Explicit Expression for the Steady-State Translation Rate in the Infinite-Dimensional Homogeneous Ribosome Flow Model

Gene translation is a central stage in the intracellular process of protein synthesis. Gene translation proceeds in three major stages: initiation, elongation, and termination. During the elongation step, ribosomes (intracellular macromolecules) link amino acids together in the order specified by messenger RNA (mRNA) molecules. The homogeneous ribosome flow model (HRFM) is a mathematical model of translation-elongation under the assumption of constant elongation rate along the mRNA sequence. The HRFM includes n first-order nonlinear ordinary differential equations, where n represents the length of the mRNA sequence, and two positive parameters: ribosomal initiation rate and the (constant) elongation rate. Here, we analyze the HRFM when n goes to infinity and derive a simple expression for the steady-state protein synthesis rate. We also derive bounds that show that the behavior of the HRFM for finite, and relatively small, values of n is already in good agreement with the closed-form result in the infinite-dimensional case. For example, for n = 15, the relative error is already less than 4 percent. Our results can, thus, be used in practice for analyzing the behavior of finite-dimensional HRFMs that model translation. To demonstrate this, we apply our approach to estimate the mean initiation rate in M. musculus, finding it to be around 0.17 codons per second.