Optimal control of a microbial growth model by means of substrate concentration and resource allocation

Resource allocation models have been proven to be a highly Effective tool to study the growth of microorganisms. Here, we use one such growth model describing the metabolism of a given bacterium in an artificial (e.g. biotechnological) environment. Tis model involves two controls, one quantifying the protein precursors allocation (i.e. the cellular internal control) and the other representing the nutrient concentration in the culture. We seek to determine the controls that maximize the resulting growth rate of cells living in this controlled environment. We first carry out a theoretical analysis of this optimal control problem (OCP) by means of the Pontryagin’s Maximum Principle (PMP). We show the bang-bang structure of the optimal resource allocation control. We fnd the environmental control to follow a bang-singular-bang structure, and give an expression for its singular arc depending on the state and costate given by the PMP. We solve the OCP in fixed final time using a direct optimization method, implemented on the BOCOP software. Tis resolution reveals an intrinsic period of the optimal control, corresponding to the solution of the periodic OCP in free final time. Moreover, we fnd a singular arc of the environment coinciding with the analytical expression given before. Our study highlights the optimality of periodic, non-constant environments in maximizing bacterial growth.