Optimal control of metabolic networks with saturable enzyme kinetics

This note addresses the optimal control of non-linear metabolic networks by means of time-dependent enzyme synthesis rates. The authors consider networks with general topologies described by a control-affine dynamical system coupled with a linear model for enzyme synthesis and degradation. The problem formulation accounts for transitions between two metabolic equilibria, which typically arise in metabolic adaptations to environmental changes, and the minimisation of a quadratic functional that weights the cost/benefit relation between the transcriptional effort required for enzyme synthesis and the transition to the new phenotype. Using a linear time-variant approximation of the non-linear dynamics, the problem is recast as a sequence of linear-quadratic problems, the solution of which involves a sequence of differential Lyapunov equations. The authors provide conditions for convergence to an approximate solution of the original problem, which are naturally satisfied by a wide class of models for saturable enzyme kinetics. As a case study the authors use the method to examine the robustness of an optimal just-in-time gene expression pattern with respect to heterogeneity in the biosynthetic costs of individual proteins.