  The ribosome flow model with input and output (RFMIO) is a deterministic
dynamical system that has been used to study the flow of ribosomes during mRNA
translation. The RFMIO and its variants encapsulate important properties that
are relevant to modeling ribosome flow such as the possible evolution of
"traffic jams" and non-homogeneous elongation rates along the mRNA molecule,
and can also be used for studying additional intracellular processes such as
transcription, transport, and more. Here 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, using connection weights, between several inputs of other
RFMIOs. We show that under quite general feedback connections the network has
two important properties: (1) it admits a unique steady-state and every
trajectory converges to this steady-state, and (2) the problem of how to
determine the connection weights so that the network steady-state output is
maximized is a convex optimization problem. These mathematical properties make
these networks highly suitable as models of various phenomena: property (1)
means that the behavior is predictable and ordered, and property (2) means that
determining the optimal weights is numerically tractable even for large-scale
networks. For the specific case of a feed-forward network of RFMIOs we prove an
additional useful property, namely, that there exists a spectral representation
for the network steady-state, and thus it can be determined without any
numerical simulations of the dynamics. We describe the implications of these
result to several fundamental biological phenomena and biotechnological
objectives.
