Fig. 3

Schematic of the dynamic optimization approach. Example optimum solution for maximum production of succinate in A. succinogenes. Time-course trajectories in a are broken down into three stages, each of which is composed of a number finite elements (4 in this example, indicated by light/dark shading). Within each finite elements, state variables are represented by a lagrange interpolating polynomial at each of 4 collocation points (\(\tau _0 \rightarrow \tau _3\)), with continuity constraints between finite elements. The step size of the finite elements, h, within each stage is also optimized. Metabolic fluxes are optimized by simultaneously optimizing the fractional expression of each elementary mode by stage (b) with the overall activity of all elementary modes (a, lower plot). In this example, high-growth modes are replaced by high succinate yielding modes in later fermentation stages