Pathway identification using parallel optimization for a nonlinear hybrid system in batch culture

The complex bio-process for the bioconversion of glycerol to 1,3-propanediol (1,3-PD) can be characterized by a nonlinear hybrid dynamical system of enzyme-catalytic kinetics and genetic regulation. In this paper, in consideration of the possible ways that 3-hydroxypropionaldehyde (3-HPA) inhibits the cell growth, various possible transports and inhibition mechanisms, we consider 576 possible metabolic pathways and establish a fourteen-dimensional nonlinear hybrid dynamical system with uncertain system parameters and pathway parameters for describing the process of batch culture. Some important properties of the hybrid system are discussed. Taking into account the difficulty in accurately measuring the concentration of intracellular substances and the absence of equilibrium points for the hybrid system, we quantitatively define biological robustness of the intracellular substance concentrations for the overall process of batch culture. Our goal is to determine the most possible metabolic pathway and corresponding system parameters. To this end, taking the relative error between the experimental data and the computational values of the extracellular substances together with the proposed biological robustness of the intracellular substances as a cost function, we establish an identification model, in which 17 280 system parameters and 5760 pathway parameters are involved. Furthermore, the identification model is subject to the hybrid system, continuous state constraints and parameter constraints. As such, it is a very complicated task to solve the identification model by a serial program. With this in mind, we propose a parallel improved differential evolution algorithm (P-IDE), based on the constraint transcription and smoothing approximation techniques, for solving the identification model. An illustrative numerical example shows the appropriateness of the most possible metabolic pathway and the corresponding system parameters as well as the effectiveness of the parallel algorithm.