  TASEP modeling was shown to offer a parsimonious explanation for the
experimentally confirmed ability of a single upstream Open Reading Frames
(uORFs) to upregulate downstream translation during the integrated stress
response. As revealed by numerical simulations, the model predicts that
reducing the density of scanning ribosomes upstream of certain uORFs increases
the flow of ribosomes downstream. To gain a better insight into the mechanism
which ensures the non-monotone relation between the upstream and downstream
flows, in this work, we propose a phenomenological deterministic model
approximating the TASEP model of the translation process. We establish the
existence of a stationary solution featuring the decreasing density along the
uORF for the deterministic model. Further, we find an explicit non-monotone
relation between the upstream ribosome density and the downstream flow for the
stationary solution in the limit of increasing uORF length and increasingly
leaky initiation. The stationary distribution of the TASEP model, the
stationary solution of the deterministic model and the explicit limit are
compared numerically.
