# The above set of ordinary differential equations ODEs for

The above set of ordinary differential equations (ODEs) for three independent variables, viz. X, G and E, and the parameters therein characterize the S peptide process. These equations have total of 10 parameters, viz. K3, K1, K3E, kd, μm, a, b, YX/G, m and K4. In order to get the physical insight into the ultrasound induced enhancement of fermentation, the numerical solution of above equations was compared with the experimental time profiles of X, G and E. The unknowns in this model are the kinetic and physiological parameters, whose optimum values need to be determined, so as to match the time profiles of the substrate, cell mass and ethanol concentrations calculated using the model with the experimental data. As per the findings of Philippidis et al. (1992), null value has been assigned to the constant K4. The main model of three ordinary differential equations was solved using Runge–Kutta 4th order method, and optimization of the parameters was done by calculating root mean square (RMS) error between experimental and model results using Genetic Algorithm (GA). The objective function (Obj) for the optimization was defined as follows:Obj = min∑i=1neri where n is the number of experimental data points for glucose concentration, ethanol concentration and cell mass concentration. The error (er) is defined as:eri=Giexp-Gipred2+Xiexp-Xipred2+Eiexp-Eipred21/2