Yoram Zarai

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.

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.

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.

Ribosome Flow Model on a Ring

The asymmetric simple exclusion process (ASEP) is an important model from statistical physics describing particles that hop randomly from one site to the next along an ordered lattice of sites, but only if the next site is empty. ASEP has been used to model and analyze numerous multiagent systems with local interactions including the flow of ribosomes along the mRNA strand. In ASEP with periodic boundary conditions a particle that hops from the last site returns to the first one. The mean field approximation of this model is referred to as the ribosome flow model on a ring (RFMR). The RFMR may be used to model both synthetic and endogenous gene expression regimes. We analyze the RFMR using the theory of monotone dynamical systems. We show that it admits a continuum of equilibrium points and that every trajectory converges to an equilibrium point. Furthermore, we show that it entrains to periodic transition rates between the sites. We describe the implications of the analysis results to understanding and engineering cyclic mRNA translation in-vitro and in-vivo.

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.

Controlling mRNA Translation

The ribosomal density along the coding region of the mRNA molecule affect various fundamental intracellular phenomena including: protein production rates, organismal fitness, ribosomal drop off, and co-translational protein folding. Thus, regulating translation in order to obtain a desired ribosomal profile along the mRNA molecule is an important biological problem. We study this problem using a model for mRNA translation, called the ribosome flow model (RFM). In the RFM, the mRNA molecule is modeled as chain of n sites. The n state-variables describe the ribosomal density profile along the mRNA molecule, whereas 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, determine the time-varying values of the transition rates that steer the RFM to this density profile, if they exist. Specifically, we consider two control problems. In the first, all transition rates can be regulated 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, a single transition rate is controlled and the goal is to steer the production rate to a desired value. In the first case, we show that the system is controllable, i.e. the control is powerful enough to steer the RFM to any desired value, and we provide closed-form expressions for constant control functions (or transition rates) asymptotically steering the RFM to the desired value. For the second problem, we show that the production rate can be steered to any desired value in a feasible region determined by the other, constant transition rates. We discuss some of the biological implications of these results.

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.

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.

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.

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.