  Translation of proteins is a fundamental part of gene expression that is
mediated by ribosomes. As ribosomes significantly contribute to both cellular
mass and energy consumption, achieving efficient management of the ribosome
population is also crucial to metabolism and growth. Inspired by biological
evidence for nutrient-dependent mechanisms that control both ribosome active
degradation and genesis, we introduce a dynamical model of protein production,
that includes the dynamics of resources and control over the ribosome
population. Under the hypothesis that active degradation and biogenesis are
optimal for maximizing and maintaining protein production, we aim to
qualitatively reproduce empirical observations of the ribosome population
dynamics. Upon formulating the associated optimization problem, we first
analytically study the stability and global behaviour of solutions under
constant resource input, and characterize the extent of oscillations and
convergence rate to a global equilibrium. We further use these results to
simplify and solve the problem under a quasi-static approximation. Using
biophysical parameter values, we find that optimal control solutions lead to
both control mechanisms and the ribosome population switching between periods
of feeding and fasting, suggesting that the intense regulation of ribosome
population observed in experiments allows to maximize and maintain protein
production. Finally, we find some range for the control values over which such
a regime can be observed, depending on the intensity of fasting.
