# Optimality in cellular storage via the Pontryagin Maximum Principle

## TL;DR

This paper applies Pontryagin's maximum principle to a minimal cellular metabolism model to determine optimal growth strategies, revealing how growth mode depends on parameters and time scale.

## Contribution

It introduces a minimal model of cellular growth and derives optimal control strategies, highlighting the influence of parameters and time scales on growth mode selection.

## Key findings

- Optimal growth trajectories depend on model parameters.
- Short-term growth mode depends on catalytic rates and biomass weights.
- Long-term growth mode also influenced by yield coefficients.

## Abstract

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.