Efficient manipulations of synonymous mutations for controlling translation rate: an analytical approach

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.

Modeling transport of extended interacting objects with drop-off phenomenon

We study a deterministic framework for important cellular transport phenomena involving a large number of interacting molecules called the excluded flow of extended interacting objects with drop-off effect (EFEIOD). This model incorporates many realistic features of biological transport process including the length of biological "particles" and the fact that they can detach along the biological 'tracks'. The flow between the consecutive sites is unidirectional and is described by a "soft" simple exclusion principle and by repelling or attracting forces between neighboring particles. We show that the model admits a unique steady-state. Furthermore, if the parameters are periodic with common period T, then the steady-state profile converge to a unique periodic solution of period T. Simulations of the EFEIOD demonstrate several non-trivial effects of the interactions on the system steady-state profile. For example, detachment rates may help in increasing the steady-state flow by alleviating traffic jams that can exist due to several reasons like bottleneck rate or interactive forces between the particles. We also analyze the special case of our model, when there are no forces exerted by neighboring particles, and called it as the ribosome flow model of extended objects with drop-off effect (RFMEOD), and study the sensitivity of its steady-state to variations in the parameters.

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.

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.

Computational analysis of the oscillatory behavior at the translation level induced by mRNA levels oscillations due to finite intracellular resources

Recent studies have demonstrated how the competition for the finite pool of available gene expression factors has important effect on fundamental gene expression aspects. In this study, based on a whole-cell model simulation of translation in S. cerevisiae, we evaluate for the first time the expected effect of mRNA levels fluctuations on translation due to the finite pool of ribosomes. We show that fluctuations of a single gene or a group of genes mRNA levels induce periodic behavior in all S. cerevisiae translation factors and aspects: the ribosomal densities and the translation rates of all S. cerevisiae mRNAs oscillate. We numerically measure the oscillation amplitudes demonstrating that fluctuations of endogenous and heterologous genes can cause a significant fluctuation of up to 50% in the steady-state translation rates of the rest of the genes. Furthermore, we demonstrate by synonymous mutations that oscillating the levels of mRNAs that experience high ribosomal occupancy (e.g. ribosomal “traffic jam”) induces the largest impact on the translation of the S. cerevisiae genome.

The results reported here should provide novel insights and principles related to the design of synthetic gene expression circuits and related to the evolutionary constraints shaping gene expression of endogenous genes.

Each cell contains a limited number of macromolecules and factors that participate in the gene expression process. These expression resources are shared between the different molecules that encode the genetic code, resulting in non-trivial couplings and competitions between the different gene expression stages. Such competitions should be considered when analyzing the cellular economy of the cell, the genome evolution, and the design of synthetic expression circuits. Here we study the effect of couplings and competitions for ribosomes by performing a whole-cell simulation of translation of S. cerevisiae, with parameters estimated from experimental data. We demonstrate that by periodically changing the mRNA levels of a single gene (endogenous or heterologous) or a set of genes, the translation of all S. cerevisiae genes are affected in a periodic manner. We numerically estimate the exact impact of the mRNA levels periodicity on the translation process dynamics, as well as on the dynamics of the free ribosomal pool and the way it is affected by parameters such as the codon composition of the oscillating gene, its initiation rate and mRNA levels. Furthermore, we show that the codon compositions of synthetically highly expressed heterologous genes that are expected to oscillate must be carefully considered. For example, synonymous mutations resulting in “traffic jams” of ribosomes along the fluctuated mRNAs may cause significant fluctuations of up to 50% in the steady-state translation rates of all genes.

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 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.

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

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

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

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

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

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

D-Tailor: automated analysis and design of DNA sequences

Motivation: Current advances in DNA synthesis, cloning and sequencing technologies afford high-throughput implementation of artificial sequences into living cells. However, flexible computational tools for multi-objective sequence design are lacking, limiting the potential of these technologies.

Results: We developed DNA-Tailor (D-Tailor), a fully extendable software framework, for property-based design of synthetic DNA sequences. D-Tailor permits the seamless integration of multiple sequence analysis tools into a generic Monte Carlo simulation that evolves sequences toward any combination of rationally defined properties. As proof of principle, we show that D-Tailor is capable of designing sequence libraries comprising all possible combinations among three different sequence properties influencing translation efficiency in Escherichia coli. The capacity to design artificial sequences that systematically sample any given parameter space should support the implementation of more rigorous experimental designs.

Availability: Source code is available for download at https://sourceforge.net/projects/dtailor/

Contact:aparkin@lbl.gov or cambray.guillaume@gmail.com

Supplementary information:Supplementary data are available at Bioinformatics online (D-Tailor Tutorial).

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

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

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.

On the steady-state distribution in the homogeneous ribosome flow model

A central biological process in all living organisms is gene translation. Developing a deeper understanding of this complex process may have ramifications to almost every biomedical discipline. Reuveni et al. recently proposed a new computational model of gene translation called the Ribosome Flow Model (RFM). In this paper, we consider a particular case of this model, called the Homogeneous Ribosome Flow Model (HRFM). From a biological viewpoint, this corresponds to the case where the transition rates of all the coding sequence codons are identical. This regime has been suggested recently based on experiments in mouse embryonic cells. We consider the steady-state distribution of the HRFM. We provide formulas that relate the different parameters of the model in steady state. We prove the following properties: 1) the ribosomal density profile is monotonically decreasing along the coding sequence; 2) the ribosomal density at each codon monotonically increases with the initiation rate; and 3) for a constant initiation rate, the translation rate monotonically decreases with the length of the coding sequence. In addition, we analyze the translation rate of the HRFM at the limit of very high and very low initiation rate, and provide explicit formulas for the translation rate in these two cases. We discuss the relationship between these theoretical results and biological findings on the translation process.

Stability analysis of the ribosome flow model

Gene translation is a central process in all living organisms. Developing a better understanding of this complex process may have ramifications to almost every biomedical discipline. Recently, Reuveni et al. proposed a new computational model of this process called the ribosome flow model (RFM). In this study, we show that the dynamical behavior of the RFM is relatively simple. There exists a unique equilibrium point e and every trajectory converges to e. Furthermore, convergence is monotone in the sense that the distance to e can never increase. This qualitative behavior is maintained for any feasible set of parameter values, suggesting that the RFM is highly robust. Our analysis is based on a contraction principle and the theory of monotone dynamical systems. These analysis tools may prove useful in studying other properties of the RFM as well as additional intracellular biological processes.