  Translation is an important process for prokaryotic and eukaryotic cells to
produce necessary proteins for cell growth. Numerious experiments have been
performed to explore the translational properties. Diverse models have also
been developed to determine the biochemical mechanism of translation. However,
to simplify the majority of the existing models, the frameshifting of ribosome
along the mRNA transcript is neglected, which actually occurs in real cells and
has been extensively experimentally studied. The frameshifting of ribosome
evidently influences the efficiency and speed of translation, considering that
the peptide chains synthesized by shifted ribosomes will not fold into
functional proteins and will degrade rapidly. In this study, a theoretical
model is presented to describe the translational process based on the model for
totally asymmetric simple exclusion process. In this model, the frameshifting
of the ribosome along the mRNA transcript and the attachment/detachment of the
ribosome to/from the main body of mRNA codons during translation elongation
process, are explicitly included. The results show that, with ribosome
frameshifing, the speed of correctly synthesized peptide chains may increase
first and then decrease with both the translation initiation rate $\alpha$ and
the ribosome detachment rate $\omega_d$. This results indicates that regulating
the translation process to reach maximal synthesized speed of proteins is
theoretically feasible. Traffic-related problems of ribosome motion along the
mRNA transcript are also addressed theoretically. Depending on parameter
values, shock wave (or domain wall) may exist for ribosome probabilities along
the mRNA.
