  During mRNA translation, several ribosomes attach to the same mRNA molecule
simultaneously translating it into a protein. This pipelining increases the
protein production rate. A natural and important question is what ribosomal
density maximizes the protein production rate. Using mathematical models of
ribosome flow along both a linear and a circular mRNA molecule we prove that
typically the steady-state production 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.
