Mathematical models for large-scale translation in the cell

The ribosome flow model with input and output (RFMIO) is a deterministic dynamical system that has been used to model and analyze the dynamics of ribosome flow during mRNA translation. The input of the RFMIO controls its initiation rate and the output represents the ribosome exit rate (and thus the protein production rate) at the 3` end of the mRNA molecule. We consider networks of interconnected RFMIOs as a fundamental tool for modeling, analyzing and re-engineering the complex mechanisms of protein production. In these networks, the output of each RFMIO may be divided between several inputs of other RFMIOs. For the specific case of feed-forward network of RFMIOs we prove three important properties. First, the entire network converges to a steady-state that depends on all the translation rates in all the RFMIOs, but not on the initial conditions in the network. Second, there exists a spectral expression for the steady-state, and thus it can be determined without any numerical simulations of the dynamics. Third, the problem of dividing the output of an RFMIO between the inputs of other RFMIOs in a way that maximizes the total steady-state production rate of the network is a convex optimization problem. Hence, this problem is tractable even for very large networks. We describe the implications of these results to several fundamental biological phenomena and biotechnological objectives.