Optimal control strategies in a generic class of bacterial growth models with multiple substrates

Optimal control strategies are studied through the application of the Pontryagin’s Maximum Principle for a class of non-linear differential systems that are commonly used to describe resource allocation during bacterial growth. The approach is inspired by the optimality of numerous regulatory mechanisms in bacterial cells. In this context, we aim to predict natural feedback loops as optimal control solutions so as to gain insight on the behavior of microorganisms from a control-theoretical perspective. The problem is posed in terms of a control function $u_0\left(t\right)$ representing the fraction of the cell dedicated to protein synthesis, and $n$ additional controls $u_i\left(t\right)$ modeling the fraction of the cell responsible for the consumption of the available nutrient sources in the medium. By studying the necessary conditions for optimality, it is possible to prove that the solutions follow a bang–singular–bang structure, and that they are characterized by a sequential uptake pattern known as diauxic growth, which prioritizes the consumption of richer substrates over poor nutrients. Numerical simulations obtained through an optimal control solver confirm the theoretical results. Finally, we provide an application to batch cultivation of E. coli growing on glucose and lactose. For that, we propose a state feedback law that is based on the optimal control, and we calibrate the obtained closed-loop model to experimental data.