Optimal adaptation of metabolic networks in dynamic equilibrium

We consider the dynamic optimization of enzyme expression rates to drive a metabolic network between two given equilibrium fluxes. The formulation is based on a nonlinear control-affine model for a metabolic network coupled with a linear model for enzyme expression and degradation, whereby the expression rates are regarded as control inputs to be optimized. The cost function is a quadratic functional that accounts for the deviation of the species concentrations and expression rates from their target values, together with the genetic effort required for enzyme synthesis. If the network is in dynamic equilibrium along the whole adaptation process, the metabolite levels are constant and the nonlinear dynamics can be recast as a nonregular descriptor system. The structure of the reduced system can be exploited to decouple the algebraic and differential parts of the dynamics, so as to parameterize the controls that satisfy the algebraic constraint in terms of a lower-dimensional control. The problem is then solved as a standard Linear Quadratic Regulator problem for an uncon strained lower dimensional system. This solution allows for a systematic computation of the optimal flux trajectories between two prescribed dynamic equilibrium regimes for networks with general topologies and kinetics.