Developing a new approach for (biological) optimal control problems: Application to optimization of laccase production with a comparison between response surface methodology and novel geometric procedure

Laccase production by indigenous fungus, Phanerochaete chrysosporium, requires solving optimal problems to determine the maximum production of the enzyme within a definite time period and conditions specified in the solid-state fermentation process. For this purpose, parallel to response surface methodology, an analytical approach has been proposed based on the advanced concepts of Poisson geometry and Lie groups, which lead to a system of the Hamiltonian equations. Despite the dating of the Hamiltonian approach to solving biological problems, the novelty of this paper is based on the expression of a Hamiltonian system in notions of Poisson geometry, Lie algebras and symmetry groups and first integrals. In this way, all collected data and the variables are taken into account in their actual role in the Hamiltonian system without any limitation on their number and dimensions. Also, the Hamiltonian system obtained can be reduced by symmetry concepts of Lie algebras, which result in the exact solution of the initial optimal problem. In addition, it can be converted to Lagrangian and vice versa. The proposed approach applies to the mathematical models describing the production of biomass and lignocellulolytic enzymes, consumption of the lignocellulosic matrix, fermentation model of the Tequila production process, and the laccase production. Ultimately, a comparison between the approximate method for producing laccase using the response surface methodology and the proposed analytical method has been made.