Optimality in cellular storage via the Pontryagin Maximum Principle

We study an optimal control problem arising from a resource allocation problem in cellular metabolism. A minimalistic model that describes the production of enzymatic vs. non-enzymatic biomass components from a single nutrient source is introduced. The basic growth modes with this model are linear growth, where only the non-enzymatic component is produced, and exponential growth with only enzymatic components being produced. Using Pontryagin's maximum principle, we derive the optimal growth trajectory depending on the model's parameters. It turns out that depending on the parameters, either a single growth mode is optimal, or otherwise the optimal solution is a concatenation of exponential growth with linear growth. Importantly, on the short time scale, the choice of growth mode depends only on catalytic rate constants and biomass weights of the two component types, whereas on longer time scales, where the nutrient amount becomes limiting, also the yield coefficients play a role.