On the Structure of the Initiation and Elongation Rates that Maximize Protein Production in the Ribosome Flow Model

Translation is a crucial step in gene expression. During translation, macromolecules called ribosomes "read" the mRNA strand in a sequential manner and produce a corresponding protein. Translation is known to consume most of the cell's energy. Maximizing the protein production rate in mRNA translation, subject to the bounded biomolecullar budget, is thus an important problem in both biology and biotechnology. We consider this problem using a mathematical model for mRNA translation called the ribosome flow model (RFM). For an mRNA strand with $n$ sites the RFM includes $n$ state-variables that encode the normalized ribosomal density at each site, and $n+1$ positive parameters: the initiation rate and elongation rates along the chain. An affine constraint on these rates is used to model the bounded cellular budget. We show that for a homogeneous constraint the rates that maximize the steady-state protein production rate have a special structure. They are symmetric with respect to the middle of the chain, and monotonically increase as we move towards the center of the chain. The ribosomal densities corresponding to the optimal rates monotonically decrease along the chain. We discuss some of the biological implications of these results.

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