Modelling the optimal timing in metabolic pathway activation—Use of Pontryagin's Maximum Principle and role of the Golden section

The time course of enzyme concentrations in metabolic pathways can be predicted on the basis of the optimality criterion of minimizing the time period in which an essential product is generated. This criterion is in line with the widely accepted view that high fitness requires high pathway flux. Here, based on Pontryagin's Maximum Principle, a method is developed to solve the corresponding constrained optimal control problem in an almost exclusively analytical way and, thus, to calculate optimal enzyme profiles, when linear, irreversible rate laws are assumed. Three different problem formulations are considered and the corresponding optimization results are derived. Besides the minimization of transition time, we consider an operation time in which 90% of the substrate has been converted into product. In that case, only the enzyme at the lower end of the pathway rather than all enzymes are active in the last phase. In all cases, biphasic or multiphasic time courses are obtained. The biological meaning of the results in terms of a consecutive just-in-time expression of metabolic genes is discussed. For the special case of two-enzyme systems, the role of the Golden section in the solution is outlined.