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Gene translation is a central stage in the intra-cellular process of protein synthesis. Gene translation proceeds in three major stages: initiation, elongation, and termination. 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%. 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.
Ordinary Differential Equations, Biology and genetics

Y. Zarai, M. Margaliot and T. Tuller, "Explicit Expression for the Steady State Translation Rate in the Infinite-Dimensional Homogeneous Ribosome Flow Model," in IEEE/ACM Transactions on Computational Biology and Bioinformatics.
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