Availability of charged tRNAs drives maximal protein synthesis at intermediate levels of codon usage bias.

Synonymous codon usage can influence protein expression, since codons with high numbers of corresponding tRNAs are naturally translated more rapidly than codons with fewer corresponding tRNAs. Although translation efficiency ultimately depends on the concentration of aminoacylated (charged) tRNAs, many theoretical models of translation have ignored tRNA dynamics and treated charged tRNAs as fixed resources. This simplification potentially limits these models from making accurate predictions in situations where charged tRNAs become limiting. Here, we derive a mathematical model of translation with explicit tRNA dynamics and tRNA re-charging, based on a stochastic simulation of this system that was previously applied to investigate codon usage in the context of gene overexpression. We use the mathematical model to systematically explore the relationship between codon usage and the protein expression rate, and find that in the regime where tRNA charging is a limiting reaction, it is always optimal to match codon frequencies to the tRNA pool. Conversely, when tRNA charging is not limiting, using 100% of the preferred codon is optimal for protein production. We also use the tRNA dynamics model to augment a whole-cell simulation of bacteriophage T7. Using this model, we demonstrate that the high expression rate of the T7 major capsid gene causes rare charged tRNAs to become entirely depleted, which explains the sensitivity of the major capsid gene to codon deoptimization.