We started with a simple model of the Michaelis-Menten mechanism based on Keramat et al (2015) 4 to model the enzyme activity for the invertase we used in our experiment. The low substrate concentrations (Equation 1 and 2), which the Vm is calculated from the equation one above based on previous data from Keramat et al (2015) 4, and the substrate inhibition model for high concentrations of substrate (Equation 1, 3 and 4) were both used in the determination of the enzyme activity.
E = Enzyme concentration ; S = Sucrose concentration ; ES =Activate complex of enzyme–sucrose molecule ; P = Product ; V = Reaction rate ; Vm = Maximum reaction rate ; Km = Michaelis–Menten parameter ; ESS [-] Inactive complex of enzyme-two sucrose molecules ; Ki = Substrate inhibition parameter
As we can see from above the velocity rapidly accelerates as the concentration of the initial sucrose increases, but then it slowly reaches a maximum and eventually starts to decrease. The apparent decrease in reaction rate is caused by the folding of sucrose molecules from the intramolecular hydrogen bonds which is characterized by Mathlouthi et al (1998)5. From the paper, we were able to determine the inhibiting effect from the substrate reaching saturation using equation 5 and 6 which is known as the aggregation effect. The aggregation effect from the sucrose molecules can be considered using the Ks value calculated from equation 6.
Ks = Sucrose association parameter
After considering the Ks value from equation 6, we are able to combine it with equation 2 to form the following equation:
However, according to Keramat et al (2015)4, there are still several weaknesses in this model as it does not match their experimental data.4
For instease, the addition of an excess Arrhenius type term where it represents the probability of enzyme deactivation by the inhibition of products during hydrolysis. This enzyme deactivation parameter is shown by Kd from below,
EI* = Irreversible ; Kd = Irreversible enzyme deactivation parameter
However, this model still has a low accuracy for the initial time lag at the beginning of the reaction. This is considered for the enzyme sites of sucrose molecules to become available. 4 This was deduced by adding in a initial time lag facter (1-e^(-Ka * t)) which results in our final model:
Ka = Activation parameter of enzymes
Ka (= 0.1171 for the S3(III) model) and Kd (= 0.0058 and 0.0054 for S3(II) and S3(III) models, respectively) are two empirical parameters in Eqs. 11 and 12 which are calculated by the presently available data (Kermat et al. 2015).
This model is capable of predicting the reaction rate at the beginning and also at the remaining reaction pathway which includes the initial time lag and irreversible enzyme deactivation through the remaining pathway. More importantly, Keramat et al. (2015)4 have confirmed the similarity in this model and in their experimental data. Therefore, this can be a great model for kinetic investigations of the enzymatic reactions and act as good conformity when compared to our experimental data.
Figure 1. The predicted invertase reaction rate of the MM model using MATLAB.
Figure 2. The predicted invertase reaction rate of the S^2 model using MATLAB.
Figure 3. The predicted invertase reaction rate of the S^3 model using MATLAB.
Figure 4. A comparison between the predicted invertase reaction rate using the all three model.