Predicting microbial cell composition and diauxic growth as optimal control strategies

Abstract

Bacteria have evolved internal regulatory mechanisms allowing them to allocate resources to different cellular functions while dealing with the physiological limitations of the cell. In this preliminary work, we present a simple mathematical model of bacteria growing on n substitutable substrates aiming to capture these principles, focusing on the trade-off between metabolism and gene expression. The model is also able to capture a behavior known as diauxic growth, which is the sequential consumption of the nutrients in the environment resulting from the limitation of resources of the metabolic machinery. Under the hypothesis that cell regulatory mechanisms are tuned to maximize bacterial growth, we study the optimal allocation strategies through Optimal Control theory, by means of the Pontryagin's Maximum Principle. The optimal solutions are characterized by classical bang-singular-bang control structures, and can be expressed as feedback control laws, in accordance with previous results. We conclude the paper with numerical optimal trajectories of the model representing an environment with three substrates with different associated yields coefficients.