  Down regulation of mRNA translation is an important problem in various
bio-medical domains ranging from developing effective medicines for tumors and
for viral diseases to developing attenuated virus strains that can be used for
vaccination. Here, we study the problem of down regulation of mRNA translation
using a mathematical model called the ribosome flow model (RFM). In the RFM,
the mRNA molecule is modeled as a chain of $n$ sites. The flow of ribosomes
between consecutive sites is regulated by $n+1$ transition rates. Given a set
of feasible transition rates, that models the outcome of all possible
mutations, we consider the problem of maximally down regulating the translation
rate by altering the rates within this set of feasible rates. Under certain
conditions on the feasible set, we show that an optimal solution can be
determined efficiently. We also rigorously analyze two special cases of the
down regulation optimization problem. Our results suggest that one must focus
on the position along the mRNA molecule where the transition rate has the
strongest effect on the protein production rate. However, this rate is not
necessarily the slowest transition rate along the mRNA molecule. We discuss
some of the biological implications of these results.
