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

Optimality in cellular storage via the Pontryagin Maximum Principle