Optimization of metabolic pathways under stability considerations

Metabolic networks are often approached through steady-state optimization formulations that are solved to interpret and predict the behavior of the network subject to changes in external fluxes or internal enzyme activity. The major question addressed in this paper is how to ensure that solutions to these steady-state optimization models for metabolic networks are implementable from a stability point of view. The stability of a dynamic system is closely related to matrix stability. Hence, it can be determined through the computation of the largest eigenvalue of a coefficient matrix. While it is straightforward to analyze the stability of a given system, the challenge is to redesign a metabolic network in a way that guarantees that the system will be stable around the new steady-state. For this purpose, we propose to model metabolic networks through classical optimization formulations, such as the classical S-system representation, with an additional constraint to enforce stability within a prespecified neighborhood of the solution point. The proposed formulation is a bilevel optimization problem that is very difficult to solve. We develop a suitable global optimization algorithm to solve this problem after transforming it to a semi-infinite optimization problem. Computational results are presented, including application to tryptophan biosynthesis in bacteria and anaerobic fermentation in Saccharomyces cerevisiae.