Model

1. Introduction of the Project

Diabetes is a serious chronic disease and has gradually become one of the most prevalent non-communicable diseases worldwide, with its incidence rising year by year. However, current treatment methods for diabetes are time-consuming, labor-intensive, and costly, with existing medications facing issues such as high costs and numerous side effects. Our project aims to express a glycosyltransferase, Gt6CGT, from Gentiana triflora in $E. coli$, which can use low-cost luteolin as a substrate to synthesize high-cost isoorientin. Isoorientin, a natural compound found in plants, has low toxicity and side effects, and holds great potential for treating diabetes. Based on this, we have explored some characteristics of the Gt6CGT enzyme, providing a theoretical foundation for the large-scale production of isoorientin in the future.

2. Foundation of Modeling

We aim to study the glycosyl transfer capability of the Gt6CGT enzyme, which ultimately boils down to the enzyme's catalytic reaction. The relationship between the initial enzyme reaction rate and the substrate concentration should follow the Michaelis-Menten equation:

$V0​$ is the initial reaction rate,

$Vmax$ is the maximum reaction rate, achieved when the enzyme's active sites are fully occupied by the substrate,

$[S]$ is the substrate concentration,

$Km​$ is the Michaelis constant, representing the substrate concentration at which the reaction rate is half of the maximum rate.

The equation shows the relationship between substrate concentration and the initial reaction rate of the enzyme-catalyzed reaction. To develop a linear equation, this equation can be transformed into the Lineweaver-Burk equation:

Lineweaver-Burk equation establishes a linear relationship between $1/V0$ and $1/[S]$, where the slope is $Km/Vmax$ ​, and the intercept is $1/Vmax$ ​. The Km value is determined by the properties of the enzyme.

The Michaelis-Menten model describes the steady-state model of enzyme-catalyzed reactions, as shown below:

In this context, $[S]$, $[E]$, and $[P]$ represent the substrate, enzyme, and product, respectively. At low substrate concentrations, $[S]+Km≈Km$ , resulting in a direct linear relationship between the substrate concentration and the initial reaction rate v0 . As $[S]$ increases, $[S]+Km$ gradually approaches $[S]$, and when $[S]→∞[S]$, the initial reaction rate v0 approaches the maximum reaction rate Vmax . Therefore, as substrate concentration increases, the reaction transitions from a first-order reaction, where the rate is dependent on substrate concentration, to a zero-order reaction, where the rate becomes independent of substrate concentration.

Therefore, the relationship between the initial reaction rate $v0$ and the substrate concentration for a specific enzyme-catalyzed reaction can be represented graphically, as shown in Figure 1.

3. Preliminaries

Since building the model requires measuring the changes in product concentration over time, and the detection of isoorientin currently relies on HPLC, which involves high experimental costs, we employed an alternative reaction to assess enzyme activity. This reaction produces α-naphthyl glucoside, and its fluorescence reflects the glycosyl transfer capability of the enzyme.

4. Model Description and Methodology

4.1 Data processin

According to the literature, the concentration of α-naphthylglucoside shows a linear relationship with the measured fluorescence values. To determine the linear equation, we measured the fluorescence values at different concentrations of α-naphthylglucoside and obtained the following linear equation: $y=582.11x+112.38$ (where x represents the concentration of α-naphthylglucoside, and y represents the measured fluorescence values). The correlation coefficient between the fluorescence value F and the concentration of α-naphthyl glucoside is 0.9771. Specific data can be found in Figure 3.

Table 1 shows the fluorescence values measured at different substrate concentrations (α-naphthol) and various reaction times. Each treatment was repeated three times, and the average value was taken as the final result.

Based on the linear equation from Figure 3, we converted the measured fluorescence values into the concentration of α-naphthylglucoside, resulting in the data presented in Table 2.

Based on the data from Table 2, we plotted the curve of product concentration versus time at a substrate concentration of 0.1 mM. The slope of the curve, denoted as V, is given by，

As shown in Figure 4, during the initial phase of the experiment, the substrate concentration is sufficient, resulting in an approximately linear relationship between product concentration and time. Using the data from this phase, we can determine the initial reaction rate V0​, calculated as V0 ​​, where Cn​ represents the product concentration at minute n, and C0 is the initial product concentration.

Thus, we calculated the initial reaction rates at different substrate concentrations based on the data from Table 2. The detailed results are shown in Table 3.

4.2 Regression and Results

We processed the data from Table 3 to obtain the relationship between 1/v0 ​ and 1/[S] (shown in Table 4), and plotted the linear fit graph (Figure 5). From the slope and intercept of this graph, we calculated the parameters Vmax and Km ​involved in the Michaelis-Menten equation.

As shown in Figure 5, there is a strong positive correlation between 1/V01/V_01/V0​ and 1/[S] for Gt6CGT. By substituting the slope and intercept into the Lineweaver-Burk equation, we obtained the Michaelis-Menten equation parameters: $Vmax=9.83×10−8 mM/s$ and $Km=0.213 mM$. Thus, the complete equation for Gt6CGT is： $V_0 = \frac{9.83 \times 10^{-8} [S]}{0.213 + [S]}$. The results are shown in Figure 6.

5. Discussion

From Figure 6, we can determine the optimal substrate concentration, which allows the reaction to reach its maximum rate while avoiding substrate wastage. By analyzing Figure 6 with software, the optimal substrate concentration is found to be 0.91 mM. Beyond this point, further increases in [S] will lead to diminishing returns, meaning the increase in substrate concentration will have a progressively smaller effect on the reaction rate.

6. Conclusion

In this experiment, we used mathematical modeling to calculate the maximum reaction rate Vmax and the Michaelis constant Km for an enzyme-catalyzed reaction. To achieve this, we first measured the initial reaction rates V0 at various substrate concentrations and plotted the Lineweaver-Burk double reciprocal graph of 1/V0 versus 1/[S]. By fitting the data using linear regression, we obtained the slope and intercept of the line. Using the Lineweaver-Burk equation:

The slope Km/Vmax and intercept 1/Vmax were used to calculate Vmax and Km . Here, Vmax represents the maximum catalytic rate of the enzyme, and Km reflects the enzyme’s affinity for the substrate, indicating the substrate concentration at which half of the maximum reaction rate is achieved. Finally, we calculated Vmax=9.83×10−8 mM/s and Km=0.213 mM, leading to the complete Michaelis-Menten equation:

This equation effectively describes the catalytic reaction rate of the Gt6CGT enzyme at different substrate concentrations, helping us determine the optimal substrate concentration and improve reaction efficiency.