A model of protein translation including codon bias, nonsense errors, and ribosome recycling.

We present and analyse a model of protein translation at the scale of an individual messenger RNA (mRNA) transcript. The model we develop is unique in that it incorporates the phenomena of ribosome recycling and nonsense errors. The model conceptualizes translation as a probabilistic wave of ribosome occupancy traveling down a heterogeneous medium, the mRNA transcript. Our results show that the heterogeneity of the codon translation rates along the mRNA results in short-scale spikes and dips in the wave. Nonsense errors attenuate this wave on a longer scale while ribosome recycling reinforces it. We find that the combination of nonsense errors and codon usage bias can have a large effect on the probability that a ribosome will completely translate a transcript. We also elucidate how these forces interact with ribosome recycling to determine the overall translation rate of an mRNA transcript. We derive a simple cost function for nonsense errors using our model and apply this function to the yeast (Saccharomyces cervisiae) genome. Using this function we are able to detect position dependent selection on codon bias which correlates with gene expression levels as predicted a priori. These results indirectly validate our underlying model assumptions and confirm that nonsense errors can play an important role in shaping codon usage bias.