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Translation is an important stage in gene expression. During this stage,macro-molecules called ribosomes travel along the mRNA strand linkingamino-acids together in a specific order to create a functioning protein. Animportant question is how to maximize protein production. Indeed, translationis known to consume most of the cell's energy and it is natural to assume thatevolution shaped this process so that it maximizes the protein production rate.If this is indeed so then one can estimate various parameters of thetranslation machinery by solving an appropriate mathematical optimizationproblem. The same problem also arises in the context of synthetic biology,namely, re-engineer heterologous genes in order to maximize their translationrate in a host organism. We consider the problem of maximizing the proteinproduction rate using a computational model for translation-elongation calledthe ribosome flow model (RFM). This model describes the flow of the ribosomesalong an mRNA chain of length n using a set of n first-order nonlinear ODEs. Italso includes n+1 positive parameters: the ribosomal initiation rate into themRNA chain, and n elongation rates along the chain sites. We show that thesteady-state translation rate in the RFM is a strictly concave function of itsparameters. This means that the problem of maximizing the translation rateunder a suitable constraint always admits a unique solution, and that thissolution can be determined using highly-efficient algorithms for solving convexoptimization problems even for large values of n. Furthermore, our analysisshows that the optimal translation rate can be computed based only on theoptimal initiation rate and the elongation rate of the codons near thebeginning of the ORF. We discuss some applications of the theoretical resultsto synthetic biology, molecular evolution, and functional genomics.

Maximizing Protein Translation Rate in the Nonhomogeneous Ribosome Flow Model: A Convex Optimization Approach