The tri-frame model

The tri-frame model gives mathematical expression to the transcription and translation processes, and considers all three reading frames (RFs). RNA polymerases transcribe DNA in single nucleotide increments, but ribosomes translate mRNA in pairings of three (triplets or codons). The set of triplets in the mRNA, starting with the initiation codon (usually AUG) defines the open reading frame (ORF). Since ribosomes do not always translocate three nucleotide positions, two additional RFs are accessible. The −1 RF and the +1 RF are triplet pairings of the mRNA, which are accessed by shifting one nucleotide position in the 5′ and 3′ directions, respectively. Transcription is modeled as a linear operator that maps the initial codons in all three frames into other codon sets to account for possible transcriptional errors. Translational errors (missense errors) originate from misacylation of tRNAs and misreading of aa-tRNAs by the ribosome. Translation is modeled as a linear mapping from codons into aa-tRNA species, which includes misreading errors. A final transformation from aa-tRNA species into amino acids provides the probability distributions of possible amino acids into which the codons in all three frames could be translated. An important element of the tri-frame model is the ribosomal occupancy probability. It is a vector in R3 that gives the probability to find the ribosome in the ORF, −1 or +1 RF at each codon position. The sequence of vectors, from the first to the final codon position, gives a history of ribosome frameshifting. The model is powerful: it provides explicit expressions for (1) yield of error-free protein, (2) fraction of prematurely terminated polypeptides, (3) number of transcription errors, (4) number of translation errors and (5) mutations due to frameshifting. The theory is demonstrated for the three genes rpsU, dnaG and rpoD of Escherichia coli, which lie on the same operon, as well as for the prfB gene.