Analytical Solutions for Fungal Growth and Penicillin Production Dynamics with Simultaneous Product Hydrolysis in Batch Bioprocesses

Few mathematical models describing the dynamics of cell growth, production formation and substrate consumption in batch and fed-batch bioprocesses have analytical solutions. In this study, analytical solutions for a mathematical model of a batch bioprocess of penicillin production based on the Logistic law for cell growth and on the Luedeking–Piret equation for antibiotic formation are obtained using classical methods of solving ordinary differential equations. The analytical solutions were validated by substitution into the differential equations themselves, as well as by comparison with numerical solutions obtained through the fourth-order Runge–Kutta–Gill integration method, using typical fungus inoculum concentrations (X0 = 0.25; 0.75% DW) and kinetic parameters (μm = 0.5 h−1, Xm = 3.7% DW, β = 0.02 U/(mL·h·% DW) and kh = 0.027 h−1). The novelty in relation to the few studies published on the subject, which deal with the production of different metabolites, including other antibiotics, is that in the present study, the hydrolysis of penicillin is considered simultaneously with its production in the description of the dynamics of product formation. The main finding demonstrates that the hydrolysis reaction acts as a stabilizing factor, resulting in a system with two equilibrium points: an unstable point with no penicillin production, and a stable point in which a certain amount of antibiotic is produced.