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

Abstract

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 energyconsuming 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 þ 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.

A Systems Approach to Elucidate Heterosis of Protein Abundances in Yeast

“…This result was difficult to explain from the current knowledge on heterosis, since, as far as we know, there is no model for negative heterosis. To interpret this result, we relied on previous observations showing that (i) concave genotype-phenotype relationships exist at various levels of cell organization (76, 94 -97) and in particular for the protein synthesis rate (77)(78)(79) and (ii) nonadditivity can occurs at every level of cell organization, from transcript abundance to more integrated traits (24,25).…”

“…Interestingly, the protein synthesis rates seem also to be a concave function of various factors, such as mRNA amount (77), translation factor abundance (78), ribosomal initiation rate, and elongation rate (79). Therefore, the basis of heterosis put forward for metabolic fluxes could apply for protein abundances, even though the relationship is mathematically different.…”

Entrainment to subharmonic trajectories in oscillatory discrete-time systems

“…We can now state the main result in this section. from Ω converges to a u-periodic solution of (19). Thm.…”

“…Thm. 2 implies that every solution of (19) emanating from Ω converges to a T -periodic solution of (19).…”

On the Periodic Gain of the Ribosome Flow Model

“…Ref. [21] derived a spectral representation for the steady-state density e and production rate R. Given the RFM, define the (n + 2) × (n + 2) tridiagonal matrix…”