Ribosome flow model with positive feedback

Eukaryotic mRNAs usually form a circular structure; thus, ribosomes that terminate translation at the 3' end can diffuse with increased probability to the 5' end of the transcript, initiating another cycle of translation. This phenomenon describes ribosomal flow with positive feedback--an increase in the flow of ribosomes terminating translating the open reading frame increases the ribosomal initiation rate. The aim of this paper is to model and rigorously analyse translation with feedback. We suggest a modified version of the ribosome flow model, called the ribosome flow model with input and output. In this model, the input is the initiation rate and the output is the translation rate. We analyse this model after closing the loop with a positive linear feedback. We show that the closed-loop system admits a unique globally asymptotically stable equilibrium point. From a biophysical point of view, this means that there exists a unique steady state of ribosome distributions along the mRNA, and thus a unique steady-state translation rate. The solution from any initial distribution will converge to this steady state. The steady-state distribution demonstrates a decrease in ribosome density along the coding sequence. For the case of constant elongation rates, we obtain expressions relating the model parameters to the equilibrium point. These results may perhaps be used to re-engineer the biological system in order to obtain a desired translation rate.