismal fitness, ribosomal drop off, co-translational protein folding, mRNA degradation, and more. Thus, regulating translation in order to obtain a desired ribosomal profile along the mRNA molecule is an important biological problem. We study this problem by using a dynamical model for mRNA translation, called the ribosome flow model (RFM). In the RFM, the mRNA molecule is modeled as an ordered chain of $n$ sites. The RFM includes $n$ state-variables describing the ribosomal density profile along the mRNA molecule, and the transition rates from each site to the next are controlled by $n+1$ positive constants. To study the problem of controlling the density profile, we consider some or all of the transition rates as time-varying controls. We consider the following problem: given an initial and a desired ribosomal density profile in the RFM, determine the time-varying values of the transition rates that steer the system to the desired density profile, if they exist. More specifically, we consider two control problems. In the first, all transition rates can be regulated separately, and the goal is to steer the ribosomal density profile and the protein production rate from a given initial value to a desired value. In the second problem, one or more transition rates are jointly regulated by a single scalar control, and the goal is to steer the production rate to a desired value within a certain set of feasible values. In the first case, we show that the system is controllable, i.e., the control is powerful enough to steer the system to any desired value in finite time, and provide simple closed-form expressions for constant positive control functions (or transition rates) that asymptotically steer the system to the desired value. In the second case, we show that the system is controllable, and provide a simple algorithm for determining the constant positive control value that asymptotically steers the system to the desired value. We discuss some of the biological implications of these results.

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.

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.

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.

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.