Optimal allocation between nutrient uptake and growth in a microbial trichome

Abstract

A microbial trichome grows by assimilating nutrients from its environment, and converting these into catalytic macro-molecular machinery. This machinery may be divided into assimilatory machinery and proliferative machinery. The former type is involved in nutrient uptake, whereas the latter type enables the trichome to grow. The cells in the trichome are faced with an allocation problem: given the availability of nutrients in the environment, how many macro-molecular building blocks should be allocated to the synthesis of assimilatory machinery, and how many to the synthesis of proliferative machinery? We answer this question for a particular model, which is a generalization of the Droop quota model. We formulate a two-dimensional non-linear optimal control problem, corresponding to this model. An optimal allocation regime with a singular segment is derived, based on Pontryagin’s maximum principle. We give a direct proof of optimality. We discuss how actual biological cells might implement this optimal regime.